Aspects of Some Exotic Options

Transcription

1 Aspecs of Some Exoic Opions Nadia Theron Assignmen presened in parial fulfilmen of he requiremens for he degree of MASTER OF COMMERCE in he Deparmen of Saisics and Acuarial Science, Faculy of Economic and Managemen Sciences, Universiy of Sellenbosch Supervisor: Prof. W.J. Conradie December 7 i

2 Declaraion I, he undersigned, hereby declare ha he work conained in his assignmen is my own original work and ha I have no previously in is enirey or in par submied i a any universiy for a degree. Signaure: Dae: ii

3 Summary The use of opions on various sock markes over he world has inroduced a unique opporuniy for invesors o hedge, speculae, creae synheic financial insrumens and reduce funding and oher coss in heir rading sraegies. TThe power of opions lies in heir versailiy. They enable an invesor o adap or adjus her posiion according o any siuaion ha arises. Anoher benefi of using opions is ha hey provide leverage. Since opions cos less han sock, hey provide a high-leverage approach o rading ha can significanly limi he overall risk of a rade, or provide addiional income. This versailiy and leverage, however, come a a price. Opions are complex securiies and can be exremely risky. In his documen several aspecs of rading and valuing some exoic opions are invesigaed. The aim is o give insigh ino heir uses and he risks involved in heir rading. Two volailiy-dependen derivaives, namely compound and chooser opions; wo pah-dependen derivaives, namely barrier and Asian opions; and lasly binary opions, are discussed in deail. The purpose of his sudy is o provide a reference ha conains boh he mahemaical derivaions and deail in valuaing hese exoic opions, as well as an overview of heir applicabiliy and use for sudens and oher ineresed paries. iii

4 Opsomming Die gebruik van opsies in verskeie aandelemarke reg oor die wêreld bied aan beleggers n unieke geleenheid om e verskans, e spekuleer, sineiese finansiële produke e skep, en befondsing en ander koses in hul verhandelsraegieë e verminder. Die mag van opsies lê in hul veelsydigheid. Opsies sel n belegger in saa om haar posisie op enige manier aan e pas of e manipuleer soos die siuasie verander. Nog n voordeel van die gebruik van opsies is da hulle hefboomkrag verskaf. Aangesien opsies minder kos as aandele bied hulle n hoë-hefboomkrag benadering o verhandeling, wa die algehele risiko van n verhandeling aansienlik kan beperk of addisionele inkomse kan verskaf. Hierdie veelsydigheid en hefboomkrag kom eger een n prys. Opsies is komplekse insrumene wa uiers riskan kan wees. In hierdie werksuk word verskeie aspeke van die verhandeling en prysing van n aanal eksoiese opsies ondersoek. Die doel is om insig e bied in die gebruik van opsies en die risikos verbonde aan die verhandeling daarvan. Twee-volailiei afhanklike afgeleide insrumene, e wee saamgeselde- en keuse opsies; wee padafhanklike insrumene, e wee sper- en Asiaiese opsies; en laasens binêre opsies, word in diepe bespreek. Die doel van hierdie sudie is om n dokumen e verskaf wa beide die wiskundige afleidings en deail van die prysing van bogenoemde eksoiese opsies beva, sowel as om n oorsig van hul oepaslikheid en nu, aan sudene en ander belangsellendes e bied. iv

5 Acknowledgemens I would like o express my sincere graiude and appreciaion o he following people who have conribued o making his work possible: My supervisor, Professor Willie Conradie. I hank you for your ime, encouragemen, suggesions and conribuion in he preparaion of his documen. My family for all heir suppor. I hank my moher, faher and siser for heir ineres, encouragemen and suppor houghou my sudy period. My boyfriend Edré. I hank you for your moral suppor and love. All my friends in Sellenbosch who made his journey a pleasan one. v

6 Conens. Inroducion and Overview.. Inroducion.. Overview.3. Glossary of Noaion 3. Valuaion of Sandard Opions 5.. Sandard Opions 5... Wha are Opions? 6... Types of Opions Paricipans of he Opions Marke Valuaion 9.. Arbirage Bounds on Valuaion... Arbirage Bounds in Call Prices... Arbirage Bounds in Pu Prices..3. Pu-Call Pariy 3.3. Binomial Tree The One-Sep Binomial Model The Binomial Model for Many Periods The Binomial Model for American Opions The Binomial Model for Opions on Dividend-paying Sock.3.5. Deerminaion of p, u and d.4. The Black-Scholes Formula From Discree o Coninuous Time Derivaion of he Black-Scholes Equaion Properies of he Black-Scholes Equaion 9.5. Opion Sensiiviies Dela Gamma Thea Vega Rho 4 3. Volailiy-dependen Derivaives Compound Opions Definiion Common Uses Valuaion The Sensiiviy of Compound Opions o Volailiy Arbirage Bounds on Valuaion Sensiiviies Chooser Opions Simple Choosers Definiion Common Uses Valuaion The Sensiiviy of Simple Chooser Opions o Varying Time and Srike Price Arbirage Bounds on Valuaion Complex Choosers Definiion Valuaion The Sensiiviy of Complex Chooser Opions o Some of is Parameers American Chooser Opions Definiion 9 vi

7 Valuaion Summary 9 4. Pah-dependen Derivaives Barrier Opions Definiion Common Uses Valuaion Remarks on Barrier Opions Arbirage Bounds on Valuaion Sensiiviies Asian Opions 4... Definiion 4... Common Uses Valuaion Arbirage Bounds on Valuaion Remarks on Asian Opions Sensiiviies Summary Binary Opions Definiion Common Uses Valuaion Arbirage Bounds on Valuaion Remarks on Binary Opions Sensiiviies Summary 6. Summary 3 Appendix A 6 References 35 vii

8 . Inroducion and Overview. Inroducion The use of opions on various sock markes over he world has inroduced a unique opporuniy for invesors o hedge, speculae, creae synheic financial insrumens and reduce funding and oher coss in heir rading sraegies. As explained on he power of opions lies in heir versailiy. They enable an invesor o adap or adjus her posiion according o any siuaion ha arises. Opions can be as speculaive or as conservaive as preferred. This means everyhing from proecing a posiion from a decline, o ourigh being on he movemen of a marke or index, can be implemened. Anoher benefi of using opions is ha hey provide leverage. argues ha since opions cos less han sock hey provide a high-leverage approach o rading, which can significanly limi he overall risk of a rade, or provide addiional income. When a large number of shares is conrolled by one conrac, i does no ake much of a price movemen o generae large profis. This versailiy and leverage, however, does come a a price. Opions are complex securiies and can be exremely risky. This is why, when rading opions, a disclaimer like he following is common: Opions involve risks and are no suiable for everyone. Opion rading can be speculaive in naure and carry subsanial risk of loss. Only inves wih risk capial. Being ignoran of any ype of invesmen places an invesor in a weak posiion. In his documen several aspecs of rading and valuing exoic opions will be invesigaed. The aim is o give insigh ino heir uses and he risks involved in heir rading. Two volailiy-dependen derivaives, namely compound and chooser opions; wo pahdependen derivaives, namely barrier and Asian opions; and lasly binary opions, are discussed in deail.

9 The purpose of his sudy is o provide a reference ha conains boh he mahemaical derivaions and deail in valuaing some exoic opions as well as an overview of heir applicabiliy and use for sudens and oher ineresed paries.. Overview The res of his documen consiss of five chapers. In Chaper, a summary of wellknown resuls on sandard opions is provided. This is included as background for chapers 3, 4 and 5, and for he sake of compleeness. The maerial given here is expanded where necessary for each exoic opion discussed in he subsequen chapers. Chaper 3 explores he value of volailiy-dependen derivaives. These derivaives depend in an imporan way on he level of fuure volailiy. Two of he mos common forms, compound and chooser opions, are described. The focus hen urns o pricing cerain pah-dependen derivaives in Chaper 4. Two of he mos common ypes of pah-dependen opions, barrier and Asian opions, are described. These have in common he fac ha he payoff of each is deermined by he complee pah aken by he underlying price, raher han is final value only. Finally, in Chaper 5, i is illusraed ha binary opions are opions wih disconinuous payoffs. Three forms of his ype of opion are discussed, namely cashor-nohing binary opions, asse-or-nohing binary opions and American-syle cashor-nohing binary opions. For each exoic opion he opion is firs defined, before an overview of is applicabiliy and use is given and compared o sandard opions. The opion valuaion is hen derived in deail. This is followed by a discussion on noable aspecs of ha opion. Where applicable, he arbirage bounds on valuaion of he opions are given. These are he limis wihin which he price of an opion should say, because ouside hese bounds a risk-free arbirage would be possible. They consrain an opion price o a limied range and do no require any assumpions abou wheher he asse price is normally or oherwise disribued. Lasly, he sensiiviies or Greeks of he opions are

10 given. Each Greek leer measures a differen dimension of he risk in an opion posiion, and he aim of a rader is o manage he Greeks so ha all risks are accepable..3 Glossary of Noaion c C d D E H K L M Price of an European call opion Price of an American call opion The size of he downward movemen of he underlying asse in a binomial ree Cash dividend Expecaion operaor Rand knock-ou or knock-in barrier (only used for barrier opions) Predeermined cash payoff or srike price Lower barrier in a barrier opion Curren Rand minimum or maximum price of he underlying asse experienced so far during he life of an opion (only needed for barrier opions) n Number of seps in a binomial ree N(.) Area under he sandard normal disribuion funcion N (.) Area under he sandard bivariae normal disribuion funcion p Price of an European pu opion Up probabiliy in ree model P Price of an American pu opion PV () Presen Value a ime of he quaniy in brackes r Risk-free ineres rae S S T Curren price of underlying asse Price of an underlying asse a he expiraion ime of an opion T Time o expiraion of an opion in number of years T* Time o expiraion of he underlying opion in number of years (only needed for compound opions) X u U μ π Predeermined payoff from an a all-or-nohing opion The size of he up movemen of he underlying asse in a binomial ree Upper barrier in barrier opion Drif of underlying asse The consan Pi

11 ρ σ Correlaion coefficien Volailiy of he relaive price change of he underlying asse 4

12 . Valuaion of Sandard Opions In his chaper well-known resuls on sandard opions are summarized; i is concerned wih he heory of opion pricing and is applicaion o sock opions. I is included as background for chapers 3, 4 and 5, and for he sake of compleeness. The resuls are essenial o an undersanding of he laer chapers on exoic opions. The maerial given here is expanded where necessary for each exoic opion ha is discussed in he laer chapers.. Sandard Opions.. Wha Are Opions? An opion is a conrac ha gives he buyer he righ, bu no he obligaion, o buy (call opion) or sell (pu opion) an underlying asse a a specific price (srike price) on or before a cerain dae (expiraion dae). The pary selling he conrac (wrier) has an obligaion o honour he erms of he agreemen and is herefore paid a premium. The buyer has a 'long' posiion, and he seller a 'shor' posiion. The underlying asse is usually a bond, sock, commodiy, ineres rae, index or exchange rae. Throughou his paper a reference o one of hese underlying asses is also a reference o any of he ohers, and he erms are herefore used inerchangeably. Because his is a conrac, he value of which is derived from an underlying asse and oher variables, i is classified as a derivaive. I is also a binding conrac wih sricly defined erms and properies. Once an invesor owns an opion, here are hree mehods ha can be used o make a profi or avoid a loss; exercise i, offse i wih anoher opion, or le i expire worhlessly. By exercising an opion she has bough, an invesor is choosing o ake delivery of (call) or o sell (pu) he underlying asse a he opion's srike price. Only 5

13 opion buyers have he choice o exercise an opion. Opion sellers have o honour he agreemen if he opions hey sold are exercised by he opion holders. Offseing is a mehod of reversing he original ransacion o exi he rade. This means ha an invesor holds wo opion posiions wih exacly opposie payoffs, leaving her in a risk-neural posiion. If she bough a call, she would have o sell he call wih he same srike price and expiraion dae. If she sold a call, she would have o buy a call wih he same srike price and expiraion dae. If she bough a pu, she would have o sell a pu wih he same srike price and expiraion dae. If she sold a pu, she would have o buy a pu wih he same srike price and expiraion dae. If an invesor does no offse her posiion, she has no officially exied he rade. If an opion has no been offse or exercised by expiraion, he opion expires worhlessly. The opion buyer hen loses he premium she paid o inves in he opion. If he invesor is he seller of an opion she would wan i o expire worhlessly, because hen she ges o keep he opion premium she received. Since an opion seller wans an opion o expire worhlessly, he passage of ime is an opion seller's friend and an opion buyer's enemy. If he invesor bough an opion he premium is nonrefundable, even if she les he opion expire worhlessly. As an opion ges closer o expiraion, i decreases in value. The syle of he opion deermines when he buyer may exercise he opion. Generall,y he conrac will eiher be American syle, European syle or Bermudan syle. American syle opions can be exercised a any poin in ime, up o he expiraion dae. European syle opions can only be exercise on he expiraion dae. Bermudan syle opions may be exercised on several specific daes up o he expiraion dae. I is ineresing o noe ha Bermuda lies halfway beween America and Europe... Types of Opions A call opion is a conrac ha gives he buyer he righ, bu no he obligaion, o buy an underlying asse a a specific price on or before a cerain dae. - If a call opion is exercised a some fuure ime, he payoff will be he amoun by which he underlying asse price exceeds he srike price. 6

14 T - I is only worh exercising he opion if he curren marke price of he underlying asse is greaer han he srike price. - Breakeven poin for exercising a call opion equals he srike price plus a premium. - The value of he opion o he buyer of a call will increase as he underlying asse price increases wihin he expiraion period. A pu opion is a conrac ha gives he buyer he righ, bu no he obligaion, o sell an underlying asse a a specific price on or before a cerain dae. - If a pu opion is exercised a some fuure ime, he payoff will be he amoun by which he srike price exceeds he underlying asse price. - Call wriers keep he full premium, unless he underlying asse price rises above he srike price. - Breakeven poin is he srike price plus a premium. - The value of he opion o he buyer of a pu will increase as he underlying asse price decreases wihin he expiraion period...3 Paricipans in he Opions Marke The wo ypes of opions lead o four possible ypes of posiions in opions markes:. Buyers of calls : long call posiion. Sellers of calls : shor call posiion 3. Buyers of pus : long pu posiion 4. Sellers of pus : shor pu posiion These rades can be used direcly for speculaion. If hey are combined wih oher posiions hey can also be used in hedging. 7

15 ..4 Valuaion The oal cos of an opion is called he opion premium. This price for an opion conrac is ulimaely deermined by supply and demand, bu is influenced by five principal facors: The curren price of he underlying securiy (S). The srike price (K). - The inrinsic value elemen of he opion premium is he value ha he buyer can ge from exercising he opion immediaely. For a call opion his is max( S K,), and for a pu opion max( K S,). This means ha for call opions, he opion is in-he-money if he share price is above he srike price. A pu opion is in-he-money when he share price is below he srike price. The amoun by which an opion is in-he-money is is inrinsic value. Opions a-he-money or ou-ofhe-money has an inrinsic value of zero. The cumulaive cos required o hold a posiion in he securiy, including he risk-free ineres rae (r) and dividends (D) expeced during he life of he opion. The ime o expiraion (T). - The ime value elemen of he premium is he chance ha an opion will move ino he money during he ime o is expiraion dae. I herefore decreases o zero a is expiraion dae and is dependen on he syle of he opion. The esimae of he fuure volailiy of he securiy's price (σ). The effec of hese facors on he prices of boh call and pu opions is explained by Reilly and Brown (6) in Invesmen Analysis and Porfolio Managemen and is summarised as follows: 8

16 Will Cause an increase / decrease in: An Increase in: Call Value Pu Value. S. K 3. T / 4. r 5. σ Call opion:. An increase in S increases he call s inrinsic value and herefore also he value of he call opion.. An increase in K decreases he call s inrinsic value and herefore also he value of he call opion. 3. If T increases i means ha he opion has more ime unil expiraion, which increases he value of he ime premium componen, because greaer opporuniy exiss for he conrac o finish in-he-money. The value of he call opion increases. 4. As he value of r increases, i reduces he presen value of K. The value of K is an expense for he call holder, who mus pay i a expiraion o exercise he conrac. Since i is decreased, i will lead o an increase in he value of he opion. 5. An increase in σ increases he probabiliy ha he opion will be deeper in-hemoney a expiraion. The opion becomes more valuable. Pu opion:. An increase in S decreases he pu s inrinsic value and herefore also he value of he pu opion.. An increase in K increases he pu s inrinsic value and herefore also he value of he pu opion. 3. If T increases, here is a rade-off beween he longer ime over which he securiy price could move in he desired direcion and he reduced presen value of he exercise price received by he seller a expiraion. 4. As he value of r increases, i reduces he presen value of K. This hurs he holder of he pu, who receives he srike price if he conrac is exercised. 9

17 5. An increase in σ increases he probabiliy ha he opion will be deeper in-hemoney a expiraion. The opion becomes more valuable. There are wo basic mehods of deermining he price of an opion using hese facors; he Black-Scholes pricing model and he Binomial pricing model.. Arbirage Bounds on Valuaion Arbirage bounds define he bounds wherein an opion should rade o exclude he possibiliy of arbirage opporuniies in he marke. From Gemmill (993), Hull (6), and Reilly and Brown (6) he following summary was consruced... Arbirage Bounds on Call Prices Upper bound Boh an American and European call opion gives he holder he righ, bu no he obligaion, o buy one uni of he underlying asse for a cerain price a some fuure dae. Therefore, where c is he European call value and C is he American call value c Sand C S. American-syle and European-syle Call opions I is imporan ha an American pu or call has o be a leas as valuable as is corresponding European syle conrac: c C. Lower bound Any opion, call or pu, canno be worh less han zero: c and C. Lower bound for American Calls on Non-Dividend-Paying Socks

18 The minimum value for an American call opion ha can be exercised immediaely is he curren underlying asse price minus he srike price: C S K. Lower bound for European Calls on Non-Dividend-Paying Asses rt c+ Ke S or c S Ke rt. I is never opimal o exercise an American call opion on a non-dividend-paying asse before he expiraion dae. Since he lower bound for a European call opion ( S Ke rt ) lies above he inrinsic value bound ( S K), as applicable o he American call opion, he second is redundan. This is rt because ( S Ke ) ( S K). This means ha for an underlying asse which does no pay dividends, C and c will be equal o one anoher. In summary, he arbirage bounds for call opions are: rt max[, S K] max[, S Ke ] c C S. This expression says ha. he American call is a leas as valuable as he European conrac;. neiher call can be more valuable han he underlying sock, and 3. boh conracs are a leas as valuable as heir inrinsic values, expressed on boh a nominal and discouned basis.

19 .. Arbirage Bounds on Pu Prices Upper bound Boh an American and European pu opion gives he holder he righ, bu no he obligaion, o sell one uni of he underlying asse for he srike price K a some fuure dae. No maer how low he sock price becomes, he opion can never be worh more han K. Hence, where p is he European pu value and P is he American pu value, p K and P K. For he European opion, we know ha he opion canno be worh more han K a mauriy. I follows ha i canno be worh more han he presen value of K oday: p Ke rt. American-syle and European-syle Pu opions An American pu has o be a leas as valuable as is corresponding European syle conrac: p P. Lower bound Any opion, call or pu, canno be worh less han zero: p and P. Lower bound for American Pus on Non-Dividend-Paying Asses The minimum value for an American pu opion ha can be exercised immediaely is he curren srike price minus he underlying asse price: P K S. Lower bound for European Pus on Non-Dividend-Paying Asses

20 S p+ Ke rt or rt p Ke S. The American lower bound o he pu price lies above he European bound since rt ( Ke S) ( K S). I can be opimal o exercise an American pu opion on a non-dividend-paying underlying asse before he expiraion dae. Similarly o a call opion, a pu opion can be seen as providing insurance. A pu opion, when held in conjuncion wih he sock, insures he holder agains he price falling below a cerain level. However, a pu opion is differen from a call opion in ha i may be opimal for an invesor o forego his insurance and exercise early in order o realise he srike price immediaely. In summary, he arbirage bounds for pu opions are: rt max[, Ke S] p P K. This expression says ha. he American pu is a leas as valuable as he European conrac;. neiher pu can be more valuable han he srike price, and 3. boh conracs are a leas as valuable as he inrinsic value expressed on a discouned basis...3 Pu-Call Pariy Pu-Call Pariy for European Opions on Non-Dividend Paying Asses There exiss an imporan relaionship beween European pu and call prices in efficien capial markes. Pu-call pariy depends on he assumpion ha markes are free from arbirage opporuniies. This relaionship is given by 3

21 rt c+ Ke = p+ S. (..) I shows ha he value of a European call, wih a specific srike price and mauriy dae, can be deduced from he value of a European pu wih he same srike price and mauriy dae, and vice versa. This relaionship is useful in pracice for wo reasons. Firsly, if here does no exis he desired pu or call posiion in he marke, an invesor can replicae he cashflow paern of he pu or call by using inerrelaed asses in he correc forma. By rearranging (..) i follows ha and c= p+ S Ke rt (..) p = c S+ Ke rt. (..3) Secondly, i is useful in idenifying arbirage opporuniies in he marke. A relaive saemen of he prices of pus and calls can be made if hey are compared o one anoher. If a call is overpriced relaive o he pu, he call can be sold and he pu bough o make a riskless profi, and vice versa. Pu-Call Pariy for American Opions on Non-Dividend Paying Sock The pu-call pariy relaionship for American calls and pus on non-dividend-paying sock is given by rt S Ke C P S K. Adjusing Arbirage Bounds for Dividends Assume he sock pays a dividend D(T) immediaely before is expiraion dae a ime T. Also assume ha when a dividend is paid, he share price will fall by he full amoun of he dividend. The presen value of he fall is D(). An imporan assumpion ha we are making is ha he dividend paymen is known a ime, when he opion conrac is enered ino. This is reasonable assumpion, since in pracise he 4

22 dividends payable during he life of he opion can usually be prediced wih reasonable accuracy. To adjus he previously derived bounds for dividends we can simple adjus he sock price downwards for he presen value of he dividends. This means ha we subsiue S D ( ) for S o ge: Lower bounds for European calls on dividend-paying asse c S D() Ke rt. (..4) When he underlying asse pays dividends i is no longer rue ha he American call opion and European call opion have exacly he same value. Then he argumen ha an American opion mus be a leas as valuable as is European counerpar because i allows more choice, becomes relevan again. This choice can be used o preserve value when he European conrac canno. Lower bounds for European pus on dividend-paying asse p D() + Ke rt S. (..5) Deciding o exercise a pu before expiraion does no depend on he presence of dividends. I is known ha a dividend paymen increases he value of a pu opion by reducing he value of he underlying sock wihou an offse in he srike price. This is irrelevan in deermining wheher o exercise early when compared o he liabiliy of he sock iself. This means ha having he choice o exercise early and receive he inrinsic value immediaely is he only deerminisic facor. Pu-Call Pariy for European opions on dividend-paying asse rt c+ D() + Ke = p+ S. 5

23 Pu-Call Pariy for American opions on dividend-paying asse rt S Ke C P S D() K..3 Binomial Tree If opions are correcly priced in he marke, i should no be possible o make definie profis by creaing porfolios of long and shor posiion in opions and heir underlying socks. We herefore price opions using risk-neural valuaion. In a risk-neural world, all securiies have an expeced reurn equal o he risk-free ineres rae. Also, in a risk-neural world, he appropriae discoun rae o use for expeced fuure cashflows is he risk-free ineres rae. As shown by Gemmil (993), Hull (6) and Reilly e al. (6), he Binomial Tree mehod can be used o find he fair value for opions and shares. A number of simplifying assumpions are made:. The underlying asse price follows a binomial random process over ime.. The disribuion of share prices is muliplicaive binomial. 3. The upward (u) and downward (d) mulipliers are he same in all periods. 4. There are no ransacion coss, so ha a riskless hedge can be consruced for each period beween he opion and he asse a no exra cos. 5. Ineres raes are consan. 6. A firs we assume ha early exercise is no possible. 7. There are no dividends. 8. No riskless arbirage opporuniies exis..3. The One-Sep Binomial Model To derive he value for an opion we se up a hedged posiion wih boh an opion and is underlying share. This creaes a riskless posiion ha mus pay he risk-free rae. Suppose he opion expires a he end of he nex period of lengh T. Le S be he iniial share price, which in he nex period will eiher rise by an upward facor u o us or fall by a downward facor d o ds, where u > and d <. The corresponding payoffs o he opion is f u and f d. 6

24 Share price moves Call opion values Pu opion values us f u = c u = us K f u = p u = K us S c p ds f d = c d = ds K f d = p d = K ds Figure : Sock and opion prices in a general one-sep ree The value of he opion is given by where ( ) rt f = e pfu + p fd (.3.) rt e d p = u d. (.3.) In Eq. (.3.) he value of he opion is given by he presen value of he weighed average of he pay-offs o higher and lower share prices. The weighs p and (-p) are inerpreed as he implici probabiliies of an up movemen in he sock price and a down movemen in he sock price respecively. The value of he opion hen simply becomes he presen value of he probabiliy weighed pay-offs. Therefore f = PV E pay off..3. The Binomial Model for Many Periods Consider he wo-period ree in Fig. below, where he objecive is o calculae he opion price a he iniial node of he ree. Using he same assumpions as before, wih he lengh of each ime sep se equal o δ years, we apply our binomial formula Eq. (.3.) o he op wo branches of he ree which gives ( ) rt f = e pf u uu + p fud. (.3.3) Repeaing his procedure for he boom wo branches leads o 7

25 ( ) rδ f = e pf d ud + p fdd. (.3.4) Solve f by subsiuing Eq. (.3.3) and Eq. (.3.4) ino Hence ( ) rδ f = e pfu + p fd. r f = e p fuu + p p fud + p fdd δ. (.3.5) The opion price is again equal o is expeced pay-off in a risk-neural world discouned a he risk-free rae. This remains rue as we add more seps o he binomial ree. Su f uu Su f u S Sud f f ud Sd f d Sd f Figure : Sock and opion prices in a general wo-sep ree To calculae he value of an opion in erms of he price of he underlying sock a ree is consruced ha comprises of many successive wo-branch segmens. Valuaion begins wih he known final pay-off and works backwards sep by sep unil he presen ime is reached. The mehod can be exended o opions ha have any number of discree ime periods o mauriy. This allows an invesor o make each period arbirarily shor by dividing he ime o mauriy ino enough ime seps in order o obain reasonably accurae resuls. dd 8

26 The many-period binomial opions-pricing formula is obained for a call opion as n rnδ n k n k k n k c= e p ( p) max { u d S K, } k = k and for a pu opion as (.3.6) n n rnδ k n k k n k p = e p ( p) max { K u d S, } k = k. This shows ha European opions can be valued by. calculaing for each possible pah he payoff a expiraion (afer n ime seps);. weighing his by he risk neural probabiliy of he pah, 3. adding he resuling erms, and 4. discouning his back o he presen a he risk-free rae of ineres..3.3 The Binomial Model for American Opions To value American opions he possibiliy of early exercise has o be considered. The opion will only be exercised early if he pay-off from early exercise exceeds he value of he equivalen European value a a specific node. Therefore, work back hrough he ree from he end o he beginning in he same way as for he European opions, bu es a each node wheher early exercise is opimal. The value of he opion a he final node is he same as for European opions. A earlier nodes he value of he opion is he greaer of rt.) The value given by = + ( ) f e pfu p fd..) The pay-off from early exercise given by he inrinsic value. 9

27 .3.4 The Binomial Model for Opions on Dividend-paying Sock The Binomial model can be used for dividend-paying sock. When a dividend is paid, he price of a share will fall by he amoun of he dividend. If i is less han he dividend, he rader could buy he share jus prior o he ex-dividend dae, capure he dividend, and sell he share immediaely afer i has fallen. I is herefore assumed ha he fall in share price is equal o he full dividend amoun. Boh he cases where he dividend is a known Rand amoun or a known dividend yield are considered If i is assumed ha he Rand amoun of he dividend is known in advance, he shareprice binomial ree will be knocked side-ways a he ex-dividend dae. If a given dividend is paid in Rand, he ree will sar o have branches ha do no recombine. This means ha he number of nodes ha have o be evaluaed, paricularly if here are several dividends, becomes large. As shown in Fig. 3 a single dividend of size D resuls in a new, separaely developing ree being formed for each node ha exised a he ime of he dividend paymen. This process is compuaionally slow. Su ( Su ) D u S Su D Sd ( Su ) D d ( Sd ) D u Sd D ( Sd ) D d Figure 3: Two-period sock-price ree wih a dividend afer one period To avoid his problem he assumpion is made ha he dividend is some proporion δ of he share price a ha poin in he ree. I is herefore assumed ha he dividend yield is known. The share price afer he dividend paymen will hen eiher increase o

28 Su ( δ) or decrease o Sd ( δ) afer one sep. The whole ree is again a geomeric process and he nodes recombine. Su Su ( δ ) Su ( δ ) S Sud ( δ ) Sd Sd ( δ ) Sd ( δ ) Figure 4: Two-period sock-price ree wih a dividend-yield paymen afer one period.3.5 Deerminaion of p, u and d I is necessary o consruc a binomial ree o represen he movemens in a general sock price in he marke. This is done by choosing he parameers u and d o mach he volailiy of he sock price and making i consisen wih normally disribued reurns. The Binomial ree of share price, as described, is boh symmerical and recombines in he sense ha an up movemen followed by a down movemen leads o he same sock price as a down movemen followed by an up movemen. In order for his o hold we choose he down muliplier (d) as he inverse of he up muliplier (u). This means ha if u =, hen he reurns o holding he asse will be symmerical. d The widh of he binomial ree is relaed o he size of u, he up muliplier per sep, and he number of seps ha have occurred. The equivalen assumpion for an asse

29 ha has normally disribued reurns is ha he variance is consan per period. If he variance for a ime sep of δ years is given by σ δ, hen he sandard deviaion or he volailiy of he asse is equal o σ δ. If we assume ha prices are lognormally disribued, we can imagine he disribuion widening as ime goes by, jus as he binomial ree widens a successive branches. The acual values o use for he up and down mulipliers in a binomial ree should be consisen wih normally disribued reurns. Le μ be he expeced reurn on a sock and σ he volailiy in he real world. Imagine a one sep binomial ree wih a sep of lengh δ. The binomial process for asse prices gives normally disribued reurns in he limi if and u = e σ δ d = = e u σ δ. Afer a large number of seps his choice of u and d leads o a variance of σδ. Assume ha he expeced reurn of an up movemen in he real world is q. In order o mach he sock price volailiy wih he ree s parameers, he following equaion mus be saisfied e μδ μδ σδ = ( u+ d) ud e. One soluion o his equaion is u = e σ δ σ δ d = e.

30 .4 The Black-Scholes Pricing Formula.4. From Discree o Coninuous Time The binomial formulas derived for he muli-period model is he discree-ime version of he coninuous-ime Black-Scholes formula. From he wo-sep binomial ree i can be shown ha if i is assumed ha he underlying sock prices are lognormally disribued, and u and d are defined in order o be consisen wih he volailiy of he sock price reurns, he Binomial opion values will converge o he Black Scholes values as n. This is explained by Gemmill (993), who carries on o show he similariy beween he Binomial and Black Scholes models. If we assume ha prices are lognormal, is disribuion widens as ime goes by, jus as he binomial ree widens a successive branches. Beginning a a share price S a ime zero, he disribuion widens unil a par of i exceeds he srike price, K. A mauriy, he pay-off o he opion is he shaded area above he srike price for a call opion. The Black-Sholes value of he opion oday is he presen value of his shaded area. Figure 5: Call price rising as he price disribuion widens over ime.4. Derivaion of he Black-Scholes Equaion The Black-Scholes equaion is derived as shown in Gemmill (993), Hull (6), Smih (976) and Chappel (99). The derivaion of he Black-Scholes equaion consiss of wo pars. Firsly, i is shown ha a riskless hedge can be consruced when he sochasic process for he underlying asse price is lognormal. This is done by 3

31 seing up a porfolio conaining sock and European call opions. In he absence of arbirage opporuniies, he reurn from his porfolio mus be he risk-free rae. The reason for his is ha he sources of change in he value of he porfolio mus be he prices, since i affecs he value of boh he sock iself and he derivaive in he porfolio. This follows also from he fac ha a a poin in ime he quaniies of he asses are fixed. If he call price is a funcion of he sock price and he ime o mauriy, hen changes in he call price can be expressed as a funcion of he changes in he sock price and changes in he ime o mauriy of he opion. Thus, in a shor period of ime, he price of he derivaive is direcly correlaed wih he price of he underlying sock. Therefore, a any poin in ime, he porfolio can be made ino a riskless hedge by choosing an appropriae porfolio of he sock and he derivaive o offse any uncerainy. If quaniies of he sock and opion in he hedge porfolio are coninuously adjused in he appropriae manner as he asse price changes over ime, hen he reurn o he hedge porfolio becomes riskless and he porfolio mus earn he risk-free rae. Secondly, i shows ha he call opion price is deermined by a second order parial differenial equaion. The Black-Scholes equaion makes exacly he same assumpions as he binomial approach, plus one addiional one; i is also assumed ha he underlying asse price follows a lognormal disribuion for which he variance is proporional o ime. The assumpions used o derive he Black-Scholes equaion are as follows:. The sock price follows a geomeric Brownian moion, wih μ and σ consan. Therefore he disribuion of possible sock prices a he end of any finie inerval is lognormal and he log reurns are normally disribued.. Shor selling is allowed and no penalies imposed. 3. There are no ransacion coss, so ha a riskless hedge can be consruced for each period beween he opion and he asse a no exra cos. 4. The risk-free ineres rae is consan and he same for all mauriies. 5. There are no dividends. 6. No riskless arbirage opporuniies exis. 7. Securiies rade coninuously in he marke. 8. The opion is European and can only be exercised a mauriy. 4

32 9. All securiies are perfecly divisible so ha i is possible o borrow any fracion of he price of a securiy or o hold i a he shor-erm ineres rae. The Black-Scholes-Meron differenial equaion is given by c c rc = + rs + σ S S S c (.4.) (Hull, 6). I can be derived and solved for many differen derivaives ha can be defined wih S as he underlying variable, no only a European call opion. I is imporan o realise ha he hedge porfolio used o derive he differenial equaion is no permanenly c riskless. I is riskless only for a very shor period of ime. As S and change, S also changes. To keep he porfolio riskless, i is herefore necessary o change he relaive proporions of he derivaive and he sock in he porfolio frequenly. (Hull, 6) The differenial equaion defines he value of he call opion subjec o he boundary condiion, which specifies he value of he derivaive a he boundaries of possible values of S and. I is known ha, a mauriy, a call opion has he key boundary condiion: ( S ) c= max, K. (.4.) Black and Scholes used he hea-exchange equaion from physics o solve he differenial equaion for he call price, c, subjec o he boundary condiion. A more inuiive soluion is suggesed in he paper by Cox and Ross (975). To solve he equaion, wo observaions are made: Firs, whaever he soluion of he differenial equaion, i is a funcion only of he variables in (.4.) and (.4.). Therefore, he soluion o he opion pricing problem is a funcion of he five variables: ) he sock price, S; ) he insananeous variance rae on he sock price, σ; 3) he srike price of he opion, K; 4) he ime o mauriy of he opion, T, and 5

33 5) he risk-free ineres rae, r. The firs four of hese variables are direcly observable; only he variance rae mus be esimaed. Secondly, in seing up he hedge porfolio, he only assumpion involving he preferences of he individuals in he marke is ha wo asses which are perfec subsiues mus earn he same equilibrium rae of reurn; no assumpions involving risk preference are made. This suggess ha if a soluion o he problem can be found which assumes one paricular preference srucure, i mus be he soluion of he differenial equaion for any preference srucure ha permis equilibrium (Smih, 976). This leads o he principle of risk-neural valuaion. I says ha he price of an opion or oher derivaive, when expressed in erms of he price of he underlying sock, is independen of risk preferences. Opions herefore have he same value in a risk-neural world as hey have in he real world. I may herefore be assumed ha he world is risk-neural for he purpose of valuing opions, o simplify he analysis. In a risk-neural world all securiies have an expeced reurn equal o he risk-free ineres rae. Also, in a risk-neural world, he appropriae discoun rae o use for expeced fuure cashflows is he risk-free ineres rae. (Hull, 6) The expeced value of he opion a mauriy in he risk-neural world is ( ) Eˆ max ST K,, where Ê denoes he expeced value in a risk-neural world. From he risk-neural argumen, he European call opion price, c, is he expeced value discouned a he risk-free ineres rae, ha is rt c= e Eˆ c T rt = e Eˆ max ST K,. (.4.3) Sar by looking a he value of c T. If he sock price S follows he process 6

34 ds = μ Sd +σ Sdz, (.4.4) hen, using Io s lemma, i can be found ha he process followed by ln S is σ d ( ln S ) = μ d +σdz. (.4.5) Because μ and σ are consan, ln S follows a generalised Wiener process wih consan drif rae and consan variance. The change in ln S beween ime zero and some fuure ime, T, is herefore normally disribued wih mean and variance given respecively by This means ha or σ μ Τ and σ T. σ ln ST ln S φ μ T, σ T σ ln ST φ μ T + ln S, σ T (.4.6) where φ(m,s) denoes he normal disribuion wih mean m and sandard deviaion s. This leads o he well known resul for a European call opion: rt c= e SN d e KN d rt. (.4.7) The proof can be found in he mos financial mahemaical exbooks (cf. Hull, 6). The funcion N( x) is he cumulaive probabiliy funcion for a sandardised normal disribuion. In oher words, i is he probabiliy ha a variable wih a sandard normal disribuion will be less han x. The expression N( d ) is he probabiliy ha he 7

35 opion will be exercised in a risk-neural world, so ha KN ( d ) imes he probabiliy ha he srike price will be paid. The expression is he srike price he expeced value of a variable ha equals ST if S T > K and is zero, oherwise, in a risk-neural world. (Hull, 6) SN d e rt is Since rt c= e Eˆ c T, he expeced pay-off a mauriy, can also be rewrien as rt = Eˆ c SN d e KN d T ( ) N d rt = N d Se N d K. In his expression N( d ) is he probabiliy ha he call finishes in-he-money and is muliplied by he expeced in-he-money pay-off (Gemmill, 993). To find he European pu price, pu-call pariy can be used: p = c S+ Ke rt. Subsiuing from he Black-Scholes equaion for c gives rt ( ) rt ( rt. p = SN d Ke N d S + Ke ) rt = S N d Ke N d (.4.8) = Ke N d SN d The expression in (.4.8) can also be derived direcly from he parial differenial equaion solved subjec o he primary boundary condiion for pu opions given by p = max, K S. 8

36 .4.3 Properies of he Black-Sholes Equaion The properies of he Black-Scholes equaion is given by Gemmill (993), Hull (6), Smih (976) and Chappel (99). American Opions on Non-Dividend Paying Sock The expressions above were derived for European pu and call opions on nondividend-paying sock. Because he European price equals he American price when here are no dividends, (.4.7) also gives he value of an American call opion on nondividend-paying sock. There is no exac analyic formula for he value of an American pu opion on non-dividend-paying sock. Adjusing he Black-Scholes Equaion for Dividends The Black-Scholes model, like he Binomial model, can be used o value dividendpaying sock. I is assumed ha he amoun and iming of he dividends during he life of he opion can be prediced wih cerainy. When a dividend is paid, he price of a share will fall by an amoun reflecing he dividend paid per share. In he absence of any ax effecs, he fall in share price is equal o he full dividend amoun. A dividend is a pay-ou o a shareholder which he holder of a call opion does no ge, ye he holder suffers he fall in share price. From he oher perspecive, he holder of a pu opion will benefi from he fall in share price ha follows a dividend. Dividends ha will be paid ou over he lives of opions herefore reduce call prices. (Gemmill, 993) European Opions Consider he value of a European opion when he sock price is he sum of wo componens. The one componen is ha par of he price accouned for by he known dividends during he life of he opion and is considered riskless. The riskless componen, a any given ime, is he presen value of all he dividends during he life of he opion discouned from he ex-dividend daes o he presen. By he ime he opion maures, he dividends will have been paid and he riskless componen will no longer exis. The Black-Scholes formula can be used, provided ha he sock price is 9

37 reduced by he presen value of all dividends during he life of he opion, discouned from he ex-dividend daes a he risk-free rae. (Hull, 6) American Call Opions If a dividend is sufficienly large, i will be profiable o exercise a call jus before he dividend is due. Shares go ex-dividend before he acual paymen is made, so he fall in share price occurs on he ex-dividend dae. I is opimal o exercise only a a ime immediaely before he sock goes ex-dividend, because exercising a his ime yields an exra dividend, bu resuls in he loss of he ime-value on he call. (Gemmill, 993) Assume ha here are n ex-dividend daes expeced during he life of he opion and ha,,, n are he imes immediaely before he n ex-dividend daes where < <... < n. D, D,..., D n. Le he dividend paymens corresponding o hese imes be (Hull, 6) Suppose here is no volailiy, ha is σ =. Le S be he share price and K be he srike price. Sar by considering early exercise jus before he las ex-dividend dae, ime n. I is known ha he share price will fall o S( n) dae. Assume he call is in-he-money ( S( n ) > K). Then: D n on he las ex-dividend. Exercising jus before he ex-dividend dae gives a payoff afer he dividend paymen equal o ( ( n) n) n ( n) S D K + D = S K ;. no exercising, bu waiing unil mauriy, resuls in a value oday, given ha σ =, ha is equal o he lower bound given by (..4) of S D Ke n n rt ( ) n. Wih zero volailiy, exercise will herefore be worh while if rt ( n) S K > S D Ke. n n n 3

38 Hence, for opimal exercise a ime n, i is required ha D > K PV ( K); n ha is n ( rt n ) D > K e. (.4.9) Similarly, i can be shown ha (.4.9) holds for any one of he n ex-dividend daes during he life of he opion. This means ha early exercise is opimal if i ( r ) i i + D > K e. This implies ha early exercise is more likely if:. The dividend (D i ) is large relaive o he srike price (K);. he ime unil he nex ex-dividend dae is fairly close so ha PV ( K) K, and 3. he volailiy is low, so ha he ime value given up o be exercising he opion is low. To value American call opions on dividend-paying sock he pseudo-american approach, firs oulined by Black (975), can be used (Gemmill, 993). This involves calculaing he price of European opions ha maure a ime T and, and seing he American price equal o he greaer of he wo (Hull, 6). To demonsrae he procedure, consider he case where here is only one ex-dividend dae during he life of he opion. The share price will fall a ime, when he share goes ex-dividend, bu he opion poenially coninues unil mauriy a ime T. The buyer of he American call is now considered o have wo separae European call opions. The firs call opion, worh c shor, expires a ime, immediaely afer which he sock pays a dividend of D. The call price equaion can be wrien as a funcional relaionship: r c = f S De,, r, σ, K D. shor 3

39 The sock price is discouned by he presen value of he dividend, bu his is offse as he srike price is reduced by he dividend paymen. The second call, worh, expires a T and pays no dividend. I can be wrien as c long long ( r,,,, ) c = f S De T r σ K. As before, he sock price is discouned by he presen value of he dividend, bu his ime here is no receip of dividend o reduce he exercise price. As he holder of he American call effecively has wo muually exclusive European call opions, he call will be valued oday as he higher of he wo, ( shor long ) C = max c, c. Two Black-Scholes evaluaions are made and he larger value is chosen as he correc call value (Gemmill, 993). As he approach is exended o siuaions where here are n dividend paymens during he life of he opion, he number of Black-Scholes evaluaions will increase o ( n +).The correc value of he American call opion will hen be he maximum of hese. The pseudo-american adjusmen is relaively accurae, bu will slighly undervalue he call. The reason is ha i assumes ha he holder has o decide oday when he call will be exercised. In pracice, he choice remains open unil jus before each of he ex-dividend daes. American Pu Opions The Black-Scholes model is inadequae when valuing American pu opions. The problem is ha early exercise may be profiable for a pu, especially in he absence of dividends. There is no analyic equivalen of he Black-Scholes equaion ha allows for his, because in principle exercise could occur a almos any dae beween oday and he mauriy of he opion. One suggesion is o abandon he Black-Scholes mehod and use he binomial model insead. The binomial model is accurae because he exercise value is considered a each node of he ree. For he same reason he mehod is compuaionally very slow. (Gemmill, 993) 3

40 Following he explanaion given by Gemmill (993), i is found ha he Macmillan (986) mehod gives a relaively simple equaion which is reasonably accurae and quick o calculae. This is an approximaion based upon he Black-Scholes equaion. Sar wih he equaion P = p + FA, (.4.9) where P is he American pu price, p is he European pu price and FA is he difference beween he European and American pus. This follows from he arbirage bounds on pu opions, where i was argued ha an American pu is a leas as valuable as he European conrac. To calculae FA, i is regarded as he value of anoher opion, he opion o exercise early and i is analysed as follows (Gemmill, 993). ** Early exercise will occur if he sock price, S, falls below some criical level S. Below his level he pu is simply given by is inrinsic value, P= K S for S S **. Above he criical sock price, he pu value is given by P= p+ FA for S > S **. The correcion facor (FA) depends on how far he sock price, S, is above he criical level, ** S. I is where q S, ** FA = A S M q =.5 ( M ) ( M ) + 4 W S A = ** N d ** q, 33

41 ** in which is he Black-Scholes d value a S = S d ** M W r = σ r = e. The ieraive sock price ** S is found by an ieraive procedure as follows:. Calculae q from he equaion above, which is a consan.. Guess a value for S **, he criical sock price, and calculae A from he equaion above. 3. See wheher he exercise value, K S, exacly equals he approximaed unexercised value, ha is wheher q S. ** K S = p+ A S If his equaion holds, hen he criical price ** S has been found. If he equaliy does no hold, a new is chosen and he algorihm is coninued a sep (). This mehod can be implemened using a sofware searching algorihm, for example Solver in Microsof Excel. ** S.5 Opion Sensiiviies The sensiiviy of opion prices o is five inpus is measured. This can be done since closed form soluions exis for sandard opion prices..5. Dela Dela measures he sensiiviy of he opion price o he share price. I is he raio of change in he price of he sock opion o he change in he price of he underlying sock in he limi. We find dela by aking he firs parial derivaive of he opion price, which also represens he slope of he curve ha relaes he opion price A o he curren underlying asse price B. 34

42 Figure 6: Calculaion of dela The dela of a European call (c) opion on non-dividend paying socks is given by where d c Δ c = = N d S, + + ( σ ) ln S/ K r / T = σ T. Dela of a call opion has a posiive sign ( N( d ) ) for he buyer of a call and a negaive sign ( N( d ) ) for he seller of a call. For a European pu on non-dividend paying socks, dela is p Δ p = = N( d ) S. This dela is negaive ( ) for he buyer of a pu and posiive ( N d ) for he seller of a pu. N( d ) 35

43 Dela ranges beween zero and approximaely one and changes as an opions goes more ino- or ou-of-he-money. A-he-money opions have a dela of approximaely.5, while he dela of in-he-money opions ends owards one and he dela for ouof-he-money opions ends owards. These delas or hedge raios can be used o consruc a riskless porfolio consising of a posiion in an opion and a posiion in he sock. This is known as dela hedging, where hedgers mach heir exposure o an opion posiion. I is imporan o remember ha because dela changes, he invesor s posiion remains dela hedged only for a relaively shor period of ime. Therefore he hedge has o be rebalanced periodically. Consider he following: Long call posiion Figure 7: Long call posiion A hedger owns a call opion. If he sock price increases, he value of he call opion also increases. We know ha he dela of a long call posiion is posiive, i.e. c Δ long call = = N( d ) >. S In order o hedge he posiion, he hedger sells dela shares of he sock. His porfolio herefore consiss of + : opion, and - : shares of he sock. 36

44 If he share price increases, he invesor makes a profi on he call posiion equal o he loss he makes on he shares. The dela for a porfolio of opions dependen on a single asse, which price is S, is Π Δ= S, where П is he value of he porfolio. If he porfolio consiss of a quaniy w i of opion f i ( i n), ha is n Π= wf i i, i= hen he dela of he porfolio is given by weighed sum of he individual delas: Π f n n i Δ= = wi = wiδ S i= S i= i, where Δ i is he dela of he i h posiion..5. Gamma The sensiiviy of dela o a change in he share price is known as gamma. I is he rae of change of he porfolio s dela wih respec o he price of he underlying asse. Gamma is calculaed as he second parial derivaive of he opion price wih respec o he share price: Π Π Π Π Γ= = = Δ S S S S. The absolue value of he gamma for a European pu or call on non-dividend-paying sock is given by 37

45 where Γ= N '( d ) Sσ, d N' ( d ) = e. π The sign is deermined by he posiion aken. Since he dela of a long call is posiive, he gamma of a long call will also be posiive. The dela of a shor call is negaive, so i also has a negaive gamma. Similarly, a long pu has a posiive gamma and a shor call has a negaive gamma. Gamma can also be seen as he gap beween he dela slope and he curve of he opion price, relaive o he underlying sock price. In Fig. 8 i can be seen ha dela is an inaccurae measuremen of he relaive movemen of asse price and opion value. When he sock price moves from S o S, using dela assumes he opion price moves from C o C, when i really moves from C o C. Gamma compensaes for his error by measuring he curvaure of he relaionship beween he opion price and he sock price or he rae a which dela changes. Figure 8: Hedging error inroduced by curvaure If he absolue value of gamma is small, he rae of change in dela is small and he dela error is small. If he absolue value of gamma is large, dela changes quickly and he dela error becomes large. 38

46 Gamma is greaes for a-he-money opions and falls o zero for deeply in-he-money or ou-of-he-money opions..5.3 Thea The sensiiviy of he opion price o he ime o expiry, T, is known as hea. I is he rae of change in he value of he porfolio wih respec o ime. Derived from he Black-Scholes equaion for a call opion, hea is given as where c T Sσ rt = N' ( d) rke N( d), T Θ c = d + ( σ ) ln S/ K r / T =, σ T and for a pu opion i is given by p T Sσ rt = N' ( d) + rke N( d). T Θ p = The formula calculaes he reducion in price of he opion for a decrease in ime o mauriy of one year. In pracice, hea is usually quoed as he reducion in price for a decrease in ime o mauriy of a single day. To calculae hea per day, divide he formula for hea by 365. Thea has a negaive sign for long opions and a posiive sign for shor opions. This is because as he ime o mauriy decreases, wih all else remaining he same, he ime value of he opion decreases. Thea measures his decrease in value and, since he opion value decreases a an increasing rae over he lifeime of a-he-money opions, hea is lowes jus before expiraion (Hull, 6). This is because a-he-money opions may become eiher in-he-money or ou-of-he-money on he las day. An ou- 39

47 of-he-money opion has some chance of becoming in-he-money before he las few days bu, if i is sill ou-of-he-money in hose las few days, has lile chance of any pay-off and i has lile ime value lef o decrease (Gemmill, 993). An in-he-money opion follows a similar paern as he ou-of-he-money opion, bu if i is in-hemoney a expiraion i will lose a posiive amoun of ime premium..5.4 Vega The sensiiviy of he opion price o volailiy is called vega. I measures he change in opion premium for a % change in volailiy. For a European call or pu opion on non-dividend paying sock, vega is given by ν f = σ = S T N' >. ( d ) Vega has a posiive sign for long opions and a negaive sign for shor opions. This means ha if you are an opion buyer, vega works for you, bu if you are an opion seller, vega works agains you. If he absolue value of vega is high, he derivaive s value is very sensiive o small changes in volailiy. Similarly, if he absolue value of vega is low, changes in volailiy have lile impac on he value of he derivaive. Vega is greaes for a-he-money opions and decreases o zero for exreme in-hemoney or ou-of-he-money opions Rho The sensiiviy of call prices o ineres raes is measured by rho. I is he rae of change of he value of a derivaive wih respec o he ineres rae. For a European call opion on non-dividend paying sock, i is given by 4

48 f ρ= r = KTe >, rt N d and for a European pu opion on non-dividend paying sock Therefore f ρ= r = <. rt KTe N d ρ long call >, ρ shor call <, ρ long pu <, ρ >. shor pu 4

49 3. Volailiy-dependen derivaives As Clewlow and Srickland (997) explain, he value of volailiy dependen derivaives depends in an imporan way on he level of fuure volailiy. Of course, he value of all opions is dependen on volailiy, bu hese opions are special in ha heir value is paricularly sensiive o volailiy over a period which begins no immediaely, bu in he fuure. As such hey are viewed, in some sense, as forwards or opions on fuure volailiy. These opions are paricularly useful when here is some even which occurs in he shor erm which will hen poenially affec oucomes furher in he fuure. They are herefore ofen used as a kind of pre-hedge o lock ino he curren levels of pricing unil more informaion is known a a laer dae. For boh compound and chooser opions he opion is firs defined, before an overview of heir applicabiliy and use is given and compared o sandard opions. The opion valuaions are hen derived in deail. A discussion follows on noable aspecs of boh opions. For compound opions he arbirage bounds on valuaion of he opions are given. These are he limis wihin which he price of an opion should say, since ouside hese bounds a risk-free arbirage would be possible. They allow an invesor o consrain an opion price o a limied range, and do no require any assumpions abou wheher he asse price is normally or oherwise disribued. Lasly, he sensiiviies or Greeks of he compound opions only are given. Simple chooser opions decompose exacly ino a porfolio of a call opion and a pu opion and heir Greeks can be calculaed from his porfolio. Each Greek leer measures a differen dimension of he risk in an opion posiion, and he aim of a rader is o manage he Greeks so ha all risks are accepable. 4

50 3. Ordinary Compound Opions 3.. Definiion A compound opion is a sandard European opion on an underlying European opion. From his definiion here are four basic compound opions: A call on a call, a call on a pu, a pu on a call, and a pu on a pu. If he compound opion is exercised, he holder receives a sandard European opion in exchange for he srike price; oherwise, nohing. 3.. Common Uses This ype of opion usually exiss for currency or fixed-income markes, where an uncerainy exiss regarding he opion's risk proecion capabiliies. Compound opions are also used when here is uncerainy abou he need for hedging in a cerain period. When valuing a compound opion here are wo possible opion premiums. The firs premium is paid up fron for he compound opion. The second premium is paid for he underlying opion in he even ha he compound opion is exercised. Generally, he premium for he compound opion is modes. Compound opions are also useful in siuaions where here is a degree of uncerainy over wheher he underlying opion will be needed a all. The small up-fron premium can be viewed as insurance agains he underlying opion no being required and, since i is a known cos, i can be budgeed for (Clewlow and Srickland, 997). Therefore, he advanages of compound opions are ha hey allow for large leverage and are cheaper han sandard opions. However, if he compound opion is exercised, he combined premiums will exceed wha would have been he premium for purchasing he underlying opion ourigh a he sar. ( 43

51 Consider he following wo examples from A major conracing company is endering for he conrac o build wo hoels in one monhs ime. If hey win his conrac hey would need financing for R3.5 million for 3 years. The calculaion used in he ender uilizes oday's ineres raes. The company herefore has exposure o an ineres rae rise over he nex monh. They could buy a 3yr ineres rae cap saring in one monh bu his would prove o be very expensive if hey los he ender. The alernaive is o buy a one monh call opion on a 3yr ineres cap. If hey win he ender, hey can exercise he opion and ener ino he ineres rae cap a he predeermined premium. If hey lose he ender hey can le he opion lapse. The advanage is ha he premium will be significanly lower. Compound Opions can also be used o ake speculaive posiions. If an invesor is bullish on R/USD exchange rae, hey can buy a 6 monh call opion a say 7. for 4.%. Alernaively, hey could purchase a monh call on a 4 monh R/USD 7. call a.5%. This will cos say.% upfron. If afer monhs he R/USD is a 7.5, he compound call can be exercised and he invesor can pay.5% for he 4 monh 7. call. The oal cos has been 4.5%. If he R/USD falls, he opion can lapse and he oal loss o he invesor is only.% insead of 4.% if hey had purchased he sraigh call Valuaion Closed form soluions for compound opions in a Black-Scholes framework can be found in he lieraure (cf. Geske, 979). For hese soluions he following assumpions are made:. Securiy markes are perfec and compeiive.. Unresriced shor sales of all asses are allowed wih full use of proceeds. 3. The risk-free rae of ineres is known and consan over ime. 4. Trading akes place coninuously in ime. 5. Changes in he value of he underlying opion follow a random walk in coninuous ime. 6. The variance rae is proporional o he square of he value of he underlying opion. (Geske, 979) 44

52 As wih he valuaion of sandard European opions, he principle of risk-neural valuaion is used. The discouning of he expeced payoff of he opion a expiraion by he risk-free ineres rae is hus allowed. Also, in a risk-neural world he underlying asse price has an expeced reurn equal o he risk-free ineres rae minus any payous. Firs, K and T are defined as he srike price and mauriy of he compound opion. The underlying opion ( c SΤ, K, T ) has a srike price K and mauriy dae T > T. Compound opions, herefore, have wo srike prices and wo exercise daes. S T is he value of he underlying asse a ime T. (.) PVΤ indicaes he presen value afer ime T of he quaniy in brackes. Two binary variables, φ and Ψ, which are defined below, are used in he derivaives: + if he underlying opion is a call, if he underlying opion is a pu, + if he compound opion is a call, if he compound opion is a pu. φ= Ψ= The combined payoff funcion for a compound opion is hen given by: Τ max Τ ( Τ ) max, ΨPV,φS φk ΨK = max, Ψc S, K, T, φ ΨK. (3..) Consider a call on a call. On he firs exercise dae, T, he holder of he compound opion is eniled o pay he srike price, K, and receive a call opion. The call opion gives he holder he righ o buy he underlying asse for he second srike price, K, on he second exercise dae, T. The compound opion will be exercised on he firs exercise dae only if he value of he opion on ha dae is greaer he firs srike price. From (3..) he payoff of a call on a call a ime T is: 45

53 Τ max Τ ( Τ ) max, PV, S K K = max, c S, K, T K. This is he maximum of he value of he payoff of he underlying opion, discouned o he ime of expiraion of he compound opion, T, and he srike price of he compound opion. Furher define S as he value of he underlying opion s underlying wih a volailiy σ. The coninuously compounded dividend yield of he underlying asse is q and r is he coninuously compounded risk-free ineres rae. The payoffs of he four basic European compound opions are given in Fig. 9 for S =, K = 3, K = 9, r = 5%, q = 3%, T =, T = and σ = %. T Payoff of a call on a call Payoff of a pu on a call 4 4 Payoff Payoff S S 5 4 Payoff of a call on a pu 3.5 Payoff of a pu on a pu Payoff 3 Price S S Figure 9: Payoff diagrams for compound opions. Parameers: S =, K = 3, K = 9, r = 5%, q = 3%, T =, T =, σ = % T The resuls in Lemma, considered below, are necessary for he derivaion of he value of compound opions in Theorem. Lemma : (Wes, 7) 46

54 Define he following for he normal disribuion: n x e π x f ( x) = e σ π N h = h = n x dx., x μ σ, Then: a A B n( z) N( A+ Bz) dz = N a,, + B + B, (3..) a A Az AB + C B e N( C+ Bz) n( z) dz = e N a A, ; + B + B, (3..3) where which is defined by ( N a, b; ρ ) is he cumulaive bivariae sandard normal disribuion funcion a b u uv+ v N ( a, b; ρ ) = exp dudv, π ρ ( ρ ) where ρ is he correlaion coefficien beween he wo bivariae sandard normal random variables. Theorem : (Geske, 979 and Rubinsein, 99) Assume ha invesors are unsaiaed, ha securiy markes are perfec and compeiive, ha unresriced shor sales of all asses are allowed wih full use of proceeds, ha he risk-free rae of ineres is known and consan over ime, ha rading akes place coninuously in ime, ha changes in he value of he underlying opion follow a random walk in coninuous ime wih a variance rae proporional o he square roo of he value of he underlying opion, and ha invesors agree on his variance σ. Then he curren value a ime of a compound opion is given by 47

55 ( + ) qt ( ) C S, K, K, T, T, σ, r, q, φ, Ψ =φψse N φψ X σ Τ, φd ; Ψρ T ( ) rt ( ) rt Ke N X (, ; ) ( φψ ) φψke N φψx φd Ψρ Ψ (3..4) where d + S σ ln + r q+ T K = σ T S σ ln + r q ( T ) K d d+ T, 3..5 = = σ, 3..6 σ T and ρ= ( T ) ( T ) X is he unique sandardised log-reurn, saisfying where σ r q ( T ) +σ T X H( X) = c Se, K, T, σ, r, q, φ K ( ) τ ( ) τ =φ Se e N( φd+ ) K e N( φd ) K = σ r q ( T ) +σ T X qt T rt T and Proof: d τ ± S T σ ln + r q± τ K = σ τ τ = T T. (3..8) 48

56 To uniquely deermine a price for an opion we need a model for he evaluaion of he underlying S. In he heorem, he Black-Scholes model wih consan volailiy is assumed, where S follows a geomeric Brownian moion. The process for S is given by, ds = r q S d + σ S dw S >, (3..9) where { } W denoes a sandard Wiener process, r he coninuously compounded risk-free ineres rae, q he coninuously compounded dividend yield and σ he volailiy. The formula for he value a ime T of he underlying opion is derived by Wysup (999) and is given by a generalizaion of he Black-Scholes formula as: qt ( T) ( ) (,,,,,, ) ( τ rt T τ Τ Τ + ) c S K T σ r q φ =φ S e N φd K e N φ d, (3..) d τ ± S T σ ln + r q± τ K =, σ τ τ = T T, where φ = for a call and φ = - for a pu. The price S T, a he curren ime is a random variable. Using risk neural valuaion he curren value (a ime ) of he compound opion C is he discouned expecaion of he payoff: rt ( ) E ( Τ ) C ST, K, K, T, T, σ, r, q, φ, Ψ = e T max, Ψc S, K, T, σ, r, q, φ ΨK. Since S = T ST, he price of he underlying asse, is lognormal, he log reurn u ln S is normally disribued. The probabiliy densiy funcion of u follows as 49

57 ( T ) v u T μ σ f ( u) = e wih v= and μ= r q. σ π σ T Hence C can be wrien as he inegral of he payoff over he probabiliy densiy of S T a ime : rt ( ) ( T ) ( Τ ) C S, K, K, T, T, σ, r, q, φ, Ψ = e max, Ψc S, K, T, σ, r, q, φ ΨK f u du. - (3..) Transforming u μ( T ) y = (3..) σ T leads o dy = du. (3..3) σ T Therefore du =σ T dy. (3..4) Noe ha he ransformaion in (3..) implies S log S y = T σ r q T σ T, where y is in fac he sandardised log-reurn. Hence S σ T log = r q ( T ) +σ T y, S so ha he price of he underlying a ime T is given by S T σ r q ( T ) +σ T y = Se. (3..5) 5

58 Using (3..3), (3..4) and (3..5), equaion (3..) can be wrien in erms of he sandardised log-reurn: σ r q ( T ) +σ T y rt ( ) C = e max c Se, K, T,, r, q, K, Ψ σ φ n( y) dy. - (3..6) Noaion is simplified by seing ( ) r q σ T +σ T H y c Se y =, K, T, σ, r, q, φ K. Where here is no confusion, he value of a compound opion ( ) C ST, K, K, T, T, σ, r, q, φ, Ψ will be simplified o C. Before valuing C i is firs wrien in he form rt ( ) C = e max ΨH( y), n( y) dy e ( ) rt = Ψ { y: ΨH( y) } H y n y dy. To evaluae he inegral i is noed ha he payoff is only posiive when Ψc( S, K, T, σ, r, q, φ ) > ΨK, Τ since he underlying opion price is monoonic in he * asse price. The variable S is he asse price a ime T, for which he opion S T price a ime T equal K. If he acual asse price is more han opion will be exercised; if i is less han obain he value * S * S, he following equaion is solved * S a ime T, he firs, he opion expires worhlessly. To * * qt ( T) ( ) ( τ rt T τ + ) ( ) c S, K, T, σ, r, q, φ Κ =φ S e N φg K e N φg K =, where 5

59 g τ ± * ln S σ + r q± τ K =. σ τ Alernaively, he unique sandardised log-reurn X is found which solves he following equaion : ( ) r q σ T +σ T H X c S X = T e, K, T, σ, r, q, φ K =. This can be solved using he Newon-Raphson procedure. Also, because he funcion H(y) is sricly increasing if φ = and sricly decreasing if φ = -, he value of he compound opion can be wrien as ( ) ( ) ( ) { y: ΨH( y) } { y: ΨH( y) H( X) } { y: φψy φψx} rt C = e ΨH y n y dy rt = e Ψ rt = e Ψ H y n y dy H y n y dy, simply by checking he four possible cases for he pair ( φ,ψ ). Now, by subsiuing z = φψy, he following is obained for φψ =: and for φψ =, rt ( ) C = e Ψ H( φψz) n( z) d φψx φψx rt ( ) dz, = e ΨH φψz n z z φψx rt ( ). C = e ΨH φψz n z dz 5

60 By subsiuing z = φψyino (3..6) and using he resuls in (3..) and (3..), he value of he compound opion follows as: y=+ σ r q ( T ) +σ T y rt ( ) C = e c Se, K, T,, r, q, K Ψ σ φ n( y) dy y=- z= φψx σ r q ( T ) σ T φψz rt ( ) = e c Se, K, T,, r, q, K Ψ σ φ n( z) dz z= ( ) rt = e z= φψx z= ST σ σ ( ) ln r q T T r q T σ T φψz + + qt ( T) K φse e N φ σ T T Ψ Ψ K n( z) dz ST σ ln + r q ( T T) rt ( T) K Ke N φ σ T T +. To evaluae he inegral, i is broken down ino hree componens corresponding o he hree payoff variables S, K and K inside he square brackes. Then (3..5) is subsiued in and he familiar forms of (3..5), (3..6) and (3..7) are recognized. σ rt + r q T qt T [] φψ Se z= φψx ln T φψ z+ r q ( T ) + r q+ ( T T) σ T φψz K e N n z z= σ rt + r q T qt T =φψ Se z= φψx z= e σ T φψz φ N Ψz S ( ) (, + ; ) ( ) =φψ φψ σ Τ φ Ψρ qt Se N X d σ σ σ T T S σ σ ln + r q T + r q+ T T Τ K +φ T T σ T T n dz z dz 53

61 For par [] he ideniy in (3..3) is applied in he final sep wih = z [ ] = φψx [] φ [] x in 3..3 in, a in 3..3 in, A in 3..3 = σ Τ Ψ in, ( Τ ) ( T T ) [] B in 3..3 = Ψ in, and C S σ σ ln + r q ( T ) + r q+ ( T T) K in 3..3 in. =φ σ T T [] Then ( φ ) ( X ), a A= φψx σ Τ Ψ = φψ σ Τ S σ σ ln Τ K σ σ ln r q ( T ) r q ( T T ) + r q T + r q+ T T ( ) ( σ Τ φψ) Ψ + φ AB + C ( T T) σ T T = + B ( Τ ) + Ψ ( T T) S σ Τ +φ +φ +φ + K = σ T =φd +, ( Τ ) ( T T) ( ) ( T T ) ( Τ ) ( T T ) Ψ B = + B ( Τ ) + Ψ ( T T) =Ψ Τ =Ψρ. 54

62 [] rt ( ) φψ Ke z= φψx ln T φψ z+ r q ( T ) + r q+ ( T T) σ T φψz K e N φ n z z= ( ) rt = φψ Ke z= φψx z= ( ) S S σ σ ln + r q T + r q+ T T K N Ψ z ( ) σ σ T T Τ + φ T T σ T T = φψke N φψx φd Ψρ rt, ; σ n( z) dz dz For par [] he ideniy in (3..) is applied along he same lines as for [] in he final sep wih a = φψx, A S σ σ ln + r q ( T ) + r q+ ( T T) K =φ, B = Ψ ( Τ ) ( T T ). σ T T ( ) [3] ΨKe rt N φψ X, where N ( a, b ; ρ) is he bivariae sandard normal disribuion funcion, which is defined in (3..3). Sandard resuls for he inegral of he produc of a normal densiy and a cumulaive normal disribuion was used for he firs wo inegrals as given in Lemma. Puing hese ogeher, he curren value of he compound opion is 55

63 where (, + ; ) qt ( ) C =φψse N φψ X σ Τ φd Ψρ ( ) rt ( ) rt Ke N X (, ; ) ( φψ ), φψke N φψx φd Ψρ Ψ d d + S σ ln + r q+ T K = σ T S σ ln + r q ( T ) K = = d+ σ T, σ T, and ρ= ( T ) ( T ). The formula for he compound opion involves he bivariae cumulaive normal disribuion. This comes from he fac ha he opion price depends on he join disribuion of he asse price a he mauriy daes of he compound and underlying opions (Clewlow and Srickland,997). Hence, using + if he underlying opion is a call, if he underlying opion is a pu, + if he compound opion is a call, if he compound opion is a pu. φ= Ψ= leads o he following: The value of a European call on a call is qt rt rt + ( X) C = Se N σ Τ X, d ; ρ K e N X, d ; ρ K e N. call on call The value of a European pu on a call is qt rt rt + ( X) C = Se N X σ Τ, d ; ρ + K e N X, d ; ρ + K e N. pu on call 56

64 The value of a European call on a pu is qt rt rt + ( X) C = Se N X σ Τ, d ; ρ + K e N X, d ; ρ K e N. call on pu The value of a European pu on a pu is qt rt rt + ( X) C = Se N σ Τ X, d ; ρ K e N X, d ; ρ + K e N. pu on pu 3..4 The Sensiiviy of Compound Opions o Volailiy Compound opion values are exremely sensiive o he volailiy of volailiy ( This follows from he fac ha he price of he underlying opion is firsly deermined using he implied fuure raes and volailiies. Then his opion value is used as he underlying for he compound opion. As wih sandard opions, he volailiy of he underlying will be a key facor of he value. However, wih compound opions, i is more significan as i has a double effec. If volailiy rises, his raises he value of he opion. Wih a compound opion, an increase in volailiy will also increase he value of he underlying asse (anoher opion) ( The analyic formulas derived above incorporae he Black-Scholes assumpion of consan volailiy, so hey end o undervalue he opions significanly. Research ino pricing mehodologies in his regard is ongoing ( In Fig., below S =, S = 9, K = 9, K = 6, T =, T =, =.5, r = 5% T and q = 3%. As he volailiy increases from. o.55, he value of he sandard call remains unchanged a R For a similar change in volailiy, he value of he compound call on call opion changes wih R.8 or 584.6%, increasing from R.39 o R.47. The change in he compound pu on call is R.8 or 3.9%, increasing from R7.7 o R7.78. The higher he value of he underlying sandard call opion (deeper in-he-money), he greaer he sensiiviy of he compound opions o changes in he volailiy, compared o he sandard opion. In Fig. where K = 9, K =, he underlying opion is ou-of-he-money and he sandard opion is more sensiive o changes in volailiy han he compound opions. The oal increase in he value of he sandard call opion is R 9.3 or 669.5%, from R.95 o R.69. The oal increase in he value of he compound call on call is R.57 from R o R

65 The value of he compound pu on call decreases wih R7.99 or.9%, from R85.98 o R Price Volailiy Call Call on call Pu on call Figure : The prices of a sandard call and compound opions on a call as a funcion of volailiy where if K = 9, K = 6. Price Volailiy Call Call on call Pu on call Figure : The prices of a sandard call and compound opions on a call as a funcion of volailiy where K = 6, K =. Fig. shows ha he paern does no hold for underlying sandard pu opions and he compound opions on a pu. 58

66 Price Volailiy Pu on Pu Pu Call on pu Figure : The prices of a sandard pu and compound opions on a pu as a funcion of volailiy where K = 6, K =. We noe in Fig. 3 ha for compound calls, he higher he second opional paymen defined as he srike price of he compound call opion K, he lower he iniial compulsory paymen. This means ha as K increases, he value of a call on a pu and a call on a call will decrease. The opposie holds for compound pus: he higher he value of K, he higher he price of boh a pu on a call and a pu on a pu. Price Call on call Pu on call Call on pu Pu on pu Call Pu K Figure : The prices of compound opions as a funcion of he srike price. Parameers: S =45, ST =, K = 3, K = 5, r = %, q = %, T =, T =, σ = %. 59

67 3..5 Arbirage Bounds on Valuaion Shilling () derives upper and lower bounds for he value of he compound opions in erms of he underlying sandard opions. They are based on he assumpion ha he marke is free of arbirage opporuniies. Define ( c S,, T, K, r, q, φ): he value of a sandard European opion, ( c c S,, T, K, r, q) : he value of a sandard European call opion, ( p c S,, T, K, r, q) : he value of a sandard European pu opion, ( c C S,, T, T, K, K, r, q, φ): he value of a European compound call opion, and ( p C S,, T, T, K, K, r, q, φ ): he value of a European compound pu opion. Theorem (Pu-call pariy for compound opions): Shilling () c p Given a compound call and a compound pu C wih he same srike K and he C same mauriy T on he same underlying opion (,,,,,, ) relaionship holds for [, ] T : c S T K r q φ he following c (,,,,,,,, φ ) + p ( C S T T K K r q K e rt ( ) = C S,, T, T, K, K, r, q, φ + c S,, T, K, r, q, φ). (3..7) Proof: We derive his relaionship by consrucing wo porfolios. If hey pay he same amoun under all condiions a mauriy, and canno be exercised before he expiraion dae, hen hey mus cos he same oday. c Porfolio A: Buy one European compound call opion a a price of. A he same ime deposi enough money o give he srike price a he ime of expiraion of he call opion. This is he cash amoun equal o rt ( ) Ke. C 6

68 p Porfolio B: Buy one European compound pu opion a a price of C. Buy one share of he underlying sandard European opion a is curren price c. The value of he sraegies a mauriy of he opion, ime T, if (,,,,,, ) c SΤ K T σ r q φ > K : Porfolio A: The compound call opion is exercised and he porfolio is worh ( Τ,,,,,, ) ) (,,,,,, ) Τ T ( c S K T σ r q φ K + K = c S K T σ r q φ = c. Porfolio B: The compound pu opion expires worhlessly and he porfolio is worh c T. Thus in his case Porfolio A = Porfolio B a ime T. The value of he sraegies a mauriy of he opion if (,,,,,, ) c S K T r q K Τ σ φ < : Porfolio A: The compound call opion expires worhlessly and he porfolio is worh K. Porfolio B: The compound pu opion is exercised and he porfolio is worh ( Τ,,,,,, ) ( Τ,,,,,, ) c S K T σ r q φ c S K T σ r q φ + K = K. Thus Porfolio A = Porfolio B a ime T in his case also. Therefore Porfolio A = Porfolio B a he exercise dae T in boh cases. This means c SΤ K T σ r q φ > Kor ha he resul holds independenly of wheher (,,,,,, ) ( Τ,,, σ,,, φ ) < c S K T r q K, hence independenly of he value of. Since he values are he same a ime T, hey mus also be equal a ime. I follows ha S T c (,,,,,,,, φ ) + p ( C S T T K K r q K e rt ( ) = C S,, T, T, K, K, r, q, φ + c S,, T, K, r, q, φ). To derive no-arbirage bounds on he value of compound opions, he following lemma (see Meron,973 ) is useful. 6

69 Lemma : Shilling () Suppose here are wo sandard opions c (,,,,,, ) (,,,,,, ) S T K r q φ and c S T K r q φ. Le < K < K. Then he following inequaliy holds [ ] for, T : rt ( ) <φ,,,, φ,,,, φ < c S T K c S T K e K K. (4..8) Proof: The lef inequaliy is obviously rue: If φ =: The lower he srike, he more valuable a call opion becomes since he payoff from a call is given by ( S K) max,. Therefore c (,,, ) (,,, ) c S T K > c S T K. c If φ= -: The higher he srike, he more valuable a pu opion becomes since he payoff from a pu is given by max, ( K S ). Therefore p (,,, ) (,,, ) c S T K < c S T K. p whereas for pu opions he conrary is rue. For φ = +, he righ inequaliy can be shown by comparing he payoff profiles of c c rt ( ) c S,, T, K and e ( K K ) on he one hand and c ( S,, T, K ) on he oher c c hand. In Fig. 3 he payoff profiles of (,,, ) ( ( e rt ( ) K ) is shown. K c S T K c S,, T, K ) and 6

70 Values S Difference beween call opions RHS Figure 3: Comparing he payoff profiles of he righ inequaliy for φ = + Parameers: S =5, K = 6, K =, r = %, q = %, T =, σ = %, =.5. The proof for φ = - is analogous: In Fig. 4 he payoff profiles of p p c ( S,, T, K ) ( S,, T, K ) and rt ( ) c e ( K K ) are shown. Values S Diffenrence beween pu values RHS Figure 4: Comparing he payoff profiles of he righ inequaliy for φ = - Parameers: S =5, K = 6, K =, r = %, q = %, T =, σ = %, =.5. Like sandard opions, compound opions canno be more valuable han heir underlying (in he case of compound calls) or heir srike (in he case of compound rt ( ) pus). Hence c S,, T, K, φ) is a upper bound for compound calls and e K is a ( upper bound for a compound pu. I is possible o improve hese rivial bounds. 63

71 Theorem 3 (Upper bound on he value of compound opions): Shilling () Suppose here is a compound opion (,,,,,,,, ) underlying c( S,, T, K, r, q, ) C S T T K K r q φ, Ψ wih he φ. Then he following inequaliies hold for [, T ], c rt ( T) C S,, T, T, K, K, φ < c S,, T, K + φk e, φ 3..9 ( S T K ) ( < c,,,, φ, 3..) rt ( T) ( C S, T,, T, K, K, φ < c S, T,, K + φke, φ c S, T,, K, φ 3.. p ( ) rt < Ke. ) ( 3.. ) Proof: Proof of (3..9) and (3..). I is shown ha he following relaionship holds a T : rt ( T) max, c ST, T, T, K, φ K < c S,, T, K +φke, φ 3..3 < c ( S,, T, K, φ ). ( 3..4) Using Lemma leads o he inequaliy rt ( T) rt ( T) rt ( T) rt ( T) c( ST T ) T K c ST T T K Ke K < c S, T, T, K, φ c S, T, T, K +φk e, φ < e K +φk e K T T <,,,, φ,,, + φ, φ < 3..5 which implies (3..4). On he oher hand, inequaliy (3..5) can be ransformed o rt ( T) (, ),,, φ <,,, + φ, φ c S T T K K c S T T K K e T T As he value of a sandard opion is always posiive, (3..6) is equivalen o (3..3). Proof of (3..) and (3..). These inequaliies are derived by using in his order he pu-call pariy for compound opions (3..7), inequaliy (3..9), he pu-call pariy for sandard opions and Lemma. 64

72 C p ( S,, T, T, K, K, φ) c (,,,,,, ) (,,,, ) = C S T T K K φ c S T K φ + K e (,,,, ) (,,,, ) rt ( T) (,,,, ) (,,,, ) ( ) rt < c S T K +φk e φ c S T K φ + K e rt ( ) < Ke ( 3..7) rt T rt = c S T K +φk e φ c S T K φ Remark. Shilling () For T T he value of a compound opion converges owards he value of is upper no-arbirage bound. Proof: Remark implies ha he payoff of a compound opion C( S,, T, T, K, K, φ, Ψ) for T ( + φ ) c S,, T, K K, φ, using he following limi: T is equal o ha of a sandard opion T T ( ) ( ) T T lim C S, T, T, K, K, σ, r, q, φ,ψ = max max S K, K Ψ φ. (3..8) If he following ideniies hold (3..8) mus be rue. For compound calls (Ψ = +), T ( T ) max max φ S K, K = max φ S K K,, and for compound pus (Ψ = -), K φ( ST K ) ( = K+φ K ST K φ K ST ) max max, max, max,. These ideniies can be verified by examining all possible cases. Consider as an example Ψ = +, φ = +: Firsly, T T ( ST K ) max, S K if S > K φ = if ST > K. 65

73 Then ( ( ST K ) K) ST K Kif ST > K and ST K > K max max, Ψ φ = oherwise ( T ) = max φ S K K,. The oher combinaions of Ψ and φfollow analogously. As for sandard opions, he lower arbirage bound of compound opions is equal o is discouned inrinsic value. Theorem 4 (Lower arbirage bound on he value of compound opions): Shilling () Given a compound opion C( S,, T, T, K, K, r, q, φ,ψ) wih he underlying opion c S,, T, K, r, q, φ ), he following inequaliy holds for [, T ) (, rt ( ) Ψ ( c( S T K φ) e K ) < C S T T K K φ, Ψ. max,,,,,,,,,,, 3..9 Proof: In he case of a compound call (Ψ = +), c le porfolio A consis of a compound call (,,,,,, ) invesmen of e ( ) rt K in bonds, le porfolio B consis of a sandard opion (,,,, ) A ime T porfolio A is worh ( T φ) worh ( T c S, T, T, K, φ ) C S T T K K φ and an c S T K φ. max c S, T, T, K,, K, whereas porfolio B is. Using he no-arbirage argumens, porfolio A mus be more valuable han porfolio B a ime < T. This can be ransformed o yield ( ) c S,, T, K, φ e K < C S,, T, T, K, K, φ. rt c As he value of he compound opion is always posiive, he lower bound is given by (3..9). In he case of a compound pu (Ψ = -), p le porfolio C consis of a compound pu (,,,,,, ) sandard opion (,,,, ) c S T K φ ; C S T T K K φ and a 66

74 le porfolio D consis an invesmen of e ( ) rt K in bonds. A ime T porfolio C is worh ( T φ) max c S, T, T, K,, K, whereas porfolio D is worh K. Using he no-arbirage argumens, porfolio C mus be more valuable han porfolio D a ime < T. This can be ransformed o yield ( ) (,,,, ) (,,,,,, ) e K c S T K φ < C S T T K K φ. rt p As he value of he compound opion is always posiive, he lower bound is given by (3..9). The arbirage bounds on he four basic compounds opions are illusraed in Fig. 5 o Fig. 9 below. Arbirage Bounds for a Call on a Call 9 Value Call on call Trivial upper bound Tigh upper bound Lower bound S Figure 5: Arbirage bounds for a Call on a Call. Parameers : %, q = %, T =, T =, =.5, σ = %. ST =5, K = 3, K = 5, r = Arbirage Bounds for a Pu on a Call 4 3 Payoff Pu on call Trivial upper bound Lower bounds Tigh upper bound - -3 S Figure 6: Arbirage bounds for a Pu on a Call. Parameers: q = %, T =, T =, =.5, σ = %. S =5, K = 3, K = 5, r = %, T 67

75 Arbirage Bounds for a Pu on a Pu Price Pu on Pu Tivial upper bound Tigh upper bound Lower bound S Figure 7: Arbirage bounds for a Pu on a Pu. Parameers: q = %, T =, T =, =.5, σ = %. S =5, K = 3, K = 5, r = %, T Arbirage Bounds for a Call on a Pu Payoff Call on pu Tigh upper bound Trivial upper bound Lower bound S Figure 8: Arbirage bounds for a Call on a Pu. Parameers: q = %, T =, T =, =.5, σ = %. S =5, K = 3, K = 5, r = %, T Wih hese bounds i is easy o derive he well-known fac ha buying a compound call is always cheaper han immediaely buying he underlying opion, bu exercising he compound call (and hus having o pay addiional K a mauriy) is more expensive han immediaely buying he underlying opion. The analogous resul for compound pus: buying a compound pu is always cheaper han he immediae acquisiion of a bull or a bear spread, bu exercising he compound pu leads o higher coss. 68

76 3..6 Sensiiviies The Greeks in his secion were aken direcly from Wysup (999). Define τ = T, τ = T, τ= T T X τ + and g = f = φ τ n( d τ + ) τ τ d,. Dela C S ( + ) qτ =φψe N φψ X σ τ φd Ψρ, ; Gamma qτ C e = n X N d n d N σ τ φ + S σs τ τ τ Ψ ( ) ( + ) ( + ) ( φψγ ) Thea C (, + ; ) =φψ φψ σ τ φ Ψρ qτ rse N X d rτ rτ qke N X (, ; ) ( φψ ) φψqk e N φψ φd Ψρ Ψ σ σ τ φ + φψ τ Ψ ( ) ( + ) ( + ) qτ Se n X N d n d N e τ τ Vega C σ τ ( + ) ( + ) = τ σ τ φ +Ψ τ φψ qτ Se n X N d n d N e 69

77 Rho r C rτ (, ; ) ( = φψτ Ke N φψx φd Ψρ + Ψτ Ke N φψx rτ ) Analysis of he sensiiviies of a call on a call and a call on a pu show ha hey have similar sensiiviies o he underlying call and pu respecively as shown below in Fig. 9 o Fig.. Figure 9: Comparison of he dela and gamma profiles for a call on call and a sandard call. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T 7

78 Figure : Comparison of he vega and hea profiles for a call on call and a sandard call. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T 7

79 Figure : Comparison of he dela and gamma profiles for a call on pu and a sandard call. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T 7

80 Figure : Comparison of he hea and vega profiles for a call on pu and a sandard call. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T Pus on pus and pus on calls are more complex. Figures 3 o 8 show he price, dela, gamma, hea, vega and rho surfaces wih respec o he underlying asse of a pu on a call and a pu on a pu. 73

81 Figure 3: The price and dela profiles of a pu on a pu. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T Figure 4: The gamma and hea profiles of a pu on a pu. Parameers: =, r = 5%, q = 3%, T =, T =, =.5, σ = %. ST =, K = 6, K 74

82 Figure 5: The rho and vega profiles of a pu on a pu. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T The delas are quie differen o sandard call or pus in ha hey are peaked near he srike price. As a resul he gammas of hese opions can be posiive or negaive, depending on he level of he underlying asse relaive o he srike price. The mos criical aspec of hese opions is he speed a which heir vega changes. This confirms heir exreme sensiiviy o volailiy changes (Clewlow and Srickland, 997). Figure 6: The price and dela profiles of a pu on a call. Parameers:, r = 5%, q = 3%, T =, T =, =.5, σ = %. S =, K = 6, K = T 75

83 Figure 7: The gamma and hea profiles of a pu on a call. Parameers: S =, K = 6, K =, r = 5%, q = 3%, T =, T =, =.5, σ = %. T Figure 8: The vega and rho profiles of a pu on a call. Parameers:, r = 5%, q = 3%, T =, T =, =.5, σ = %. S =, K = 6, K = T 76

84 3.. Chooser Opions 3... Simple Choosers 3... Definiion A sandard chooser opion, also known as an as you like i opion, has he feaure ha, afer a specified period of ime, he holder can choose wheher he opion is a call or a pu (Hull, 6). If he srucure of a chooser opion is considered, one finds ha i is idenical o ha of consrucing a sraddle, or a posiion in a call and a pu simulaneously, wih he excepion ha chooser opions are comparaively cheaper ( 7) Common Uses Boh a sraddle and a chooser can be hough of as a way of speculaing on an exreme move in he marke. A chooser opion is herefore valid for cliens who expec srong volailiy in he underlying, bu who are uncerain abou he direcion. This makes i an ideal mechanism o ake posiions on volailiy, as seen in Fig. 9 below. A chooser is more appropriae han a sraddle when he invesor believes informaion will become available in he fuure which will indicae he direcion of he marke move. The advanages of a chooser lie in he flexibiliy of choosing wheher i is a pu or a call opion and in ha he invesor does no need o ake a direcional view. I will herefore always be more expensive han a single sandard pu or call. A chooser opion will be cheaper han a sraddle sraegy (buying a call and a pu a he same srike) since afer he chooser dae, he buyer has only one opion (my.dreamwiz.com, 7; Clewlow and Srickland,997) Simple Chooser 4 35 Opion Price Volailiy Figure 9: Simple chooser wih varying volailiy. Parameers: S =, K = 5,r = %, q = 5%, T =, T =, =.5. T 77

85 Consider he following example: A privae invesor who rades mainly on echnical daa is convinced ha a major movemen is abou o happen in he FTSE index. On he chars ha are available i is clear ha he FTSE index is currenly rading very close o a major suppor line a 3. The invesor believes ha he suppor level will no be broken and ha he FTSE will move up srongly. On he oher hand, a breach of he suppor level is seen as a major urn in marke senimen and will mos likely be followed by a sharp drop in he index. The invesor radiionally would ener ino a sraddle (bough call and bough pu). However, a poenially beer sraegy is o ener ino a monh chooser opion on a 5 monh FTSE opion wih a srike of 3. A he end of he monh, he invesor has he choice of a 5 monh 3 pu or a 5 monh 3 call (my.dreamwiz.com, 7) Valuaion A simple chooser opion is purchased in he presen, bu, afer a predeermined elapsed ime T in he fuure, i allows he purchaser o choose wheher he opion is a European sandard pu or call wih a predeermined srike price K and remaining ime o mauriy T -T (Rubinsein, 99). The payoff from a simple chooser opion a he choice dae is Choosersimple [ c p] = max,, where c and p denoe he respecive European call and pu values underlying he opion. For a simple chooser, he underlying opions are boh European wih he same srike price and mauriy dae. Suppose ha S is he asse price a ime T, K is he srike price, T is he mauriy of he opions, r is he risk-free ineres rae and q is he dividend yield. Using he pu-call pariy relaionship, we can re-wrie he payoff as simple () = max [, ] Chooser c p = max cc, + Ke Se rt T qt T qt T rt T qt T = c+ e max, Ke Se. 78

86 This shows ha he simple chooser opion is a package and will have he same payoff oday as he payoff from:. Buying a call wih underlying asse price S, srike price K and mauriy T.. Buying a pu wih underlying asse price ( T) rt Ke and mauriy T. ( T) qt Se, srike price We can herefore wrie he formula for a simple chooser as () ( ) ( τ τ ) ( ) + ( ) ( ) ( τ τ ) ( + ) = qt rt Choosersimple Se N d Ke N d + Ke N d Se N d rt qt, (3..) where τ = T, τ = T, d τ + S σ ln + r q + T K = σ T τ τ, d d+ = σ τ, d τ + S σ ln + r q + T K = σ T τ τ, d d+ = σ τ The sensiiviy of Chooser Opions o Varying Time and Srike Price Define he variables similarly o hose defined earlier, wih T T being he ime o mauriy and T being he ime o choice. Fig. 3 below shows how he ime o choice affecs he opion value. As he ime o choice decreases, he value of he opion also decreases. 79

87 Opion Value Time o choice (Years) Figure 3: Simple chooser wih varying ime o choice. Parameers: S =, K = 5,r = %, q = 5%, T =, T =, σ = %. T Fig. 3 shows ha chooser opions are generally quie expensive; by varying he srike price, i is shown ha even when he asse price is equal o he srike price, he value is sill high ( 7). Simple Chooser 5 Opion Value Srike Price Figure 3: Simple chooser wih varying srike price. Parameers: S =, =.5,r = %, q = 5%, T =, T =, σ = %. T Arbirage Bounds on Valuaion Consider he five simple chooser opions valued in Table below, using he Black- Scholes framework. All he opions have curren underlying asse price =, srike price K =, ime o expiraion T The opions only differ by he ime o choice. =, r = %, q = 5% and σ = 3%. S 8

88 Table : Comparison of Opion Prices Time o choice Call Pu Sraddle Chooser The choice dae is obviously he key parameer. If he choice dae is oday, T =, hen he value of he chooser is he same as he value of he call. For a simple chooser, if he choice dae is equal o he mauriy daes of he call and pu, T = T, hen he value of he chooser is he sum of he values of he call and pu, since he opion has become a sraddle (Clewlow and Srickland,997). These wo exreme cases place a minimum and maximum value on he value of he chooser. For all oher cases he value of he simple chooser always lies beween he value of a single pu or call opion and he value of a long sraddle posiion Complex Choosers 3... Definiion More complex choosers can be defined where he call and he pu do no have he same srike price and ime o mauriy. Because of his propery, a complex chooser canno be broken down in erms of vanilla opions. They are no packages and have feaures ha are somewha similar o compound opions (Hull 6) Valuaion The derivaion of he value of complex chooser opions are given by Clewlow and Srickland (997). Define he srike price of he chosen call (pu) as K c (K p ) and mauriy daes T c (T p ). The payoff for a complex chooser on he choice dae, T, can be wrien as ( T c c ) ( T p p ) max c S, K, T T, p S, K, T T. Using risk-neural valuaion, he curren value of a complex chooser opion is he discouned expecaion of is payoff: 8

89 () ( ) ( ) ( ) rt Choosercomplex = e Ε max c ST, K,,,,. c T c p ST K p T p Since S = T ST, he price of he underlying asse, is lognormal, he log reurn u ln S is normally disribued. Le f ( u) denoe he normal probabiliy densiy funcion as given in he proof of Theorem in secion Hence C can be wrien as he inegral of he payoff over he probabiliy densiy of S T a ime : () rt u u complex = c c p p Chooser e c Se K T p Se K T f u du. ( ) max (,, ), (,, ) To evaluae he inegral we noe ha since he underlying call and pu are monoonic funcions of he asse price, he inegraion space can be divided ino wo regions. In he lower region we inegrae over he pu price and in he upper region we inegrae over he call price. The regions are divided a ha value of he asse price which makes he call and pu prices equal. Therefore * S ln S rt u u Choosercomplex () = e p Se K p T p f u du + c Se Kc T c f u du * S ln S ( ) (,, ) (,, ), where * S is he soluion o ( *, ) ( * c, c, p, p) c S K T = p S K T. The value of S* can be solved ieraively, using he Newon-Raphson search mehod which saisfies he condiion: * qtc rtc + c S e N z K e N z + * qt ( p ) rt ( p ) S e N z K e N z = + p, 8

90 where * S σ ln + r q+ ( Tc T) Kc +, + z = z = z σ T T σ T T c * S σ ln + r q+ ( Tp T ) K p z+ =, z = z+ σ Tp T. σ T T p c The complex chooser can also be valued in erms of he sandardised log-reurn, similarly o he compound opion. Define S σ ln + r q+ ( T c ) Kc c+ = c = c+ σ c c p d, d d T, 3.. σ T S σ ln + r q+ ( T p ) K p dp+ =, dp = dp+ σ T p, 3..3 σ T and ρ= c ( T ) ( T ) c, ρ = p ( T ) ( T p ). (3..4) In his case, sar wih σ r q ( ) T +σ T y c Se, Kc, T c, σ, r, q, rt ( ) Choosercomplex () = e max n( y) dy σ - r q ( T ) +σ T y p Se, Kp, T p, σ, r, q. Again, he inegral is divided ino wo regions, 83

91 σ r q ( T ) +σ T y rt ( ) Choosercomplex () = e c Se, Kc, T c,, r, q σ n( y) dy + X X σ r q ( T ) +σ T y rt ( ) e p Se, Kp, T p,, r, q σ n( y) dy, - where X is he unique sandardised log-reurn which solves he following equaion: r q σ ( ) ( ) T T X,,,,, r q σ +σ c Se T +σ T X Kc T c σ r q= p Se, Kp, T, σ, r, q p. This can be solved using he Newon-Raphson procedure. Before valuing form (,,,,,, σ,, ) Chooser S K K T T T r q complex T c p c p i is firs wrien in he X σ r q ( T ) +σ T y rt ( ) Choosercomplex () = e c Se, Kc, T c,, r, q σ n( y) dy + X σ r q ( T ) +σ T y rt ( ) e p Se, Kp, T p,, r, q σ n( y) dy, - since he value of he underlying call is sricly increasing. Now he value of a complex chooser opion is deermined as follows. 84

92 X σ r q ( T ) +σ T y rt ( ) Choosercomplex () = e c Se, Kc, T c,, r, q σ n( y) dy - X σ r q ( T ) +σ T y rt ( ) + e p Se, Kp, T p,, r, q σ n( y) dy - ( ) rt = e X e Se Ke σ r q ( T ) T y σ qt ( c T ) e ( ) rt X ( ) ST σ ln + r q T T K N σ Tc T rtc T c ST σ ln + r q+ T T Kc N σ Tc T c c S T σ ln + rt ( p T) K r q ( Tp T) p Ke N σ Tp T S T σ σ ln + r q+ Tp T r q ( T ) σ T y qt ( ) p T K p Se e N σ Tp T n( y) dy+ n( y) dy To evaluae he inegral, i is broken down ino four componens, he firs wo corresponding o he underlying call and he las wo corresponding o he underlying pu. Then (3..5) is subsiued in and he familiar forms of (3..), (3..3) and (3..3) are recognized. 85

93 σ rt ( ) + r q ( T ) qtc ( T) [] Se e ln T y r q ( T ) r q X ( Tc T ) σ T y Kc e N n y dy S σ rt ( ) + r q ( T ) qtc X σ T y e N y ( T) = Se e ( Τ ) ( T T ) c S ln K + ( c ) (, + ; ) qt c c = Se N σ Τ Xσ Τ d ρ σ σ σ T T c σ σ r q T r q Tc T σ T T c n( y) dy The ideniy in (3..3) is applied in he final sep wih z = y, a = X, A= σ Τ, B = S ln K C = ( Τ ) ( T T ) c, σ σ c + r q T + r q+ T T σ T T c. 86

94 [] rt ( ) c Ke c ln T y r q ( T X ) r q ( Tc T ) σ T y Kc e N n y ( ) = Ke X rtc c N y S ( ) σ σ c σ σ r q T r q Tc T S ln + + Τ Kc + T T σ T T ( ) c c = Ke N X, d ; ρ rtc c c c σ T T n( y) dy dy The ideniy in (3..) is applied in he final sep wih z = y, a= X, A = ( Τ ) ( T T ) c, σ σ c S ln + r q T + r q+ T T Kc B = σ T T c. σ rt ( ) + r q ( T ) qt ( T) e [3] Se ln X + T y+ r q T + r q+ T T σ T y K p e N n y dy σ rt ( ) + r q ( T ) qt ( T) e = Se S X σ T y e N y ( Τ ) ( Tp T) S ln K ( p ) (, + ; ) qt = Se N X σ Τ d ρ p p σ σ σ T ( p ) T p σ σ + r q T + r q+ T T σ T T ( ) ( p ) p n( y) dy 87

95 The ideniy in (3..3) is applied in he final sep wih z = y, a= X, A=σ Τ, B = ( Τ ) ( Tp T) S ln K C = p, σ σ + r q T + r q+ T T σ T T ( ) ( p ) p. rt ( p ) [4] Ke p S σ σ ln ( ) X + T y+ r q T + r q+ ( Tp T) σ T y K p e N n σ Tp T rt ( p ) = Ke p S σ σ ln + r q T + r q+ T T X K p N Τ y ( Tp T) σ Tp T rt ( p ) = Ke N X, d ; ρ p p p ( ) ( p ) n( y) dy y dy The ideniy in (3..) is applied in he final sep wih z = y, a = X, S ln K A = B = p ( Τ ) ( Tp T) σ σ + r q T + r q+ T T σ T T. ( ) ( p ) p, 88

96 The value of he complex chooser opion can hen be evaluaed in a similar way o he compound opion o give where Chooser complex () = qtc rt c c+ c c c Se N σ Τ X, d ; ρ Ke N X, d ; ρ + rt ( p ) qtp Ke N X, d ; ρ Se N X σ Τ, d ; ρ, ( p+ p) p p p c d d S σ ln + r q+ ( T c ) Kc =, dc = dc+ σ T c, σ T c+ c S σ ln + r q+ ( T p ) K p =, dp = dp+ σ T p, σ T p+ p and ρ= c ( T ) ( T ) c, ρ = p ( T ) ( T p ) The Sensiiviies of Complex Chooser Opions o Some of is Parameers The formula for he valuaion of complex chooser opions is quie similar o he formula for valuing compound opions. Observe ha Chooser complex () = qtc rt c c+ c c c Se N σ Τ X, d ; ρ Ke N X, d ; ρ + rt ( p ) qtp Ke N X, d ; ρ Se N X σ Τ, d ; ρ ( p+ p) p p p c he firs wo erms of he complex chooser formula are he same as he firs wo erms of he formula for a call on a call, 89

97 ( ) C = Se N σ Τ X, d ; ρ K e N X, d ; ρ K e N X call on call qt rt rt + he 3 rd and 4 h erms of he complex chooser formula are he same as he firs wo erms of he formula for a call on a pu ( X) C = Se N X σ Τ, d ; ρ + K e N X, d ; ρ K e N. call on pu qt rt rt + The only difference is ha he criical underlying asse price S*, or is corresponding unique sandardised log-reurn, is se o he level a which he value of he sandard call will equal he value of he sandard pu afer elapsed ime T : ( *, ) ( * c, c, p, p) c S K T = p S K T (Rubinsein, 99) i.e. r q σ ( ) ( ) T T X,,,,, r q σ +σ c Se T +σ T X Kc T c σ r q= p Se, Kp, T, σ, r, q p. Wih varying srike price in Fig. 3, he complex chooser opion has he lowes price when he underlying call and pu opions have equal values. The price of he opion increases as he difference in value beween he wo underlying opions increase. Complex Chooser 7 6 Opion Price S Figure 3: Complex chooser wih varying srike price. Parameers: =.5, T =,T c =,T p =,,r = 5%, q = 3%, T =, T =, σ = % The closer he choice dae, T, is o he mauriy daes of he underlying opions, T c and T p, he higher he opion price. This is because he invesor has a lo of ime o accumulae informaion on evens no very far in he fuure. As he ime o he choice 9

98 dae (T -) decreases, he value of he opion also decreases. This is illusraed in Fig. 33 and 34. Complex Chooser 3 5 Opion Price T Figure 33: Complex chooser wih varying choice dae. Parameers: =.5, S = 85, T c =,T p =,,r = 5%, q = 3%, T =, T =, σ = %. Complex Chooser 3 5 Opion Price (Years) Figure 34: Complex chooser wih varying valuaion dae. Parameers: T =, S = 85, T c =,T p =,,r = 5%, q = 3%, T =, T =, σ = % American Chooser Opions Definiion Chooser opions can be American, in he sense ha he choice of a call and pu a he choice dae is an American opion raher han European in exercise ( 7). 9

99 3..3. Valuaion In he Black-Scholes world, allowing he invesor o choose a any ime, up o some dae, does no add any value o he opion. This can be seen by recognizing ha he value of he chooser is an increasing funcion of he ime o he choice dae, so i is opimal for he invesor o wai as long as possible. In he real world, however, being able o choose a any ime is valuable, in he sense ha i allows an immediae profi from a move in he marke a any ime. (Clewlow and Srickland, 997.) An American chooser is priced in a similar fashion o he valuaion given for European choosers, bu replaces he European payoff funcion wih an American one o find an approximae price ( 7). 3.3 Summary In his chaper wo volailiy dependen derivaives, compound and chooser opions, were discussed. They were firs defined, before an overview of heir applicabiliy and use was provided and compared o sandard opions. The opion valuaions were hen derived in deail in he Black-Scholes framework, using properies of he normal disribuion. The sensiiviy of compound opions o volailiy was illusraed and he arbirage bounds on is valuaion were given. These are he limis wihin which he price of an opion should say, since ouside hese bounds a risk-free arbirage would be possible. They allow an invesor o consrain an opion price o a limied range and do no require any assumpions abou wheher he asse price is normally, or oherwise, disribued. For simple chooser opions he sensiiviy o varying ime and srike price was illusraed. The sensiiviies or Greeks of he compound opions only were provided and illusraed. They were no provided for simple chooser opions, since hese decompose exacly ino a porfolio of a call and pu opion and heir Greeks can be calculaed from his porfolio. Each Greek leer measures a differen dimension o he risk in an opion posiion, and he aim of a rader is o manage he Greeks so ha all risks are accepable. In he case of complex chooser opions, here is no an exac decomposiion, bu i is shown ha here are similariies beween complex chooser opions and compound opions. 9

100 4. Pah-Dependen Derivaives Barrier opions and Asian opions are examples of pah-dependen opions. Pahdependen opions are opions wih payoffs which depend on he complee pah aken by he underlying price o reach is expiraion value (Clewlow and Srickland,997). Barrier opions have weak pah dependence. This is since he payoff a expiry depends boh on wheher he underlying hi a prescribed barrier value a some ime before expiry, and on he value of he underlying a expiry. Srongly pah-dependen conracs have a payoff ha depends on some propery of he asse pah in addiion o he value of he underlying asse a he presen momen in ime. Asian opions are srongly pah-dependen since heir payoff depends on he average value of he underlying asse from incepion o expiry (Wilmo, 998). 4. Barrier Opions 4.. Definiion Basic barrier opions differ in hree ways:. Kind of opion: Call or pu.. Does he opion cancel or come ino exisence when he underlying price his or crosses a predeermined barrier? A knock-ou opion ceases o exis immediaely when he underlying asse price reaches a cerain barrier. A knock-in opion comes ino exisence only when he underlying asse price reaches a barrier. If he underlying price does no hi or cross he barrier, he opion does no come ino exisence and herefore expires worhlessly. In eiher case, if he opion expires inacively, hen here may be a cash rebae paid ou. This could be nohing, in which case he opion ends up worhless, or i could be some fracion of he premium. 93

101 3. Does he opion knock-in or -ou when he underlying price ends up above or below he barrier? A down-barrier opion knocks in or ou when he underlying price ends up below he barrier. An up-barrier opion knocks in or ou when he underlying price ends up above he barrier. The four basic ypes of barrier opions are herefore down-and-ou, up-and-ou, downand-in and up-and-in opions. 4.. Common Uses Barrier opions are aracive o purchasers seeking o pay he lowes possible Rand premium for an opion. The weak pah-dependency of barrier opions makes barrier opions less valuable and herefore less expensive han sandard opions. Their cheapness, relaive o sandard opions, is ofen why hey are used by invesors who believe an asse or index will move in a specific manner and who wish o speculae or hedge heir porfolios based on heir percepion of such poenial movemens. Alhough here is a greaer risk of loss, barrier opions are less expensive han sandard opions, bu provide similar poenial invesmen reurns. (Braddock, 997.) Consider he following examples of scenarios where insiuional invesors can use barrier opions: Knock-ou calls are used o capure upside sock price movemens under he assumpion ha he underlying asse price will no decline and remain below he barrier level. Knock-in pus are used o lock-in profis if upside price moves appear o have peaked. Knock-in call opions can be used as an inexpensive sraegy o paricipae in he poenial for volaile sock price movemens. Knock-in pus ac as insurance for bondholders fearing inflaion and lower bond prices. (Braddock, 997.) 94

102 4..3 Valuaion We follow he derivaion of barrier opions as given by Reiner and Rubinsein (99). Barrier opions are valued in a Black-Scholes environmen ha assumes:. The underlying asse follows a joinly lognormal random walk.. No arbirage opporuniies exis in he marke. From. above, risk-neural valuaion can be applied as follows: The riskless ineres rae is used as he discoun rae. The underlying asse price is expeced o appreciae a he same riskless rae. The expeced payoff of he opions a expiraion are discouned by seing he value of he opion a any period equal o he discouned expeced value of he opion one period laer, or is early exercise value, whichever is greaer. To find closed form soluions of barrier opions, he densiy of he naural logarihm of he risk-neural underlying asse reurn, u, is needed: f u = σ π ( T ) e v wih and u T v μ r q σ = μ=. σ T This is he normal densiy, where r is he risk-free rae of ineres, q is he dividend yield, σ is he volailiy of he underlying asse and T is he expiraion ime of he opion. This is he same densiy used in secion 3..3 for he valuaion of compound opions and in secion 3... for he valuaion of complex chooser opions. If he underlying asse price firs sars a S above he barrier H, he densiy of he naural logarihm of he underlying asse reurn, when he underlying asse price breaches he barrier, bu ends up above he barrier a expiraion, is no equal o he densiy given above, bu is given by 95

103 wih μα σ g( u) = e e σ π ( T ) ( T ) v u ηα ημ T H v = and α= log. σ S This is he normal densiy pre-muliplied by μα σ e wih η =. Alernaively, given ha he underlying asse price firs sars below he barrier, he densiy of he naural logarihm of he underlying asse reurn, when he underlying asse price breaches he barrier, bu ends up a expiraion below he barrier, is given by he same expression, bu wih η = -. In order o disinguish beween hese wo siuaions, define: + for he case when he underlying asse price sars above he barrier, η= for he case when he underlying asse price sars below he barrier. Also define some inermediae values prior o considering he six payoff or probabiliy expressions ha cover boh in- and ou-barrier opions: + for a barrier call opion, φ= for a barrier pu opion. r q+.5σ λ= σ μ= r q.5σ μ a =, b= σ μ + rσ σ ln S K w = +λσ σ T w ( T ) ln S H = +λσ σ T ( T ) 96

104 w 3 w w 4 5 ln H SK = +λσ σ T ( T ) ln H S = +λσ σ T ( T ) H ln S = + bσ σ T ( T ). The prices of he basic barrier opions are combinaions of he following six expressions denoed by [ i], i =,...,6. qt rt qt rt [] =φse N φw φke N φw φσ Τ [] =φse N φw φke N φw φσ Τ λ qt ( ) H rt ( 4) 4 λ H [3] =φse N ηw φke N ηw ησ Τ S S λ μ H qt ( ) H rt =φ Se N( w4) Ke N w4 η η ησ Τ σ S S S K λ λ H qt qt H H φ e N( w4) e n( w η φη 4) S S σ Τ λ qt ( ) H rt ( 3) 3 λ H [4] =φse N ηw Ke N ηw ησ Τ S S rt [5] = Re N ηw ησ Τ N ηw ησ Τ S λ ( ) H ( ) ( 4 ) [6] = R N η w + N ηw ηbσ a+ b a b H H ( 5) ( 5 T ) S S In heorem 5 below he prices of he basic barrier opions are informally derived as combinaions of expressions [] o [6]. 97

105 Theorem 5: The prices of he basic barrier opions are given by he following combinaions: Table : Prices of he basic barrier opions Opion ype φ η in/ou Reverse Combinaion Sandard up-and-in call K > H []+[5] Reverse up-and in call K H []-[4]+[3]+[5] Sandard down-and-in call K > H [4]+[5] Reverse down-and-in call K H []-[]+[3]+[5] Sandard up-and-ou call K > H [6] Reverse up-and ou call K H []-[]+[4]-[3]+[6] Sandard down-and-ou call K > H []-[4]+[6] Reverse down-and-ou call K H []-[3]+[6] Sandard up-and-in pu K > H []-[]+[3]+[5] Reverse up-and in pu K H [4]+[5] Sandard down-and-in pu K > H []-[4]+[3]+[5] Reverse down-and-in pu K H []+[5] Sandard up-and-ou pu K > H []-[3]+[6] Reverse up-and ou pu K H []-[4]+[6] Sandard down-and-ou pu K > H []-[]+[4]-[3]+[6] Reverse down-and-ou pu K H [6] Consider he up-and-in call opion given in he firs wo rows of Table. Here φ = + indicaes a barrier call opion, η = - indicaes ha he underlying asse price sars above he barrier, and he knock-in feaure is indicaed by a binary variable (in/ou) se o negaive one in he able for disincion from a similar barrier wih a knock-ou feaure. The up-and-in call has wo separae price combinaions. The price of he firs one, he sandard up-and-in call, i.e. when K > H, is given by expression [] plus expression [5]. This is considered o be he sandard par. The price of he second one, he reverse up-and-in call, i.e. when K H, is given by expression [] minus expression [4] plus expression [3] plus expression [5]. Informal Proof: European Knock-in Barrier Opions Down-and-in Call Alhough he paymen for a down-and-in call opion is made up fron, he call is no received unil he underlying asse price reaches a pre-specified barrier level H. If, afer elapsed ime T, he underlying asse price his he barrier, he invesor receives a sandard call wih srike K and ime o expiraion T-. If, hrough elapsed ime, he barrier is never hi, he rebae R a ha ime is received. Le S be he price of he underlying afer elapsed ime, and S T he price of he underlying asse a 98

106 expiraion; hen he payoff for a down-and-in call opion, i.e. where S > H, is given by: [ T ] max, S K if for some T, S H; R (a expiry) if for all T, S > H. There are alernaive ways ha hese various payoffs may be earned, paricularly when i is possible for he exercise price o be eiher above or below he barrier price. To price a down-and-in call, five price pahs and heir associaed payoffs are considered as given by Kolb (3):. K S T H; payoff is S T -K.. K H S T, and he barrier was ouched; payoff is S T -K. 3. K H S T, and he barrier was never ouched; payoff is R. 4. H K S T, and he barrier was ouched; payoff is S T -K. 5. H K S T, and he barrier was never ouched; payoff is R. For each possible payoff and price pah here are associaed probabiliies and expressions. Consider firs he sandard case where K > H, i.e. 4 and 5 above. The value of he opion in his case is he sum of wo erms. The firs is a call payoff corresponding o price pah 4 above. The probabiliy ha his realises is wrien as probabiliy (4) = P[S T K, S H for some T]. This means ha price pah 4 realises if and only if S T K and S H for some T. The second erm is a rebae and follows accordingly: probabiliy (5) = P[S T H] - P[S T H, S H for some T]. For price pahs 3 and 5 above he probabiliy of being realised is he same, since i is independen of he relaionship beween S and K. This follows since receiving he T rebae is independen of he relaionship beween S and K. I only depends on wheher he barrier has been breached or no, i.e. if S H for some T. T 99

107 Mahemaically, he presen value of he pay-off ha corresponds o price pah 4 follows as rt rt u E ST K ST K, S H e = e φ Se K g u du. η K ln S The value of his inegral is given by Reiner and Rubinsein (99) as λ qt ( ) H rt ( 3) 3, λ H ( Τ ) φse N ηw Ke N ηw ησ S S which is denoed as equaion [4] in Theorem 5. The presen value of he pay-off ha corresponds o price pah 5 follows as rt ( ) rt ( ) T ln E R S K H e = Re f u g u du. η H S The value of his inegral is given by Reiner and Rubinsein (99) as λ H ( η ησ Τ ) ( η ησ Τ ) rt Re N w N w4, S which is denoed in Theorem 5 as equaion [5]. Hence, he curren value of he down-and-in call given ha K > H is he presen value of he wo payoffs expressed as E ST K ST K, S H and E RS K H T DIC > = + η= φ=. [4] [5],{, K H } can be Consider he second case where K < H. In his case i is necessary o find erms corresponding o he probabiliies ha he firs hree price pahs realise, i.e.

108 and probabiliy () = P[H S T > K] = P[S T > K] - P[S T > H], probabiliy () = P[S T > H, S H], probabiliy (3) = probabiliy (5). Price pah is simplified by he fac ha since he underlying asse sar ou above he barrier, if he underlying price hen finishes below he barrier, i mus have breached he barrier a some ime. Therefore i is spli up ino wo erms, P[S T > K] and P[S T > H], and considered as wo separae price pahs. The presen values of hese wo payoffs, ogeher wih heir soluions, are given by Reiner and Rubinsein (99) as φ rt ( ) rt ( ) u E ST K ST > K e = e φ( Se K) f ( u) du K ln S = [] qt rt =φse N φw φke N φw φσ Τ and φ rt ( ) rt ( ) u E ST K ST > H e = e φ( Se K) f ( u) du H ln S qt ( ) rt = []. =φse N φw φke N φw φσ Τ The presen value of he payoff corresponding o price pah above, ogeher wih is soluions, is given by Reiner and Rubinsein (99) as

109 rt rt u E ST K ST > H, S H e = e φ Se K g u du η ln H S λ qt ( ) H rt ( 4) 4 λ H ( σ Τ ) = φse S N ηw φke S N ηw η = [3]. Using his, he presen value of he down-and-in call can be wrien as DIC < = [] [] + [3] + [5], η=, φ=. ( K H) { } Up-and-in Call This opion is idenical o a down-and-in call, excep ha he underlying asse price sars ou below, insead of above, he barrier. The payoff from an up-and-in call opion, i.e. where S < H, is given by [ T ] max, S K if for some T, S H; R (a expiry) if for all T, S < H. An up-and-in call can be priced using he five price pahs and heir associaed payoffs as given for a down-and-in call. For he K > H case here is again a payoff erm corresponding o he rebae and a payoff erm corresponding o he call payoff. The rebae payoff is received if or K S T H and he barrier was never ouched S T H K, and he barrier was never ouched, wih he probabiliy given by P[S T H] - P[S T H, S H]. The densiy corresponding o he P[S T H] is of course f(u). The densiy corresponding o P[S T H, S H] is idenical o g(u), bu wih η = -. Therefore, he rebae erm is represened by equaion [5] wih η = - in g(u).

110 The call payoff is received if H K S T, wih he probabiliy P[S T > K], which is given by equaion []. Therefore, UIC > = [] + [5], η=, φ=. ( K H) { } For he K < H case, here is a rebae payoff equal o he one derived for he case where H < K. The call payoff is received if S T K H, payoff is S T -K; H S T K, and he barrier was ouched; payoff is S T -K, wih associaed probabiliies given by P[S T H] and P[H > S T > K, S H] respecively. The second probabiliy can be resaed as P[H > S T > K, S H] = P[S T < H, S H]- P[S T > K, S H]. Then immediaely i can be wrien ha UIC < = [] + [3] [4] + [5], η=, φ=. ( K H) { } The remaining knock-in opions follow similarly: Down-and-in Pu A down-and-in pu is a pu opion ha ceases o exis when a barrier less han he price is reached (Hull, 3). The payoff of a down-and-in pu, i.e. S > H, is given by [ T ] max, K S if for some T, S H; R (a expiry) if for all T, S > H. When K > H: Receive pu payoff equal o ( K ST ) [ H] + P[ H < < K S H] T wih associaed probabiliy PS S,. T 3

111 When H > K: Receive pu payoff equal o ( K ST ) wih associaed probabiliy PS < K. [ ] When K > H or H > K: Receive he rebae payoff equal o R wih associaed probabiliy PS > H P S > H, S H. T [ ] [ ] T T Therefore, DIP DIP = + + { η= φ= K> H } = [] + [5], { η=, φ= }. K< H [] [3] [4] [5],,. Up-and-in Pu A down-and-in pu is a pu ha comes ino exisence only if he barrier, H, ha is greaer han he curren asse price is reached (Hull, 3). The payoff of an up-and-in pu, i.e. where S > H, is given by [ T ] max, K S if for some T, S H; R (a expiry) if for all T, S < H. When K > H: Receive pu payoff equal o K-S T wih associaed probabiliy P[ H ST < K] + P[ S T < H, S H ]. When H > K: Receive pu payoff equal o K-S T wih associaed probabiliy PS < KS, H. [ ] T When K > H or H > K: Receive he rebae payoff equal o R wih associaed probabiliy PS < H P S < H, S H. [ ] [ ] T T Therefore, UIP UIP = + + { η= φ= K> H } = [4] + [5],{ η=, φ= }. K< H [] [] [3] [5],,. For knock-ou barrier opions here is a sixh price payoff o consider, which will now be presened. 4

112 European Knock-ou Barrier Opions Down-and ou Call A down-and-ou call is one ype of knock-ou opion. I is a regular call opion ha ceases o exis if he asse price reaches a cerain barrier level. This barrier is below he iniial asse price (Hull, 6). The payoff from a down-and-ou call opion, i.e. where S > H, is given by [ T ] max, S K if for all T, S > H; R (a hi) if for some T, S H. If he rebae equals zero he following pariy relaionship makes i easy o wrie down he values of European knock-ou barrier opions: Payoff from sandard opion = Payoff from down-and-ou opion +Payoff from down-and-in opion To show ha his ideniy holds, suppose an invesor owns oherwise idenical downand-ou and down-and-in opions wih no rebaes. If he common barrier is never hi, he receives he payoff from a sandard opion. If he common barrier is hi, as he down-and-ou opion ceases o exis, he down-and-in opion delivers him a sandard opion idenical o he one he los when he down-and-ou opion was cancelled. Therefore, even in his case, he invesor ends up receiving he payoff from a sandard opion. If he rebae is no equal o zero, i is necessary o consider ha for knock-in opions, i is no possible o receive he rebae prior o expiraion, since one coninues o remain in doub abou wheher or no he barrier will never be hi. However, for a knock-ou opion, i is possible, as well as cusomary, for he rebae o be paid he momen he barrier is hi. This complicaes he risk-neural valuaion problem since he rebae may now be received a a random, raher han pre-specified, ime. Therefore, he densiy of he firs passage ime (τ) for he underlying asse price o hi he barrier is given by Reiner and Rubinsein (99) as 5

113 wih ηα v g u h( τ ) = = e τ στ πτ v α+μτ = σ τ. Here, η = if he barrier is being approached from above and η = - if he barrier is being approached from below. The presen value of he expeced rebae follows as he expeced rebae discouned by he ineres rae, raised o he power of he firs passage ime: T a+ b a b r H H R e h() d = R N( η w5) + N( ηw5 ηbσ T ) S S = [6]. Using his relaionship, i is now possible o wrie down he valuaion soluions for he down-and-ou call and he hree remaining knock-ou barrier opions: DOC = C DIC ( K> H) ( K> H) ( K< H) ( K< H) { } = [] [4] + [6], η=, φ=. DOC = C DOC { } = [] [3] + [6], η=, φ=. Here C indicaes he payoff from a sandard call priced in he Black-Scholes framework. Up-and-ou Call An up-and-ou call is a regular call ha ceases o exis if he asse price reaches a specified barrier level ha is higher han he curren asse price (Hull, 6). The payoff from an up-and-ou call opion, i.e. where S < H, is given by [ T ] max, S K if for all T, S H; R (a hi) if for some T, S H. 6

114 The valuaion soluion is UOC = C UIC ( K> H) ( K> H) { } = [6], η=, φ=, UOC = C UIC ( K< H) ( K< H) { } = [] [] + [4] [3] + [6], η=, φ=. The rebae provides he only conribuion o he value of an up-an-ou call when he srike price is greaer han he barrier. Since S < H < K, in order for he underlying asse price o end up above he srike price i mus firs breach he barrier, bu in his even, he call is exinguished. Down-and-ou Pu A down-and-ou pu is a pu opion ha ceases o exis when a barrier less han he curren asse price is reached (Hull, 6). The payoff from a down-and-ou pu opion, i.e. where S > H, is given by [ T ] max, K S if for all T, S > H; R (a hi) if for some T, S H. The valuaion soluion is DOP = P DIP ( K> H) ( K> H) ( K< H) ( K< H) { } { } = [] [] + [4] [3] + [6], η=, φ=. DOP = P DIP = [6], η=, φ=. Here P indicaes he payoff from a sandard pu opion priced in he Black-Scholes framework. Similarly o an up-and-ou call, he rebae provides he only conribuion o he value of a down-and-ou pu when he srike price is less han he barrier. 7

115 Up-and-ou Pu An up-and-ou pu is a pu opion ha ceases o exis when a barrier, H, ha is greaer han he curren asse price, is reached (Hull, 6). The payoff from an up-and-ou pu opion, i.e. where S < H, is given by [ T ] max, K S if for all T, S < H; R (a hi) if for some T, S H The valuaion soluion is UOP = P UIP ( K> H) ( K> H) ( K< H) ( K< H) { } = [] [3] + [6], η=, φ=. UOP = P UIP { } = [] [4] + [6], η=, φ=. In heorem 5, he prices of he basic barrier opions are informally derived as combinaions of expressions [] o [6], where equaions [] o [4] correspond o he payoff of he underlying opion and [5] and [6] correspond o he rebae. Tables 3 hrough 6 below presen he expressions for [] o [6] for down calls (φ=,η=), down pus ( φ=,η= ), up calls φ =,η= and up pus ( φ =,η = ). Table 7 shows he values of each possible barrier opion in erms of he expressions in Tables 3 hrough 6. 8

116 Table 3: Valuaion Expressions for Down Calls DC DC qt rt Se N w Ke N w σ Τ Se N w Ke N w σ Τ qt rt DC3 DC4 DC5 DC6 λ qt ( ) H rt ( 4) 4 λ H ( σ Τ ) Se N w Ke N w S S λ qt ( ) H rt ( 3) 3 λ H ( σ Τ ) Se N w Ke N w S S λ ( σ Τ ) ( 4 σ Τ ) Re N w N w S rt H a+ b H R N w N w b T S S a b H ( 5) + ( 5 σ ) Table 4: Valuaion Expressions for Down Pus DP rt ( ) qt ( ) Ke N w σ Τ Se N w DP rt ( ) qt ( ) Ke N w σ Τ Se N w DP3 DP4 DP5 DP6 λ λ rt ( ) H qt ( ) H ( 4 σ Τ ) 4 Ke N w Se N w S S λ λ rt ( ) H qt ( ) H ( 3 σ Τ ) 3 Ke N w Se N w S S λ H ( σ Τ ) ( 4 σ Τ ) S rt Re N w N w a+ b H R N w N w b T S S a b H ( 5) + ( 5 σ ) 9

117 Table 5: Valuaion Expressions for Up Calls UC UC qt rt Se N w Ke N w σ Τ Se N w Ke N w σ Τ qt rt UC3 UC4 UC5 UC6 λ qt ( ) H rt ( 4) 4 λ H ( σ Τ ) Se N w Ke N w + S S λ qt ( ) H rt ( 3) 3 λ H ( σ Τ ) Se N w Ke N w + S S λ H ( +σ Τ ) ( 4 +σ Τ ) S rt Re N w N w a+ b H R N w N w b T S S a b H ( 5) + ( 5 + σ ) Table 6: Valuaion Expressions for Up Pus UP rt ( ) qt ( ) Ke N w σ Τ Se N w UP rt ( ) qt ( ) Ke N w σ Τ Se N w UP3 UP4 UP5 UP6 λ λ rt ( ) H qt ( ) H ( 4 +σ Τ ) 4 Ke N w Se N w S S λ λ rt ( ) H qt ( ) H ( 3 +σ Τ ) 3 Ke N w Se N w S S λ H ( +σ Τ ) ( 4 +σ Τ ) S rt Re N w N w a+ b H R N w N w b T S S a b H ( 5) + ( 5 + σ )

118 Table 7: Valuaion of Barrier Opions Sandard Reverse K > H K < H Down-and-in Call (DIC) DC4+DC5 DC-DC+DC3+DC5 Up-and-in Call (UIC) UC+UC5 UC-UC4+UC3+UC5 Down-and-in Pu (DIP) DP+DP3-DP4+DP5 DP+DP5 Up-and-in Pu (UIP) UP-UP+UP3+UP5 UP4+UP5 Down-and-ou Call (DOC) DC-DC4+DC6 DC-DC3+DC6 Up-and-ou Call (UOC) UC6 UC-UC-UC3+UC4+UC6 Down-and-ou Pu (DOP) DP-DP-DP3+DP4+DP5 DP6 Up-and-ou Pu (UOP) UP-UP3+UP6 UP-UP4+UP6 These analyic formulas presen a mehod o price barrier opion in coninuous ime, bu in pracice he asse price is sampled a discree imes. This means ha periodic measuremen of he asse price is assumed, raher han a coninuous lognormal disribuion. Broadie, Glasserman and Kou (997) arrived a an adjusmen o he barrier value o accoun for discree sampling as follows: H He δσ T n, where n is he number of imes he asse price is sampled over he period. For up opions which hi he barrier from underneah, he value of δ is.586. For down opions where he barrier is hi from he op, he value of δ is Remark In Appendix A he mahemaical background, used o derive barrier opion prices formally, is given Remarks on Barrier Opions Here graphs of he shape of he value of barrier opions are shown as a funcion of boh he ime o expiraion and he underlying sock price.

119 Figure 35: The values of a reverse up-and-ou call. Parameers: K =.95, H =.5, r = 5%, q = %, T =9/36, σ = %, R =. See Table 7. Figure 36: The values of a sandard down-and-ou call opion and a reverse down-and-ou cal opion. Sandard Parameers: K =.95, H =.9, r = 5%, q = %, T =9/36, σ = %, R =. Reverse Parameers: K =.9, H =.95, r = 5%, q = %, T =9/36, σ = %, R =. See Table 7.

120 Figure 37: The values of a sandard up-and-in call opion and a reverse up-and-in call opion. Sandard Parameers: K =.95, H =.9, r = 5%, q = %, T =9/36, σ = %, R =. Reverse Parameers: K =.9, H =.95, r = 5%, q = %, T =9/36, σ = %, R =. See Table 7. Figure 38: The values of a sandard down-and-in call opion and a reverse down-and-in call opion. Sandard Parameers: K =.95, H =.95, r = 5%, q = %, T =9/36, σ = %, R =. Reverse Parameers: K =.9, H =.95, r = 5%, q = %, T =9/36, σ = %, R =. See Table 7. 3

121 Figure 39: The values of a sandard down-and-in call opion and a reverse down-and-in call opion. Sandard Parameers: K =.95, H =.95, r = 5%, q = %, T =9/36, σ = %, R =. Reverse Parameers: K =.9, H =.95, r = 5%, q = %, T =9/36, σ = %, R =. See Table Arbirage Bounds on Valuaion Plain Vanilla Pu-Call Transformaion (Haug, 999) The American plain vanilla pu-call ransformaion, where S is he asse price, K he srike price, T ime o mauriy, r he risk free ineres rae and b he cos of carry, is given by (,,,,, ) (,,,,, ) C S K T r b σ = P K S T r b b σ. I shows ha he value of an American call opion is similar o he value of an American pu opion, wih he pu asse price equal o he call srike price, he pu srike price equal o he call asse price, risk-free rae equal o r-b and cos of carry equal o b. This ransformaion also holds for European opions. Rewrie he payoff funcion from a call opion, ( S K ) max,, as he pu-call ransformaion o ge he pu-call symmery: K S max S,and combine i wih S K K S C( S, K, T, r, b, σ ) = P K,, T, r b, b, σ. S K 4

122 This equaion is useful for saic hedging and valuaion of many exoic opions on he basis of plain vanilla opions. This is because i is no possible o buy, for insance, a pu opion wih asse price K when he asse price is S (assuming K S ). However, K o buy S pu opions wih srike S K opions marke. and asse price S is a real possibiliy in he Barrier Opion Pu-Call Transformaion (Haug, 999) The only difference beween a plain vanilla pu-call ransformaion and a pu-call barrier ransformaion is he probabiliy of barrier his. Given he same volailiy and drif oward he barrier, he probabiliy of barrier his only depends on he disan ce beween he asse price and he barrier. In he pu-call ransformaion he drifs are differen for he call and he pu, b versus b. However, given ha he asse price of he call is above (below) he barrier and he asse price of he pu is below (above) he barrier, his will naurally ensure he same drif owards he barrier. In he case of a pu-call ransformaion beween a down-call wih asse price S, and an up-pu wih asse price K, i mus be ha S H p ln = ln, Hc K where he call barrier H Sand he pu barrier H > K. In he case of a pu-call c < p ransformaion beween an up-call and a down-pu he barriers and srike mus saisfy Hc K ln = ln, S H p where Hc > Sand H p < K. In boh cases he pu barrier can be rewrien as H p SK =. H c For sandard barrier opion he pu-call ransformaion and symmery beween in opion mus, from his, be given by 5

123 SK DIC ( S, K, H, r, b) = UIP K, S,, r b, b H K S S = UIP S,,, r b, b, S K H SK UIC ( S, K, H, r, b) = DIP K, S,, r b, b H K S S = DIP S,,, r b, b. S K H The pu-call ransformaion and symmery beween ou opion is given by SK DOC ( S, K, H, r, b) = UOP K, S,, r b, b H K S S = UOP S,,, r b, b, S K H SK UOC ( S, K, H, r, b) = DOP K, S,, r b, b H K S S = DOP S,,, r b, b. S K H If one has a formula for a barrier call, he relaionship will give he value for he barrier pu and vice versa (Haug, 999) Sensiiviies Here are he expressions ha correspond o he expressions [] o [4] of heorem 5 ha are relevan for he differen sensiiviies. These expressions will be used in he same combinaion used o price each barrier opion, o derive is sensiiviy o he various variables. 6

124 Dela [] =φ φ qt ( ) e N w qt ( ) qt ( ) K [] =φe N( φ w) + e n( φw) / σ T H λ Η qt ( ) Η ( ) [3] =φ Se N ηw Ke N ηw ησ T S S Η [4] = φ S λ e qt ( ) rt ( 4) ( 4 ) Gamma ( ) ( ) ( ) K qt [] = e n w / Sσ T [ ] qt [] = e n w / Sσ T w / σ T H λ λ H qt qt ( ) H 3 3 ( 4) 4 μ K A = B φ e Ν ηw φηe n w σ T σ S S S H / ( ) λ H qt ( ) H rt ( ) [ B3] =φ Se Ν( ηw4) Ke Ν ηw4 ησ T S S λ μ B3 qt H ηn w 4 [3] = A3 +φe λν ( η w λ+ 4) + σ S S S σ ( T ) λ μ H [ A4] = B 4 e σ S φ S qt ( ) K +φηe n w4 / Sσ T H ( ) ( ) qt ( w ) Ν η ( T ) λ [ B4] =φ H qt H Se w3 Ke w3 T S S Ν η rt Ν η ησ 3 λ σ w 4 7

125 Thea qt ( ) qt ( ) rt [] = σse n( w) / T +φse N ( φw) q φke N φw φσ T r qt ( ) qt ( ) rt ( ) [] = σse n( w) K /( H T ) +φse N ( φw) q φke N ( φw φσ T ) r qt ( ) K Se w4 /( ( T ) ) H λ qt ( ) ( 4) K Η [3] = φ Se ηn w w / T K / H T S + σ H λ Η qt rt + φ qse N w rke N w T S S λ Η qt ( ) [4] = φ Se ηn( w3 ) σ/ T S Η ( η 4) ( η 4 ησ ) λ Η qt ( ) Η rt +φ qse N ( ηw3) rke N ηw3 ησ T S S Vega ( ) λ rt [] = Se n w T rt ( ) K [] = Se n( w) T w / σ H λ Η qt ( ) Η rt [ A3] =φ Se N( ηw4) Ke N S S ηw ησ 4 H [3] = log 3 ( r q ) A 3 σ S ( 4) ( 4 ) ( 4 T ) Η qt ( ) K K +φ Se ηn w T w / σ + T S H H λ Η qt ( ) Η rt [ A4 ] =φ Se N ( ηw3) Ke N ηw3 ησ T S S λ 4 Η [4] = log 3 ( ) 4 +φ η ( 3) σ S S H r q A Se qt ( ) n w T 8

126 In figure 4 he sensiiviies for a reverse up-and-ou call opion is illusraed. Figure 4: The sensiiviies of a reverse up-and-ou call. Parameers: K =.95, H =.5, r = 5%, q = %, T =9/36, σ = %, R =. 9

127 4. Asian Opions 4.. Definiion As discussed in an Asian opion is an opion of which he payoff is linked o he average value of he underlying asse on a specific se of daes during he life of he opion. There are wo basic forms: An average rae opion or average price opion (ARO) is a cash-seled opion of which he payoff is based on he difference beween he average value of he underlying asse during he life of he opion and a fixed srike. Here he expiry dae is usually he same dae as he las recording dae deermining he average. An average srike opion (ASO) is a cash seled or physically seled opion. I is srucured like a vanilla opion, excep ha is srike is se equal o he average value of he asse prices recorded over he life of he opion. In his srucure i is common for he user o specify an expiry dae laer han he las recording. Boh ypes of Asian opions can be srucured as pus or calls. They are generally exercised as European, bu i is possible o specify early exercise provisions based upon an average-o-dae. 4.. Common Uses Asian opions were firs used in 987 when Banker's Trus Tokyo office used hem for pricing average opions on crude oil conracs; hence he name "Asian" opion. Asian opions are opions in which he underlying variable is he average price over a period of ime. They are aracive because hey end o be less expensive and sell a lower premiums han comparable vanilla pus or calls. This is because he volailiy in he average value of an underlying asse ends o be lower han he volailiy of he value of he underlying asse. They are commonly raded on currencies and commodiy producs which have low rading volumes. In hese siuaions he underlying asse is hinly raded, or here is he poenial for is price o be manipulaed, and Asian opion offers some proecion. I is more difficul o manipulae he average value of an underlying asse over an exended period of ime

128 han i is o manipulae i jus a he expiraion of an opion ( Consider he following example by Kolb (3): A corporae execuive is given opions on he firm s shares as par of her compensaion. If he opion payoff were deermined by he price of he firm s shares on a paricular day, he execuive could enrich herself by manipulaing he price of her shares for ha single day. However, if he payoff of he opions depended on he average closing price of he shares over six monhs, i would be much more difficul for her o profi from manipulaion Valuaion According o Asian opions are broadly segregaed ino hree caegories; arihmeic average Asians, geomeric average Asians, and boh hese forms can be averaged on a weighed average basis, resuling in he hird caegory, whereby a given weigh is applied o each sock being averaged. This can be useful for aaining an average on a sample wih a highly skewed sample populaion. In oher words, averages can be calculaed arihmeically: arimeic average s+ s s = m m or geomerically: geomeric average = m ss... s m. They can also be weighed wih some weighs w i : weighed arimeic average ws + ws wms = w + w w m m m geomeric average =.... w+ w wm w w w s s s m To his dae, here are no known closed form analyical soluions for arihmeic opions. The main heoreical reason is ha in he sandard Black-Scholes

129 environmen, securiy prices are lognormally disribued. Consequenly, he geomeric Asian opion is characerised by he correlaed produc of lognormal random variables, which is also lognormally disribued. As a resul, he sae-price densiy funcion is lognormal and hence, he analyic no-arbiragobained using risk-neural expecaions. In conras, he arihmeic Asian opion value of he opion can be depends on he finie sum of correlaed lognormal random variables, which is clearly no lognormally disribued and for which here is no recognisable closed-form probabiliy densiy (Milevsky and Posner, 998). A furher breakdown of hese opions conclude ha Asians are eiher based on he average price of he underlying asse or, alernaively, here is he average srike ype. The payoff of geomeric Asian opions is given as: m m Payoff Asian Call = max, Si K and i= m m Payoff Asian Pu = max, K S i. i= The payoff of arihmeic Asian opions is given as: Payoff Asian Call m S i i= = max, m K and Payoff Asian Pu = max, K m i= m S i. The payoff funcions for he Asian opions above can also be wrien in a more general way. For an average price Asian: ( ( ave )) V = max, η S K

130 and average srike Asian: ( ) V = max, η ST Save, where η is a binary variable which is se o for a call, and - for a pu opion. A final consideraion is how much daa o use in he calculaion of he average. If closely-spaced prices over a finie ime are used, hen he sum ha is calculaed in he average becomes an inegral of he asse over he averaging period. This would give a coninuously sampled average. More commonly, only he daa a reliable poins are aken. Closing prices are used, which is a smaller se of daa. This is discree sampling (Wilmo, 998). Formally define r q S A The coninuously compounded risk-free rae of ineres, assumed consan over he life of he opion. The coninuous yield on he asse, assumed consan over he life of he opion. The spo price a ime. The running discree arihmeic average o dae, defined for any ime poin, m < m + given by A m = S m i = i for a corresponding ineger m< N and A = for <. A N The arihmeic average of N prices. G The corresponding geomeric average given by m m G = S, S,..., S. ARO K, The value a ime of an ARO call opion. C AROP ( K, ) The value a ime of an ARO pu opion. 3

131 Characerising Valuaion Formulae Opion valuaion usually assumes insananeous asse reurns o be normally disribued so ha asse prices a some fuure dae are log-normally disribued. So in he risk-neural world, he underlying asse price is assumed o follow he sochasic differenial equaion ds = μ Sd +σ Sdz, (4..) where dz is a Wiener process, μ he drif parameer and σ he volailiy parameer. The payoff on Asian opions is based on he fuure pah of he spo prices wih he process given in (4..). Under (4..) S i can be expressed in erms of S i as: i i μ σ ( i i ) + i i Yi S = S e, (4..) where Y N(,) i. For > i, ln S N ln S, i + μ σ i σ i. Using he risk-neural ransformaion of Cox and Ross (976), he soluion o he Asia n opion a ime = may be characerised as: { } * ARO cal l:, rt ARO max C S K = e E A K, N (4..3) rt * ASO call: AROC( S ) = e E { max ST A, } N, (4..4) where * E is he expecaion condiional on S a ime = under he risk adjused densi y funcion. This means ha in (4.. ), which gives h e process for S, μ is replaced by r q. Suppose he condiional densiy funcion for, condiional ha A N * > K, is denoed by f ( w ), hen he expeca ion erm in (4..3) can be wrien as A N { max, } = N N * * E A K A K f w K dw. (4..5) 4

132 * For he ASO, he join densiy of A and is needed. Denoe his by ϒ ξ, w, hen (4..4) can be wrien as: N S T T } N T N * w E * {max S A, = S A ϒ ξ, w dξdw. Valuing Geomeric AROs and ASOs Two valuaion mehods are given for geomeric AROs and ASOs. The firs one uses discree sampling, while he second one uses coninuous sampling. D iscree Sampling Because he produc of log-normal prices is iself log-normal, he geomeric average, * * G, is also log-normal and he funcions and ϒ ( ξ, w) N f w can be deermined. Hence he valuaion of Asian opions, deermined by a geomeric average of prices, is a relaiv ely simple maer. The derivaion given by Clewlow and Srickland (997) is followed here. T he geomeric average is given by G =,,..., N S S S N N. Therefore N ln G = ln S N( μ,σ G ); N i G N i= since i is a linear combinaion of normally d isribued random variables, i is also normally disribued. I also follows ha he disribuion ln G, ln S N T is bivariae normal wih ρσ he covariance beween ln G GσT G and ln S N T. Hence he geomeric ARO call and ASO call are, respecively, given by: μ G+ σg { } rt rt max, ( ) ( x) N rt * AROC e E G K e x e K = = Φ Φ () where Φ. is he sandard normal disribuion funcion, 5

133 x x = ( μg ln K +σg) = x σ G σ ; G and μ G+ σg rt yt { max, } ( ) Φ N rt * ASOC = e E ST G = Se KΦ y e y where ln S + ( r q) T μg σ G + Σ y = Σ y = y Σ Σ =σ +σ T ρ σ σ. G G G T The ARO pu and ASO pu are, respecively, given by: and μ G+ σ rt G { max, } ( ) N = = Φ Φ rt * AROP e E K G e K x e x μ G+ σg { } rt yt ( N rt * ASOP = e E max G ST, = e Φ y Se KΦ y). To calculae he expressions for μ G, σ G and ρ G σσ, G T he mean and variance of he logarihm of he geomeric average and is covariance wih ln S T is firs derived for any given ime poin. A any ime, ln G, can be expressed as N ln G N m = ln G + ln S, N N = + N i i m where ( i ) N μ,σ m m G = S, S,..., S, for some m< N. Each ln is disribued i. I follows ha he mean of ln G is immediaely N S i 6

134 N m μ G = ln G + N N i = μ i. m + For consan r, q and σ μ i = ln S + r q σ i σ =σ i ( ) i. Hence, N m N m μ G = ln G + ln S + r q σ ( i ). N N N i= m+ For equidisan fixing inervals from, ( ) ( ) ( N ) (,..., ) h= i = m+ N, μg can be expressed as N = + i h, where m+ i m+ m N m h μ = + + σ ( ) + ( ) N N G ln G ln S r q m+ N m. The variance of ln G N is given by N N N σ G =, σ i + ρijσσ i j N i= m+ i= m+ j= i+ σ i where ρij is he correlaion beween ln S i and ln S. For i j, ρ j ij = ; hence σ j N N σ G = N σ + σ i= m+ i= m+ i ( N i) i. When σ =σ i ( ) i his becomes 7

135 N N σ ( i ) ( N i)( ) N i i= m+ i= m+ σ G = + and for ( ) ( ) m+ i = m+ + i h ( )( ) 6( N m) N m h N m N m σ G =σ ( m+ ) +. N Finally he covariance erm ρσ G GσT is given by ρσ σ = Cov N = σi. N G G T i T N i= m+ i= m+ N ln S,ln S For σ =σ ( ) i i N σ ρσ σ = ( ), Ν G G T i i= m+ and for ( ) = ( ) + ( i ) m+ i m+ h N m h ρσ G Gσ T =σ ( m+ ) + ( N m ). Ν Coninuous Sampling Kemna and Vors (99) shows ha in a risk neural world, he probabiliy disribuion of he geomeric average of an asse price over a cerain period is he same as ha of he asse price a he end of he period if he asse s expeced growh σ rae is s e equal o r q, raher han ( r q), and is volailiy is se equal o 6 8

136 σ, 3 raher han σ. The geomeric average opion can herefore be reaed like a regular opion wih he volailiy se equal o σ 3 and he dividend yield equal o σ σ r r q = r+ q The geomeric average as,..., N G = N S, S S in he coninuous case can be wrien N G N N = exp log Sτ τ. N d (4..6) The variable G N is lognormally disribued so ha is expecaion and variance values may be calculaed explicily. Define V = log S and Z = log G. (4..7) From Io s lemma, i follows ha (4..) and (4..6) give rise o he following sysem of sochasic differenial equaions: V V r d σ σ = d dz. Z + + Z N (4..8) Using Arnold (974), i follows ha since his is a linear sochasic differenial equaion, ( V Z )' mus be a Gaussian process. This means ha ( V Z )' is binormally disribued. Hence, G log G N is normally disribued. Also from Arnold (974) follows ha 9

137 V V E r d E σ = d. Z + Z E N E (4..9) The covariance marix of ( V Z )' as defined by K K K ( ) =, K( ) K (4..) which is he unique symmeric non-negaive definie soluion of he following marix differenial equaion: K K ( ) K K K K d = + N σ + σ K ( ) K K ( ) K K ( ) K N (4..) d. Solving (4..8) and (4..) gives r σ ( ε ) E ( V V ) = ( Z ) E Z r σ ( ) + V ( ) N N (4..) σ ( ) σ K K = K K σ σ N 3 N ( ) ( ) N ( ) 3 ( ) ( ). (4..3) Combining (4..7), (4..) and (4..3) immediaely gives log G n r ( ) log ;. N σ N + S σ N 3 (4..4) 3

138 From probabiliy heory i is known ha in cases where A is a random variable, such ha log A is normally disribued wih mean E and variance V, and K > is a real number, hen: E+ V E log K + V E log K E max ( A K, ) = e N KN. V V (4..5) By combining (4..4) and (4..5) he geomeric average opion can be evaluaed as follows: ( G K ) = E{ G G K} KP( G K) Emax, N N N N r σ ( N ) + log S + σ ( ) N 3 = e r σ ( N ) + log S log K + ( ) N σ 3 N σ ( N ) 3 KN σ 3 r σ N + log S log K ( ) N where S log ( ) r N + σ r σ ( N ) K 6 6 = Se N S log + r σ ( N ) K 6 KN σ N 3 σ ( N ) 3 * d = Se N d KN d σ N 3 σ 3 N, ( ) 3

139 , 6 S log + r σ N K 6 d = σ ( N ) 3 * d = r σ N. Valuing Arihmeic AROs and ASOs When, as is nearly always is he case, Asian opions are defined in erms of he arihmeic averages, exac analyic pricing formulas are no available. This is because he disribuion of he arihmeic average, which is a sum of log-normal componens has no explici represenaion or racable properies. For arihmeic Asian opions he * * funcions f w and ϒ ξ, w in he characerising valuaion formulae are nonsandard, and o evaluae he necessary inegrals, a variey of numeric and approximaion mehods have been developed. A variey of echniques have been developed in he lieraure o analyse arihmeic Asian opions. Generally, hey can be classified as follows according o Milevsky and Posner (998): I. Mone Carlo simulaions wih variance reducion echniques: Haykov (993), Boyle (977), Corwall e all. (996) and Kemna and Vors (99). II. Binomial rees and laices wih efficiency enhancemens: Hull and Whie (993), and Neave and Turnbull (993). III. The PDE approach: Dewynne and Wilmo (995), Rogers and Shi (995), and Alziary, Decamps and Koehl (993). IV. General numeric mehods: Carverhill and Clewlow (99), Curran (994), and Nielsen and Sandman (996). V. Pseudo-analyic characerisaions: German and Yor (993), Yor (993), Kramkov and Mordecky (994), Ju (997), and Chacko and Das (997). VI. Analyic approximaions ha produce closed-form expressions: 3

140 Turnbull and Wakeman (99), Levy (99), Vors (99), Vors (996), and Bouaziz, Briys and Crouhy (994), Milevsky and Posner (998). An Edgeworh Series Expansion The disribuion of he arihmeic average of a se of lognormal disribuions is approximaely lognormal and his leads o a good analyic approximaion for valuing average price opions (Hull, 6). Hull (6) proposes ha if he firs wo momens of he probabiliy disribuion of he arihmeic average in a risk-neural world is calculaed exacly, i can hen be assumed ha his disribuion is he lognormal disribuion. This means ha he arihmeic average opions can be valued similarly o geomeric average opions where pricing formulas are derived from he fac ha he produc of log-normal prices is iself log-normal. The momens of an arihmeic average opion can be calculaed using an edgeworh series expansion. * Turnbull and Wakeman (99) apply a serie s expansion for f ( w ) o adjus for higher m * omens effecs. If f w denoes he rue disribuion and a w an alernaive or approximaing disribuion which, in his case, is a log-normal * probabiliy dens iy funcion, hen we can expand f ( w ) as follows: f * () ( ) ( ) ( = + + )..., (4..6)! 3! 4! 3 4 w a w Ea w Ea w E3a w E4a w where ( i a ) ( w) is he ih derivaives of a( w) and { } E he erms involving he i difference beween cumulans implied by he log-normal fi and he rue cumulans for A N. For a given disribuion funcion F are: of a random variable X, he firs four cumulans 33

141 χ = E( X) ; χ = E X E X ; χ 3 = E X E X χ 4 = E X E X 3 χ, 3 4 ; where all expecaions are wih respec o he disribuion F. Le χ iε =χif χ ia. Then he firs four coefficiens Ei, i =,,3,4, are given by Ε =χ ε, Ε =χ +χ ε ε Ε =χ +3χ χ +χ 3 3 ε ε ε 3ε,, and Ε =χ +3χ +4χ χ +6χ χ +χ 4 4 ε ε ε 3ε ε ε 4ε. If a( w) is chosen o be he log-normal densiy wih parameers α momens of A N are approximaed by and ν, hen he rue * k E A N = exp α k+ v k. Subsiuing ino (4..5) and aking he firs four erms of (4..6), he ARO call is approximaed afer inegraing by α+ v rn rn { max, } = Φ( ) ΚΦ N rn * e E A K e x e x r () N ( e E ( x ) a( K) E a ( K) E a ) + Φ E + ( K)! 3! 4! Wh en α and ν are chosen o equae he firs wo momens of A, E = E = and he approximaion becomes N α+ v rn rn { max, } = Φ( ) ΚΦ N rn * e E A K e x e x r () N + e E3a ( K) + E4a K 3! 4!. 34

142 To apply he Edgeworh expansion, i is necessary o deermine he cumulans of he A N disribuion of he average,. Alhough he disribuion of is non-sandard, is A N momens can be found using a recursive relaionship. Le A N be deermined by fixing S for i,..., N and define he price relaives, i = i R, by S i = S R for i i i =,..., N. Fro m (4..) R i is log-normally disribued and under risk-neuraliy is given by r q σ i = ( i i ) +σ i i i, R e Y where Yi N (, ). Thus he momens for R i are given by where * k E ( Ri ) = exp α ik+ θik, α i = σ θ =σ r q i i Y. i i i i I follows by definiion ha A N can be wrien as S A = ( R RR3... RR3R4... ). N RN N Define L i as follows: LN = + RN, and Li = + RL i i+, i = N,...,. Then A N can be expressed as A N S = L N, and using S, = S R A N S = R N L. 35

143 Since R and L are independen, i follows ha S E A = N E R E L N k * k * k * k. I k is known ha ( i ) k = ( + ) * * E L E R N N * E R = exp α ik+ θ k k can be calculaed recursiv ely for i = N,...,. * k i and o find E L, noe ha * k * k and for i< N, E ( L ) = E ( + R ). i i Li + Hence, k * E L i Simple closed-form expressions for he firs wo momens are now derived. Given ( i ) ha a ime =, ln S N μ,σi, where i μ i = ln S + μ σ, i σ =σ. i i I follows ha he firs momen for A N is given by A N N μ+ i σi = e = N N N i= i= F, i where F i denoes he forward price of S i. For consan ineres raes and volailiy he following equaion holds: S N * E A e N = N i= ( r ) y i and for i = + ( i ) ) ( N N, * S g e E A e N = N e ghn gh 36

144 where ( ) N h = N g = r y. As he frequency increases, he limi as N ends o ( ) gh N * g e e N = N g N E A S. ( ) The second momen for A N is given by * N N * = N i j N i = j = E A E S S or N N N * * * E A = E S + E S N S i i j N i= i= j= i+. Noing ha * i E S S FF e = σ for j i j i j i, * N N N σi σ i E A = F. N i e + FiF j e N i= i= j= i+ For consan ineres raes and volailiy and for = + ( ) i i h, N N N * S ( g+σ ) i ( g+ σ ) i g j E A = e + e e, N N i= i= j= i+ E ( g+σ ) ( g ) hn ghn ( g ) hn Se +σ e +σ e e A = +. N ( g ) h ( g ) h gh N +σ +σ ( g ) h e e e +σ e * 37

145 In he limi as N his becomes E ( g+σ ) g( ) ( )( ) N g N Se +σ e e A =. N ( g+σ )( ) g ( g N +σ ) * F or he limi N he average for coninuous sampling is he inegral: A N A N N = S dτ τ. ( N ) In his insance i is relaively easy o provide closed-form expressions for all he A N momens for. The following resul is saed by Clewlow and Srickland (997): for any ineg er k, he kh momen for A N a curren ime is given by ks! ( kg + m( k, k ) σ )( ) k * k E A = e M N k k ( N ), where M k is given by for k = and for k > by M ( ) g N e = g M k = g + m (, k ) σ g m (, k + ) σ... ( kg + m( k, k ) σ )( N ) e M..., k g+ m k, k σ kg m( k, k) M k M + σ k wih mik (, ) = ik ( i+ ) for M k are: for i =,..., k. For example, he nex hree expressions 38

146 M ( g+σ )( N ) e = M g+σ g+σ, ( 3g+ 3σ )( N ) e M = M M g+ σ g+ 3σ 3g+ 3σ 3 ( 4g+ 6σ )( N ), and e M = M M M 4 3 g+ 3σ g+ 5σ 3g+ 6σ 4g+ 6σ. If i is assumed ha he average asse price is lognormal, an opion on he average can be regarded as an opion on a fuures conrac. Then he following equaions can be used where wih and rt c= e F N d KN d rt p = e KN d F N d d d F σ T ln + K = σ Τ F σ T ln K = σ Τ F = M M ln σ =. T M and, Coninuous Sampling In Zhang (999) an analyical approximae formula for he pricing of an arihmeic Asian opion wih coninuous sampling is derived by solving a parial differenial equaion (PDE) numerically. The mehod is shown o be more accurae han any exising mehod in he lieraure, and faser han oher PDE mehods. The mehod has a well-conrolled error, and herefore he resuls can be used as a benchmark o jusify he error compued by approximaion mehods, for which he error is unknown. 39

147 Firs AROs are considered. The resuls can be exended o deal wih ASOs. The general explicaion of Zhang (999) is followed and expanded on in his secion. The pricing formula is considered only wihin he averaging period, i.e. The price of he opion before he averaging period can be compued by solving he Black-Scholes equaion wih a paricular payoff a formula. Take ime = for simpliciy. Inroduce a new variable I; N., given by he Asian opion I = Sd τ τ, which is he sum of he underlying asse price S. Therefore I / is he Arihmeic average of he underlying over he period of [,]. Hence, he payoff of he ARO call wih coninuous sampling can be wrien as I max K,. N The following lemma is he well-known Feyman-Kac heorem (cf. Shreve 997) and will be used o prove Theorem 6. Lemma 3. Define where Then and x, ( ) v, x = E h X T, T, () = ( ) +σ ( ). dx a X d X db v(, x) + a( x) vx(, x) + σ ( x) vxx(, x) = = hx vtx,. 4

148 Theorem 6: (Zhang, 999) The price and Greeks of an arihmeic average rae call opion wih payoff I max K,, are given by he following analyical approximaion formulas: N S S ζ η AROC ( S, I, ) = f ( ζ, η ) = ζν + e N N η π Vega ζ 4η rτ ζ e ζ S η 4η rτ I ζ = S Ν + e e K Ν (4..7) r N N π η N η ARO rτ C e ζ η Δ = = Ν + e S r N η N π ARO θ = C AROC S = = σ 4 σ rτ ( e ) ζ 4η S e ζ Sσ = ( + rζ) Ν e + e e N r r Δ Γ = = ζ+ S S πη r N N η e π ζ 4η AROC S rτ ζ rho = = ( r τζ+ rτ + e ) Ν r r N η Sσ + 9 4rτ + 4 τ πη 4 8rN rτ ζ rτ rτ 4η N η 4 N πη ( r ) e ζ 4η ζ rτ rτ 4η e r e e ( ) + 3+ τ where N K I e rτ ( e rτ ζ= ), S r σ rτ rτ η= ( 3+ rτ+4e e ), 4r τ=. N Proof: Suppose he process { } S is expressed in he usual sochasic manner as used previously. For N he coninuously sampled arihmeic mean is defined as 4

149 A = Sτdτ (4..8) N where N is he mauriy dae and [, N ] is he final ime over which he average value of he sock is calculaed. Noe ha A is an average only where = N. For N <, A is defined as he par of he final average up o ime, and is a monoonically increasing funcion of. When N, he price of he opion ARO C will depend on ( S, A, ). Where < N he value of A will no be relevan. In order o deerm ine he value of he opion a =, he value of he opion in he inerval [, N ] used o calculae he value in he inerval [, is firs calculaed and a value found for ]. Since in he ime inerval [, ] he value of he opion is deermined only by and S, he sandard parial differenial equaion for he price can be derived using Black and Scholes (973) hedging argumens and Meron s (973) exension: AROC AROC AROC + σ S + rs raro. C = S S The boundary condiions for a sandard call opion which expires a expressed as can be (, ) = max (,) ARO S S K C ARO ARO C C, =,, =., For an ARO, however, he boundary condiion a ime implies ha he value of he opion is equal o ARO C S,. Recall ha before his value can be calculaed, he AROC has o be valued over he ime inerval [, N ], and in ha case AROC depends 4

150 on S, A,. A parial differenial equaion is needed for AROC ( S, A, ), < N. Noe ha (4..8) yields he equaion wher e da = β S d, β= N. (4..9) If ARO ( S, A, ) is he value of he opion a ime [ ] C Io s lemma as follows:, N, i is possible o apply AROC ARO C ARO C ARO C ARO C AROC = + σ S + rs +β S d +σs dz. S S A S ARO C Thus a coninuously-adjused porfolio consising of socks which is parially S AROC fin anced by a loan S ARO S C, bears an idenical risk o he ARO which is ARO C given by σs dz. The porfolio also has coss idenical o ARO C in paymens. S Arbirage argumens imply ha he expeced insananeous reurn on he porfolio and he opion mus be idenical, since he risks and coss are idenical. Therefore he following parial differenial equaion can be derived for he opion price ARO AROC ARO + σ S ++β S + rs S A S AROC C C raro C =, { N } which holds in he dom ain D= ( S, A, ) S, A,. Hence wihin he Black-Scholes (973) and Meron (973) framework, he price of an arihm eic average call opion ARO ( S, I, ) saisfies he following parial differenial equaion, firs derived by Kemna and Vors (99): C 43

151 AROC ARO C ARO C ARO + S + σ S + rs I S S C raro =, C and final condiion Now define I AROC ( S, I, ) = max K,. N N φ ( x, ) = E max Suμ( du) x, S =, where he process of S is given in (4..). The random variable N max Suμ( du) x,, whose condiional expecaion is being compued, does no depend on. Because o f his, he ower propery implies ha φ( x, ), N is a maringale: For, N N M = E max Suμ( du) K, I N = E max Suμ( du) K Suμ( du), I K S N u du S μ u = SE max μ( du), I S S = Sφ, ξ, ( ) (4..) where for fixed-srike Asian opions ξ = u K S μ du S, 44

152 [ ] μ du = I u du N, N and i is assumed ha he probabiliy measure μ has a densiy ρ in. From is definiion, φ is joinly coninuous and decreasing in and x. Now Io s formula is applied o ge he process for ξ. Since (, N ) S = S exp σb σ + r, he process for S is given by ds = rsd +σsdb = S ( rd +σdb ); hence Therefore ds rd db S = +σ. (, ), K c S du ξ = f S = c= S u N (, ) (, ) (, ) df S df S d f S dξ = d+ ds + σ S d d ds d K c S du K c S du cs d S S = + σ d S S S u u d 3 Similarly, ds = cd ξ +σξ S d = cd ξ rd +σ db +σξd ( ) = cd +ξ σdb rd +σ d. 45

153 (, ) (, ) (, ) (, ) dφξ dφξ d φξ dφξ = d+ dξ+ σξd d dξ d =φ d +φ cd ( db rd d) +ξ σ +σ + φσ ξ d = φ+φ ( c + rξ +σξ ) + φσξ d φξσ db, wih dφξ φ= d (, ), φ= (, ) dφξ dξ and φ = d (, ) φξ d. Assuming ha φ has enough smoohness o apply Iô s formula o (4..), i gives (, ) dm S φ =φ ds + Sdφ+ dsdφ =φ ds + S φ d +φ dξ+ φσ ξ d + dsdφ rφ Sd + S φ+φ ( ρ rξ+σ ξ ) + σ ξ φ d σsφσξ d = S rφ+φ ( ρ + rξ) φ + σ ξ φ d, where signifies ha he wo sides differ by a local maringale. A local maringale is defined as: Definiion : Local Maringale. (Eheridge,4) A process { X is a local Ρ,{ I } maringale if here is a sequence of { } } sopping imes { Tn} n such ha { X T n }, is a { } and P limtn = =. n I - Ρ I maringale for each n All maringales are local maringales bu he converse is no rue. Because M is a maringale, he sum of he d erms in dm mus be. This implies ha 46

154 =φ+ rφ+ σ ξ φ ( r ) ρ + ξ φ. (4..) Se ( ) N f x, = e r φ x,, hen i is found by Iô ha f solves f f x ( rx) f + σ ρ + =. x x The boundary condiion in he case of he fixed srike Asian opion follows from he Feyman-Kac heorem given in lemma 3 as (, ) max (,) f x = x. (4..) N Denoe he soluions o he PDE (4..), wih he fixed srike boundary condiion φ μ, N, he price of he Asian (4..), by. Then in he case where is uniform on [ ] opion wih mauriy N, fixed srike K, and iniial price S is N r du K N e Emax ( Su K), = S f, N S K,. rn = e Sφ S Noice ha wih hese parameers, for x rτ φ ( x, ) = ( e ) x. r Applying a similar ransformaion o he equaions above, o ha adoped by Roger and Shi (995) illusraed above, TK I e rτ ( e rτ ζ= ), S r τ= T, 47

155 which implies ha rτ rτ TK I =ζ Se + S ( e ) τ I S rτ S K = ζe T T T τ e S rτ rτ = ζe ( e ) T τ S f * = ( ζ,τ) T rτ ( ) I AROC ( S, I, ) = max K, T S * = max f ( ζ,τ), T S * = max { f ( ζ,τ),} T S = f ζ,τ T. ( 4..3) To find f ζ,τ proceed as follows. Equaion (4..3) is a linear diffusion equaion wih variable coefficien and an iniial condiion f, f rτ σ ζ + ( e ) = < ζ <, ( 4..4) τ r ζ f ζ, = ζ, 4..5) max. ( Iniially f ζ, = δζ ζ, herefore he effec only exiss a ζ = iniially, and will be significanly near he region of small ζ. Therefore he ζ is dropped from he (4..6). Nex solve f ( ζ,τ ), which is an analyical approxima ion of f ( ζ,τ), from he following equaions 48

156 f σ f rτ ( e ), = < ζ <, 4..6 τ r ζ f ( ζ,τ = ) = ( ζ,) max. Inroducing a new ime variable σ rτ dη= ( e ) dτ r τ σ rτ σ rτ rτ ( e ) ds 3 ( 3 r e e ), r 4r η= = + τ+4 (4..6) becomes a sandard one dimensional hea equaion: f f =, < ζ <, η ζ f ( ζ,η = ) = ( ζ,) max. The soluion can be obained by Green s funcion soluion in one dimension given by u = kuxx < x<,< < u( x,) = g( x) ( x y) u( x, ) = exp g( y) dy 4π k 4k, where k =, x=, g( x) max and u( x, ) f ζ =η, = ζ, = ζ,η, as ζ ζ ζ η f e d e 4πη η π 4η 4 ( ζ, η ) = ζ ζ = ζν +. ( 4..7) ζ η Subsiuing (4..7) in (4..3), for he r elevan, leads o (4..7), which proves he heorem. The Greeks are calculaed using he following parial derivaives (Zhang, 999): 49

157 f ζ = N ζ f f e η = = ζ ζ rτ = ( e ) S S ζ+ r ζ = + r ζ η σ = rτ rτ e e + r η η = σ σ ζ = r r ( r τζ+ r τ + e τ ) η σ = 4 9 r r 4r τ ( + τ τ 4r τ ) e + ( 3+ r τ ) e r 4r πη ζ η ζ r This proves he analyic approximaion of he opion price and is corresponding Greeks and hus complees he proof of he heorem. The analyical approximaion formulas in Theorem 6 are correced using he erms in Theorem 7. Theorem 7. (Zhang, 999) The correcion erms of he analyical approximaion formulas in Theorem 6 are given by he following: 5

158 S AROC S I f r T AROC Δ = = f ζ+ S T T r AROC S f S f θ = = ( + rζ) T ζ T ζ (,, ) = ( ζ, τ;, σ ) ( 4..8) rτ ( e ) Δ rτ Γ = = ζ+ ( e ) S ST πη r f ζ f AROC S f Vega = = σ T σ AROC S rτ f S f rho = = ( r τζ+ rτ + e ) + r r T ζ T r ζ where he funcion ( ζ τ; r ) f,, σ and is derivaives f f f f f ζ τ r σ ζ,,, and can be evaluaed by solving he following parial differenial equaion numerically wih he finie difference mehod f r f τ σ ζ 4η rτ σ ζ ( e ), τ ζ+ e = e + r ζ 4 πη r f ζ,τ = =. ζ Proof: The exac value of correcion erm f f ( ζ,τ) is equal o he analyic approximaion f ( ζ,τ) ζ,τ, i.e, plus he f ζ,τ = f ζ,τ + f ζ,τ (4..9), wih f ζ,τ saisfying (4..6), i.e. f σ rτ f ( e ) =, < ζ <, τ r ζ f ( ζ,τ = ) = ( ζ,) max, is given by (4..7) as 5

159 f ( ζ η ) = ζ e dζ = ζν + e ζ ζ ζ 4η ζ η 4η,. 4π π η η Subsiue (4..9) ino equaions (4..4) and (4..5) hen gives ( rτ f e ), f σ ζ + = < ζ <, τ r ζ f max. ( ζ, ) = ( ζ, ) rτ f rτ f σ ζ+ ( e ) = σ ζ ζ+ ( e ) f τ r ζ r ζ f ζ, τ = =. ζ σζ 4η rτ ( e ) = e ζ+ 4 πη r I is very sraighforward o implemen he presen mehod. The analyical approximaion of he average rae call is easy o compue using Theorem 6, since i is a closed-form formula in erms of he cumulaive normal disribuion funcion. In order o ge he rue value of he opion, he correcion erm mus also be compued, i.e. i is necessary o solve f (, r, ) ζ τ; σ from (4..8). Equaion (4..8) is an inhomogeneous linear diffusion equaion wih a variable coefficien. The numerical calculaions are done using he Crank-Nicolson scheme. The scheme is popular for solving parabolic parial differenial equaions Arbirage Bounds on Valuaion An arihmeic Asian Opion is Always Worh Less Than a Vanilla Opion For ime N he coninuously sampled arihmeic mean is defined as in (4..7). This expression is approximaed by he discree expression 5

160 A n = S, + i n i= where i = + i n / for large n. Using his approximaion he numerical approximaion for an arihm eic Asian opion a ime follows as: n r ( N ) * S i AROC ( S,, ) = e E max K, i= n +. (4..3) This formula enables he comparison of he value of an Asian opion wih ha of a sandard European opion which can be expressed in similar erms: n r ( N ) * S VC ( S, ) = e E max K,, i= n + (4..3) since S n = S. i= n + Equaions (4..3) and (4..3) are compared using he following lemmas given in Kemna and Vors (99): Lemma 4. If U is a random variable wih EU hen for every m Ν and K > Proof: Le m Emax + U K, E max ( U K,). (4..3) m m p U be he densiy funcion of U. I is clear ha m mk + U K iff U K =. m m m 53

161 Two cases are disinguished, namely K and K <. If K hen K K ( ) ( ) E max U K, = U K p U du ( U ) m + U K p ( U ) du m m m =Emax + U K,. m m K K p U du If K < hen K ( ) ( ) E max U K, = U K p U du ( U ) ( K ) ( U ) K m m E + U K + U K p( U) du m m m m m = E max + U K,, m m K K p U du = E U K K p U du where he las inequaliy follows because U K. I is clear ha if r > or σ>, a leas one of he above inequaliies is a sric inequaliy, which in fac esablishes he second par of Theorem 8 below. Theor em 8. If r, hen n n * S i * S E max K, E max K, i= n+ i= n+ and sric inequaliy holds if r > or σ >. (4..33) 54

162 Proof: Define R i = S i S i and R = S such ha S = R,..., R R. From (4..) i follows ha i i each R i is lognormally disribued wih: ( )( + ) r N j i E ( RR,..., Ri ) = exp. (4..34) n Hence, i is necessary o prove R + R R R R... R n + { } n E max K, max RR... Rn K, E (4..35) using lemma 4 given above. Firs i is shown how (4..35) follows from his lemma. I is enough o show ha n R + RR RR... R E E n+ n+ n { K } n max + K', max RR... Rn ', K R+ RR RR... R for each. Since E n R wih K ' = by virue of R n (4..34), lemma 4 can be applied and hence n R + RR RR... R E max n+ n+ n R+ RR RR... Rn E max K ', n E max... ',, n + { RR R K } n K ', where he las inequaliy follows from inducion on he number of variables. 55

163 Geomeric Asian Opions are Worh Less Than or Equal o Arihmeic Asian Opions Geomeric Asian opions are worh less han or equal o arihmeic Asian opions, since for a se of n posiive, real numbers x, x,..., x n, he following inequaliy holds x+ x x n n n x x... x, n and ha if and only if x = x =... = x n, x+ x x n n = n x x... x, n which is a well-known resul, see for example he proof on Arbirage Condiions for Arihmeic AROs and ASOs The pu-call pariy for arihmeic Asian opions are derived following he derivaion of Clewlow and Srickland (997). Consider he replicaion a ime of he payoff of an arihmeic average conrac ha pays A N a ime. Suppose ha a ime he running average is A and m > fixings are known. In order o replicae a ime N a paymen of S (for any i> m), he following sraegy is followed: i N A ime purchase e qi rn i e unis of he asse, so ha a ime i he holdings grow o e ( rn i unis of domesic currency. ) unis which can be sold in he marke for Invesing for ime unis yields S a ime. N i i N Se ( ) rn i The cos of his sraegy is Se i N e ; q r i hence replicaing he payoff A N a ime N would cos a ime : m V Ae S e e N rn qi rn i R = + N N i= m+. 56

164 As wih convenional European opions, a pu-call pariy condiion exiss for Asian opions. A porfolio of long he call and shor he pu has he following value a ime N : O ( K ) N ARO K, AR, = A K. C N P N Hence i follows ha, as K is valued a ime as Ke ( ) rn and A N can be replicaed a cos V R, he following pu-call pariy condiion mus hold: ( ) rn ARO K, = ARO K, V + Ke. P N C N R Once an ARO call opion valuaion mehod is adoped, he above condiion is used o value he corresponding ARO pu opion. Similarly o deriving he pu-call pariy condiion for arihmeic AROs, i can also be derived for ASOs. A porfolio of long he call and shor he pu has value a ime T: =. ASO T ASO T S A C P T N The cos a ime of replicaing qt S is Se. Replicaing he payoff a ime T T A N would cos a ime : and hence m i V Ae S e e N rt S = + N N i= m+ ( ) q ( ) rt ( ) () ASO = ASO + V S e P C S ( ). qt i, Symmery Resuls for Ari hmeic Asians Opions Pricing of fixed-srike Asian opions has been he subjec of much research over he las fifheen years. The floaing-srike Asian opion has received far less aenion in he lieraure. I is his fac ha means a relaionship beween he prices of fixed and floaing Asian opions would be exremely useful. Wih such a connecion, a floaingsrike opion could be priced using well-known mehods for he fixed-srike opion. 57

165 Define forward saring Asian opions, as Asian opions where a he curren ime, he averaging has no ye sared and where he n variables S,..., S T n+ T are random. This case saes, in conras wih he case ha T n+ where only S,..., ST remain random. This Asian opion is called in progress. Henderson and Wojakowski () use he change of numeraire echnique o obain symmery resuls beween forward saring European-syle Asian opions wih floaing and fixed srike in case of coninuous averaging. Vanmaele e al. (5) show ha hose resuls can be exended o discree averaging. The symmery resuls become very useful for ransferring knowledge abou one ype of opion o anoher. However, here does no exis such a symmery relaion for he opion in progress. Hence he procedures of Henderson and Wojakowski () are given below. A rihmeic Asian Opions wih Coninuous Averaging Define he coninuous arihmeic average as A = N Sudu. For he fixed srike Asian call opion generalised noaion is inroduced: (,,,,, ) AC x x x x x x, where x is he srike price, he iniial value of he process ( S ) x, x 3 he risk free ineres ra e, x 4 he dividend yield, x 5 he saring dae of averaging, x 6 he opion mauriy. Analogous, for a pu opion se AP x, x, x, x, x, x For floaing srike opion, inroduce a similar generalized noaion: where (,,,,, ) ACF y y y y y y, y is he iniial value of he process S, y he percenage, y 3 he risk free ineres rae, y 4 he dividend yield, y 5 he saring dae of averaging, y 6 he opion mauriy. Analogous, for a floaing srike pu opion se 58

166 (,,,,, ) APF y y y y y y The percenage y refers o he proporion of S T ha will be received in he floaing srike call opion or bough in he floaing srike pu opion. The percenage y = is he imporan case in financial opion pricing. The floaing-srike call is ypically inerpreed as a call wrien on S, wih floaing srike A T. Exercising, he holder receives or buys y unis of sock and pays he average of he pas prices, A T. For he following i is assumed ha he averaging period sars a ime, when he opion conrac is wrien, i.e. =. Hence i is assumed ha he opion is vanilla. The resuls hold, however, for he forward saring case, for prices compued up o and including ime. Theorem 9. If S follows he exponenial Brownian moion process: ds r q d dw S = +σ, he following symmery resuls hold: (, λ,,,, ) = ( λ,,,,, ) ( 4..36) ACF S r q AP S S q r N K AC K S r q N APF S q r S (,,,,, ) =,,,,,. ( 4..37) N N Proof: Equaion (4..36) is proved firs. The floaing-srike Asian call price expressed in unis of sock as numeraire is r r N N * ACP e N ACF E ( S A ) N E N ( λs A ) S e max, N N = = max,. S S λ = S S N 59

167 By changing he numeraire o S via S e N Se r N σ N +σw N = e = qn * dq, dq he measure * Q is defined. Under Girsanov heorem. Moreover, * Q, W = W σis a Brownian moion, using he * ( λs A ) max, S N N N * ( A ) = max λ, N is he erminal payo ff in unis of sock as numeraire, where A * qn * * ACF = e E max λ A, N. * N N A =. Hence, S N I can be seen ha he roles of he underlying and exercise price have swiched and he new exercise price is λ unis of sock. This is a pu wrien on a new asse coninue, rewrie A * N as N N A * N Su * A = = du = S u ( N) du S S. N N N N N * A. To For u N, a I - measurable random variable is defined as S S r q u W W u = = exp σ ( ) σ( ) * u N N N u S N = exp r q u W W ( N ) ( u ) * * + σ +σ N * * using he Q Brownian moion W. Noe ha if W = W ˆ * Brownian moion saring a zero, hen from he laws of Brownian moion * is a refleced Q - 6

168 and W * * ˆ u W = W N N u N ˆ * ˆ σ WN u+ r q σ u N N N N A = A = e du. Reversing ime via he variable chan ge s = N u, gives Aˆ N ˆ σws r q+ σ s N = e ds. N * Thus Su N are indeed log-normally disribued variables and A = A is a sum of such log-normally disribued variables. Thus * ˆ N N qn N ACF = e E max λ A, = e E max λ A,, N * qn * * * ˆ which proves (4..36). To prove (4..37), sar wih a fixed-srike call given by rn ( ) = ( ) AC K, S, r, q,, T e E max A K,, N hen (4..37) follows form pu-call pariy ha resuls. For he floaing srike, i is known qn rn APF ( S, λ, r, q,, N) ACF ( S, λ, r, q,, N) = ( e e ) S λs. q ( r ) N The analogous resul for fixed srike opions is qn rn N AC ( K, S, q,, ) AP( K, S, r,, ) = S e r r, N, q N e e K. q ( r ) N 6

169 Using hese pu-call pariy resuls he lef side of (4..37) gives qn rn ACF ( S, λ, r, q,, N) = APF ( S, λ, r, q,, N) ( e e ) S λs, q ( r ) N while he righ side of (4..37) gives AP λ S S q r = AC λs S q r e e S e λs rn qn qn,,,,, N,,,,, N. ( q r) N Therefore, APF ( S, λ, r, q,, N ) e q ( r ) qn rn ( ) = AC λs, S, q, r,, e e S e λs. N e S λs rn qn qn N ( q r) N Se K λ= and swap he dividend yield and risk-free rae, hen S which gives Κ rn qn APF S,, q, r,, N ( e e ) S K S ( q r) N = AC K, S, r, q,, e e S e K, qn rn rn N ( r q) N Κ APF S,, q, r,, N K S rn = AC K, S, r, q,, e K, N which proves (4..37). Arihmeic Asian Opions wih Discree Averaging For he fixed srike Asian call opion generalized noaion is inroduced: 6

170 where x is he srike price, (,,,,,, ) AC x x x x x x x, he iniial value of he process ( S ) x, x 3 he risk free ineres rae, x 4 he dividend yield, x 5 he opion mauriy, x 6 he number of a veraging erms and x7 he saring dae of averaging. Analogous, for a pu opion se ( x, x, x, x, x, x ) AP x., For floaing srike opion, inroduce a similar generalized noaion: w (,,,,,, ) ACF y y y y y y y, here is he iniial value of he process ( S ) free ineres y, y he percenage, y 3 he risk rae, y 4 he dividend yield, y 5 he opion mauriy, y 6 he number of a veraging erms in he srike and y7 he saring dae of averaging. Analogous, for a floaing srike pu opion se (,,,,,, ) APF y y y y y y y Theorem. and K AP + = S ACF S r q T n T n AP S S q r T n ( K, S, r, q, T, n, T n ) ACF S,, q, r, T, n, ( 4..38) (, β,,,,, + ) = ( β,,,,,,) ( 4..39) K AC K S r q T n T n + = APF S q r T n S APF S r q T n T n AC S S q r T n (,,,,,, ),,,,,, ( 4..4) (, β,,,,, + ) = ( β,,,,,,). ( 4..4) Noe ha he ineres rae and dividend yield have swiched heir roles when going from a floaing o a fixed srike Asian opion or vice versa. Proof: Only he firs symmery resul given in (4..38) is proved here, since he ohers follow along similar lines and use pu-call pariy for Asian opions. 63

171 (,,,,,, + ) AP K S r q T n T n n rt Q = e E max K ST i, n i= qt = e E e S KS S S S r q T n n Q T i max T T i, S ST n i= ST n i= n qt Q KS σ = e E max S exp r q+ +σ( BT i BT), i S n, T i= where he probabiliyq is equivalen o Q by he Radon-Nikodym derivaive, where he dividend yield q is sressed: dq S T σ = = exp T B. ( r q) T T dq Se +σ Under he probabiliy Q, B = B σ is a Brownian moion, and herefore, he dynamics of he share under Q are given by ds S ( ) = r q +σ d +σdb. Due o he independen incremens, B T i B T has he same disribuion as B * i. Therefore, aenion is focussed on he process ( ) S defined by B i and * S exp. i = S r q+σ i+σb i Indeed, hen * n qt Q KST * AP ( K, S, r, q, T, n, T n + ) = e E max Si, S n i= n qt Q KS T = e E max S i, S n i= 64

172 wih he process ( S ) defined by exp i S = S r q i B +σ +σ i wih B a Brownian moion under Q. As a conclusion, K AP ( K, S, r, q, T, n, T n + ) = ACF S,, q, r, T, n,. S 4..5 Remarks on Asian Opions Valuing Arihmeic AROs when One or More Fixing is Known An ARO srucure in which he averaging period is a relaively small proporion of he opion mauriy horizon should be valued close o he price of a convenional European opion for he corresponding period. In fac, a European opion can be viewed as he limi of an Asian opion in which he averaging period is an infiniesimal ime period prior o expiry. A anoher exreme, if jus one fixing remains o be deermined, he ARO has is erminal value deermined by a single asse price. Therefore i can immediaely be valued as /N imes a European opion on * S N wih srike K = NK ( N ) A. Exending his idea, AROs of which he recordings have begun can be reformulaed as pro porional o a new ARO of which he recordings have ye o begin. To see his, consider valuing an arihmeic ARO call when hence > m m > recordings are known, and. A new ARO can be valued by redefining he exising ARO payoff as where * max A K, =αmax M,, N K N m M = A A = N ( N m) ( N m) ( N m ) i = m + N S i. A is again he average of known recordings and he redefined srike * K is 65

173 N m * K = K A, ( N m) ( N m) and he proporionalizing facor is N m α=. N Thus when m >, AROC ( K, ) can be valued as α imes a new ARO wih N-m remaining fixings and srike whenever * K. Because * K < A ( / ) prices are assumed as always posiive,, ha is > N m K, exercise on he call opion a ime N is cerain. In his case he call opion has he value given by rn ARO K, = V Ke. C R 4..6 Sensiiviies When he sar of he averaging period of a forward saring ARO is close o he expiry dae, is premium is close o ha of a European opion wih he same mauriy. This can be seen in Fig. 4 where he value of he equivalen European call opion is R I can be explained by he fac ha he furher away he sar of he averaging period is, he higher he variance of he average rae and so he ARO will be more expensive. 66

174 Figure 4: The sensiiviy of an Average price opion o he ime unil expiraion. Parameers: K = 9, S =, r = %, q = 5%, =365, σ = %. N The effec of alering he number of equidisan fixings over a given averaging period can be seen in Fig. 4. I can be seen ha wih fewer observaions, he higher he varianc e of he average rae and herefore he higher he price. Figure 4: The sensiiviy of an Average price opion o he observaion frequency. Parameers: K = 9, S =, r = %, q = 5%, N =365, σ = %. 67

175 4.3 Summary In his lenghy chaper wo pah-dependen opions were discussed. Firs, weakly pah-dependen barrier opions were considered. The eigh basic barrier opions were shown o have prices based on combinaions of six evaluaion equaions. These six evaluaion equaions were given ogeher wih an informal proof of heir derivaion. The mahemaical background and formal derivaion on pricing barrier opion were referred o in Appendix A. In he discussion he payoff funcions of an up-and-ou call, down-and-ou call, up-and-in call and down-and-in call were illusraed. The pucall pariy relaionships for barrier opions were hen derived from he pu-call pariy relaionship for sandard opions. This led o a relaionship beween a down-and-in call and up-and-in pu, beween an up-and-in call and down-and-in pu, beween a down-and-ou call and up-and-ou call and a relaionship beween an up-and-ou call and down-and-ou pu. Lasly he sensiiviies of each of he six evaluaion equaions were given. These will be used in he same combinaion used o price each barrier opion o derive is sensiiviy o he various variables. Secondly, wo ypes of srongly pah-dependen Asian opions, namely average rae opions and average srike opions, were defined. To value hese opions general characerising valuaion formulae were derived ha showed he dependence of he valuaion funcion of AROs and ASOs on he condiional densiy funcions of he average of N prices for AROs and he join densiy of he average of N prices and he final sock price. I was shown how o value geomeric AROs and ASOs using boh discree and coninuous sampling. These valuaion mehods have closed form soluions and are derived from he fac ha he average of a se of log-normal prices is iself lognormally disribued. Therefore he join densiies menioned above are easy o evaluae when he average is defined by he geomeric average. When, as is nearly always he case, Asian opions are defined in erms of he arihmeic averages, exac analyic pricing formulas are no available. This is because he disribuion of he arihmeic average, which is a sum of log-normal componens, 68

176 has no explici represenaion or racable properies. For arihmeic Asian opions he join densiies menioned above in he characerizing valuaion formulae are nonsandard, and o evaluae he necessary inegrals a variey of numeric and approximaion mehods have been developed. Two examples of hese mehods were given, one ha approximaes he log-normal disribuion and one ha uses coninuous sampling. Several arbirage bounds were shown o hold for Asian opions: An arihmeic Asian opion is always worh less han a vanilla opion; geomeric Asian opions are worh less or equal o arihmeic Asian opions, pu-call pariy for arihmeic Asian opions and symmery resuls for arihmeic Asian opions. In he remarks on Asian opions i was shown how o value arihmeic AROs when one or more fixing is known. Finally, he sensiiviy of Asian opions o he number of days unil averaging begins and he observaion frequency in days was illusraed. 69

177 5. Binary Opions 5. Definiion A binary variable is one which is given a value of eiher or and nohing else; in he case of derivaives, a binary opion is an opion which pays eiher an asse ou a expiry, or nohing a all based, on wheher or no he opion expires in-he-money. The payoff remains he same, no maer how deep in-he-money he opion is. These binary opions are also known as digial opions, a name which reflecs he all-ornohing characer of heir payoffs. The payoff srucure for a binary is disconinuous and hese ypes of exoic opions come in one of he following formas: A Cash-or-Nohing opion pays ou a prescribed cash amoun a expiry if he opion expires in he money. The payoffs for a call and pu are shown below: Opion Type Payou of Payou of Cash Amoun Cash-or-Nohing Call S K S > K Cash-or-Nohing Pu S K S < K An American Cash-or-Nohing binary is issued ou-of-he-money and makes a fixed paymen if he underlying asse value ever reaches he srike. The paymen can be made immediaely, or deferred unil he opion's expiraion dae. An Asse-or-Nohing binary is similar o a cash-or-nohing, wih he excepion ha he posiive payoff is he asse iself, given he following payoff crieria which is he same payoff as ha of a cash-or-nohing binary: Opion Type Payou of Payou of Asse Asse-or-Nohing Call S K S > K Asse-or-Nohing Pu S K S < K 7

178 An asse-or-nohing binary migh be srucured as an American opion wih deferred paymen, bu his srucure is no common. ( American Cash-or-Nohing digial are ofen referred o as "one-ouch binary/digials", "binary-a-hi" or he rebae porion of a knock-ou barrier opion. This opion gives an invesor a payou once he price of he underlying asse reaches or surpasses a predeermined barrier. I allows he invesor o se he posiion of he barrier, he ime o expiraion and he payou o be received once he barrier is broken. Only wo oucomes are possible: ) The barrier is breached and he rader collecs he full payou agreed upon a he ouse of he conrac, or ) he barrier is no breached and he rader loses he full premium paid o he broker. ( 5. Common Uses Binary Opions are ideal for shor-erm rading, offering poenially dramaic shorreurns, bu wih sricly limied risk. A speculaor being on rising and falling erm prices can use digial opions as cheaper alernaives o regular vanilla opions. A hedger uses his cos-effecive insrumen o draw ef fecively upon a rebae arrangemen ha will offer a fixed compensaion if he marke urned he oher direcion. A digial opion can be simulaed for pricing purposes and replicaed for hedging purposes as an aggressive bull spread. A bull spread involves buying an opion a a lower srike and selling a similar opion a a higher srike; he difference in he srikes is he spread risk. Keep in mind, hough, he more aggressive he bull spread, he higher is premium, and herefore he more cosly your hedge. On he oher hand, he less igh he bull spread, he larger he exposure o spread risk. Currency markes are even-driven and i is challenging o forecas he direcion of marke movemen prior o imporan evens. Digial opions work well in hese scenarios. Technical rading does no necessarily bode very well for profi-aking before he scheduled release of key economic and rade repors. However, if you 7

179 expec increased volailiy in ligh of he announcemens, your bes choice is o rade opions and reduce reurn-relaed spikes and whipsaws. Consider he following Forex example given by A digial opion les you wager on wheher he exchange rae will rade above or below he srike price a expiraion. If exchange raes move unfavourably o he posiion, he holder exercises his opion and rims his losses by a predeermined payou amoun, whereas if he marke moves favourably, he rader coninues o deal in curren spo prices and doesn' exercise his opion. The reasoning is ha, in a volaile marke, a digial opion presens a cheaper alernaive o he radiional vanilla opion. Alernaively, if he rader is expecing a sable or relaively quie marke wih low volailiy, hen he recommended sraegy would be o wrie (sell) opions, as doing so will generae profis in an oherwise unprofiable rading environmen. Remember, he greaer he flexibiliy and higher he payou for an unfavourable marke price movemen, he larger he upfron premium associaed wih purchasing ha opion. American cash-or-nohing binary opions are useful if a rader believes ha he price of an underlying asse will exceed a cerain level in he fuure, bu is no sure ha he higher price level is susainable. 5.3 Valuaion Recall he formula of sandard European opions in he Black-Scholes-Meron environmen a ime given by c= Se N d Ke N d qt rt rt r q T = e Se N d KN d rt ( ) = e Eˆ max ST K, 6.3. p = Ke N d Se N d rt qt rt r q T = e KN d Se N d rt ( ) = e Eˆ max K ST,,

180 where d S σ ln r q T K + + = σ T d = d σ T. The expressions in (6.3.) can be decomposed ino he difference beween wo erms and inerpreed. Consider he call opion: The expression N( d ) is he probabiliy ha he opion will be exercised in a risk-neural world. Tha means ha ( ) = P( Call is exercised) = ( > ) wih N d P S K S log normal K T, = g x dx T so ha KN ( d ) is he srike price imes he probabiliy ha he srike price will be paid. The expression S T if S T Se ( r q)( T ) N d is he expeced value of a variable ha equals > K and zero oherwise in a risk-neural world. I is herefore he unproeced presen value of he he opions. underlying asse price condiional upon exercising Cash-or-Nohing Binary Opions A cash-or-nohing call pays a fixed amoun, X, if he sock price, S T, exceeds he exercise price, K; oherwise, i pays nohing. Similarly, a cash-or-nohing pu opion pays ou a fixed cash amoun, X, if he erminal sock price is below he exercise price. These opions require no paymen of an exercise price. Insead, he exercise price merely deermines wheher he opion owner receives a payoff. If he valuaion dae is, hen he value of a cash-or-nohing call will be he presen value of he fixed cash payoff muliplied by he probabiliy ha he erminal sock price will exceed he exercise price. Therefore, he value of a binary cash-or-nohing call is given by (,,,,,, ) ( ) σ, φ = = =. (6.3.3) rt v ST K T r q Callcash or nohing Xe N d 73

181 By analogous reasoning he value of a binary cash-or-nohing pu ha pays of X if he asse price is below he srike price, and nohing, oherwise, is given by (,,,,,, ) ( ) σ, φ = = =. (6.3.4) rt v ST K T r q Pucash or nohing Xe N d Asse-or-Nohing Binary Opions Asse-or-nohing opions are similar o cash-or-nohing opions, wih one major difference. Insead of paying a predeermined cash amoun, he payoff of an asse-ornohing opion is he amoun equal o he asse price a expiraion. To value hese, refer o he firs erm in he Black-Scholes formula, which gives he unproeced presen value of he underlying asse price condiional upon exercising he opions. Therefore he value of an asse-or-nohing call which pays ou nohing if he underling asse price winds up below he srike price and pays of srike price is given by S T if i ends up above he (,,,,,, ) ( ) σ, φ = = =. (6.3.5) qt w ST K T r q Callasse or nohing STe N d An asse-or-nohing pu pays off nohing if he underlying price ends up above he srike price, and an amoun equal o he asse price if i ends up below he srike price. Is value is given by (,,,,,, ) ( ) w S K T σ, r q φ = = = (6.3.6) T qt Pu. asse or nohing STe N d American-Syle Binary Opions (ASB) The derivaion of he closed form pricing formula is followed as given by The ASB pays a cash amoun of X if a barrier H is hi any ime before expiry a ime T. The binary variable is defined as if H is a lower barrier, η= if H is an upper barrier. 74

182 The sopping ime τh is called he firs hiing ime. Given ha he sock price follows he model he payoff can be wrien as The modified payoff,, ds = r q S d +σ db { τ T } H, { S } τ = inf : η ηη. H XI XI τ H { }, T describes an ASB which is paid if a knock-in-opion has no knocked in by he ime i expires and can be valued similarly by exploiing he ideniy: XI XI X τh τh { } + { } =. T T An ASB opion can furher be disinguished by wheher X is paid a he firs hiing ime or a he expiry of he opion. Denoe if X is paid a hi, ω= if X is paid a expiry. The disribuion of he firs ime a sock price his a barrier is needed o value American binary opions. Firs he hiing ime is considered for a Brownian moion wihou drif before i is expanded o he Brownian moion wih drif. Then hese resuls are used when considering he hiing ime of a sock price. Hiing Time for Brownian Moion Wihou Drif (hi a high) Now consider he properies of he firs hiing ime τ H for Brownian moion. Shreve (996) gives he mahemaics used. Le B be a Brownian moion under P wihou drif and hi level x >, hen define { B x} τ = inf : =. 75

183 τ is he firs passage ime o x. The disribuion of τ is compued based on he reflecion principle. The crucial observaion is ha B is bounded from above. Define he maximum level reached by he Brownian moion in he ime inerval [,T ] M T = max B. { T} Then, from he join disribuion of he Brownian moion and is maximum as given in Proposiion 7, i follows ha m b mb, T b= N T [ ] PM T x = exp dx, πt T m b where m>, b< m. Thus, he join densiy is Therefore, f MT, BT x, = exp dxdmdb mb π T T m b x = exp dmdb m πt T ( m b) ( m b) = exp dmdb. T πt T ( m b) ( m b) ( m b) m PM [ x] = exp dmdb x π ( m b) = exp x π m = exp dm. x π b= m dm b = Transform m z = in he inegral o ge 76

184 z PM [ x] exp = dz. x π Now he densiy of he hiing ime τ follows from he fac ha if hen F b () = g z dz, () a F a = g a ( ()) and (see Appendix A) τ M x. The densiy of he hiing ime τ is given by f ( τ ) = P{ τ } d = P{ Μτ x} d z = exp dz d x π x x = exp d π x x = exp d. π Therefore, he hiing ime densiy is given by x x p(, x) = exp. π 77

185 The Laplace ransformaion for a Brownian moion wihou drif is obained by Wes x (7). Consider Dólean s exponenial of he maringale sb, namely e x sb s τ, which is again a maringale. Since B x is bounded from above, his maringale is bounded from above by can be applied: e s x, and below by. Thus, he opional sampling heorem x sx s sbτ s( τ τ) E τ e = E e x sb s = E e =. Le p ( x, ) be he hiing ime disribuion. Le L denoe is Laplace ransformaion. Then s (, ) = (, ) L p x e p x d sτ = E e sx+ sx sτ = E e = e = e sx sx τ s sx E e. Now he hiing ime disribuion can also be wrien as - (, ) = L e p x sx x x = exp. π (6.3.7) 78

186 Hiing Time for Brownian Moion Wih Drif (hi a high) Nex, consider a Brownian moion wih drif θ. For < T, define Define B =θ + B Z = exp θb θ = exp θ B + θ. { B x} τ= inf : =. (6.3.8) Fix a finie ime T and change he probabiliy measure only up o T. More specifically, wih T fixed, define =, I. P A Z dp A A T T Under P, he process B, < T, is a Brownian moion wihou drif, so [ τ ] = [ τ ] P d P d x x = exp d, < T. π Then, for < T, 79

187 Therefore, [ ] { } P τ = E τ = E { τ } ZT = E { } exp B T T τ θ θ = E { } E exp θb T θ T I τ ( τ ) = E exp θb θ τ { τ } τ = E { } exp θx θ τ τ = exp θx θ s P = { τ ds} ( x θ) x x exp θx θ s ds s π s s x = exp ds. s π s s ( x θ) x f ( τ ) = exp, < T. (6.3.9) π Since T is arbirary, his mus be he correc formula for all >. This resul can be obained more direcly using he Laplace ransformaion as shown by Wes (7). From he definiion { B } τ= inf : θ + = x, i can be seen ha x. Again x B is he Brownian moion which is sopped when x B is bounded from above by x. Now θ+ B firs his x ( s+θ θ) x sτ= ( s+θ θ) θτ+ B τ s( τ τ) x = ( s+θ θ) Bτ ( s s+θ θ+θ )( τ τ ) x = ( s+θ θ) Bτ ( s+θ θ) ( τ τ), 8

188 and so ( s ) +θ θ x sτ E e =, as before. Again, le p ( x, ) be he hiing ime disribuion. Then s (, ) = (, ) L p x e p x d E e sτ = ( ) ( ) θ s+θ x+ s+θ θ x sτ = E e ( θ s+θ ) x ( s+θ θ) x sτ = e E e = e ( θ s+θ ) x. Thus, (, ) p x =L = e θ - e ( θ s+θ ) e x - s+θ x L θ x x x = e exp e π x ( x θ) θ x x = exp, π using he previous Laplace ransform resul in (6.3.7). Hiing Time for he Sock Price Again, consider he Laplace ransform. The following heorem by Eheridge (997) for he hiing ime of a sloping line is used. Theorem. Se { B x } τ x, θ = inf : = +θ, where τ, is aken o be infinie if no such ime exiss. Then for α>, x > and θ x θ 8

189 Proof: ( ατ x, ) θ = x( θ + α + θ ) E exp exp. (6.3.) Figure 43: In he noaion of heorem τ x+ x, θ =τ x, θ +τ x, θ, where disribuion as τ x., θ τ has he same x, θ Fix α>, and for x > and θ se (, ) exp( x, ) ψ x θ = E ατ θ. Now ake any wo values of x, x and x, and noice graphically in Fig. 43 ha τ =τ + τ τ τ +τ D, x+ x, θ x, θ x+ x, θ x, θ x, θ x, θ where τ x, θ is independen of τ x, θ indicaes equaliy in disribuion. In oher words, and has he same disribuion as τ x, θ. Here D (, ) (, ) ψ x + x θ =ψ x θ ψ x, θ, 8

190 which implies ha k x ψ x, θ = e θ, (6.3.) for some funcion k ( θ ). Since θ, he process mus hi he level x before i can hi he line x +θ.this is used o break τ x, θ ino wo pars. Wriing he random f τ for he probabiliy densiy funcion of variable τ x and condiioning on τ x, he following is obained: x (, ) τ exp(, ) x x θ ψ x θ = f E ατ τ x = d = = α ατθ, θ f () e E e τ d x α k( θ) θ fτ () e e d x ( k) α+θ θ τx = E e ( x ( k)) = exp α+θ θ. Now here are wo expressions for ψ ( x, θ). Equaing hem gives k ( θ ) = θ+θk( θ ). Since for α>, ψ( x, ) θ mus be less or equal o, choose k ( θ ) =θ+ α+θ. (6.3.) Subsiuing (6.3.) in (6.3.) leads o (6.3.), which proves he heorem. 83

191 Since B =θ + B is a Brownian moion wih a drif θ, τ, can be inerpreed as he firs hiing ime of a Brownian moion wih drif θ. Therefore he Laplaceransform of he densiy of τ for α >, x > is given by x θ ατ α ( [ τ ]) = = [ τ ] = { θ α+θ } L P d Ee e P d exp x x. (6.3.3) If τω<, hen e ατ ω α lim = ; if τω=, hen e ατ ω = for every α >, so e ατ ω α lim =. Therefore, e ατ ω α lim =. Leing α, and using he Monoone Convergence Theorem in he Laplace ransform formula, gives τ< xθ x θ x x [ ] θ θ τ< = =. P e e So, if θ, P[ τ< ] = x e θ if θ <. For upper barriers H S he firs passage ime > H τ can be rewrien equivalen o (6.3.8) as 84

192 { S H} τ = inf : = Η σ r q = inf H : B + = log, σ σ S S S B r q. Here H x = log and define σ S since = exp σ σ + ( ) σ r q θ = σ (6.3.9), is hence. The densiy of τ H, he firs hiing ime for he sock price from H H log log θ σ S P[ τ Η d] = d > π exp σ S,. Using his densiy funcion, he valuaion funcion can now be derived as given by Wysup (999). Consider he value of he paid-a-end I can be wrien as he following inegral: ( ω =) upper rebae ( η= ). rt (, ) = { τ T} vts Xe E I H H H log log T θ rt σ S = Xe d π S exp σ. (6.3.4) To evaluae his inegral, he following noaion is inroduced: e ± () = S H σ ± log σθ, (6.4.5) 85

193 wih he properies H log, σ S () () = e e + θ σ H n e = n e S e± () e () =. ( + ()) (), ( 6.3.7) ( 6.3.8) The inegral in (6.3.4) is evaluaed by rewriing he inegrand in such a way ha he coefficiens of he exponenials are he inner derivaives of he exponenials using properies in (6.3.6), (6.3.7) and (6.3.8). H log H log T rt σ S σ S vup ( T, S ) = Xe exp d π rt H = Xe log n 3 ( e () ) d σ S T rt = Xe n( e () ) e () e+ T () d θ σ + () H ( + ()) ( + ) () T rt e e = Xe n( e () ) + n e d S = + S θ θ σ rt H Xe N e T N e T Unil now a barrier hi from below was considered. Nex a barrier hi from above, when he barrier iself is considered o be a a low level, is discussed. The compuaion for lower barriers ( η = ) is similar. As given by Wes (7), when H < S, x B is bounded from below. The maringale ha needs o be considered hen x becomes sb, and i fo llows ha =. sx s E e τ I hen follows ha 86

194 = e L sx p x,. Hence, each calculaion is generalised in urn from his sage onward o obain he value of he paid-a-end ( ω = ) lower rebae ( ) η = as θ σ rt H vdown ( T, S ) = Xe N( e + ( T) ) + N( e ( T) ). S Payoff a Firs Hiing Time Nex consider an American cash-or-nohing ha pays off X a he firs hiing ime ( ) ω =, provided i occurs before he mauriy dae T. A similar mehod o he one jus shown for ( ) ω = can be used. If P[ τ d ] is he risk -neural probabiliy disri buion of he firs ime ha he sock price his H when i sars a S, a ime, hen he value of he firs-ouch digial is found by firs compleing he square; hen following he same basic sraegy as before. The soluion is given by Η θ + θ + r θ θ + r H σ H σ vη ( S,; T, H) = X N( η g+ ( T) ) + N( ηg ( T) ), S S where g ± () log S ± θ σ + σ r = H. σ Rebaes in erms of Binary Opions Ingersoll () noes ha an easier soluion can be obained by realising ha he firs-o uch digial is closely relaed o a digial share wih a barrier even. Assume he sock pays no dividend. In his case, he firs -ouch digial is idenical in value o he 87

195 fracion X H of a barrier digial share. Suppose S receives a paymen of X when he sock price firs falls o he level H. Since he sock price hen is H, his RX paymen can be used o purchase exacly > H, so he firs ouch digial sock, whenever he sock price reaches H someime during is life, or nohing if he sock price never falls o H. This is idenical o he payof f on pay off if he minimum sock price is less applies when S price resricion. Therefore, X H X H shares of he digial shares, which han or equal o H. Similar reasoning < H relaes he firs-ouch digial o a digial share wih a maximum v ( S,; T, Hw) where w( S,; T, S H) min < min S H and w S,; T, Sm even where Smax X w ( S,; T, S min < H ) for S > H, H = X w ( S, ; T, Smax > H ) for S < H, H < is he value of an asse-or-nohing call, in he even where ( > H) > H and w ax is he value of an asse-or-nohing cal,l in he is defined for a call in (6.3.5) and for a pu in (6.3.6). General Pricing Formula The general value funcion as given by Wysup (999) ha combine all differen forms of ASBs can be wrien as θ +ϑ θ ϑ r H σ H σ ω τ v(, x) = Xe N( e+ ) N( e ) η τ + η τ, (6.3.9) x x where 88

196 e ± τ =Τ, r q σ θ ± = ±, σ ( ω) ϑ = θ + r, S ± log σϑ τ τ = H, σ τ, rebae paid a hi ω =, rebae paid a end. Noe ha ϑ = θ for X paid a expiry. For X paid a hi ( ω = ) : e ± r q σ θ ± = ±, σ r q σ ϑ = θ + () r = + ( r+ q) +, σ 4 S r q σ ± log στ + ( r+ q) + H σ 4 τ = σ τ. 5.4 Arbirage Bounds on Valuaion The following bounds are given by Reib and Wysup (). Pu-Call Pariy The pu-call pariy relaionship for cash-or-nohing binary opions is given by (,,,,,,, ) (,,,,,,, ) σ φ=+ + σ φ= =. rt v x K T r q v x K T r q Xe To show his, consider 89

197 (,,,, σ,,, φ=+ ) + (,,,, σ,,, φ= ) rt ( ) rt ( ) Xe N ( d) Xe N ( d) rt ( ) Xe N ( d ) N ( d ) v x K T r q v x K T r q = + = = Xe ( ) rt. Pu-call Dela Pariy The pu-call dela pariy relaionship for cash-or-nohing binary opions is given by (, K, T,, σ, r, q, φ=+ ) (,,,,,,, ) v x v x K T σ r q φ= + =. x x To show his consider (,,,,,,, ) (,,,,,,, ) v x K T σ r q φ=+ v x K T σ r q φ= + x x n( d) n( d) rt rt = Xe Xe xσ Τ xσ Τ =. Symmeric Srike Define f as he forward price of he underlying ( r q)( T ) f = E ST S = x = xe. σ σ ( T ) r q ( T ) The choice of he srike price K = fe = xe produces idenical values rt e. a nd delas for binary calls and pus, in which case heir value is This is derived from he ideniies where ( φ ) = [ φ φκ ] N d P S T, 9

198 d S ln σ + r q T K = σ T f σ ln K = σ T f ln σ fe = σ = σ T. ( T ) ( T ) σ T ( T ) Binary call value σ v x, K = fe, T,, σ, r, q, φ=+ = Xe N σ T Binary pu value Binary call dela ( T ) rt ( ) ( ) σ v x, K = fe, T,, σ, r, q, φ= = Xe N σ T ( T ) rt ( ) σ ( T ) v x, K = fe, T,, σ, r, q, φ= + σ rt = Xe x xσ Τ Binary pu dela n ( T ) σ ( T ) v x, K = fe, T,, σ, r, q, φ= rt = Xe x xσ Τ n ( σ T ) Homogeneiy I may be necessary o measure securiies or he underlying in a differen uni. Rescaling can have differen effecs on he value of an opion ha is dependen on srikes and barrier levels. Le v( x, k ) be he value funcion of an opion, where x is he spo price and k is he srike or barrier. Le a be a posiive real number. 9

199 Definiion (Homogeneiy classes). The value funcion is called k-homogeneous of degree n if for all a > (, ) = a n v( x, k) v ax ak. An opion of which he value is srike homogeneous of degree is called a srikedefined opion and similarly an opion of which he value funcion is levelhomogeneous of degree a level-defined opion. The overall use of homogeneiy equaions is o generae double checking benchmarks when compuing Greeks. Space-Homogeneiy When he value of he underlying is measured in a differen uni he effec on he opion pricing formula, as given by Reiss and Wysup (), will be as follows ( σ, φ ) = ( σ, φ ) > ( ( σ, φ ) = ( σ, φ ) > ( v x, K, T,, r, q, v ax, ak, T,, r, q, for all a, 6.3. aw x, K, T,, r, q, w ax, ak, T,, r, q, for all a ) ) Time-Homogeneiy A similar compuaion for he ime-affeced parameers leads o T v( x, K, T,, σ, r, q, φ ) = v x, K,,, aσ, ar, aq, φ for all a>, and a a T w( x, K, T,, σ, r, q, φ ) = w x, K,,, aσ, ar, aq, φ for all a>. a a Raes Symmery Direc compuaion shows ha he raes symmery v v + = ( T ) v r q 9

200 holds for binary opions v and w. This relaionship holds for a wider class of opions, a leas for bounded smooh pah-dependen payoffs F, because in his case he value funcion v may be wrien as v= e E F xe σ σ BΤ + r q T rt ( ) ; hence v rt ( ) = ( T ) v+ ( T ) e E STF ( ST) S = x, r v rt ( ) = ( T ) e E STF ( ST) S = x. q Foreign-Domesic Symmery There exiss a relaionship beween he prices of cash-or-nohing digial opions v( x, K, T,, σ, r, q, φ ) and asse-or-nohing digial opions w( x, K, T,, σ, r, q, φ). Here φ=+ for a call opion and φ= for a pu opion. Noice ha q can also be regarded as he foreign rae of ineres given by r f and he risk-free rae of ineres is now defined as he domesic ineres rae r d he fixed amoun X = a payoff. Now,. Also, assume he cash-or-nohing opion pays v( x, K, T,, σ, rd, rf, φ ) = w,, T,, σ, rf, rd, x x K φ. (6.3.) Firs consider he lef side of (6.3.): x σ ln rd ( T ) K v( x, K, T,, σ, rd, rf, φ ) = e N φ x x σ T + rd rf ( T ). The righ side of (6.8) is given by 93

201 d w,, T,,, r,, r T σ f rd φ = e N φd x K x K σ ln + rf rd + T rd ( T ) x = e N φ x σ T x σ ln + rd rf T rd ( T ) K = e N φ x σ T. The reason is ha he value of an opion can be compued boh in a domesic and in a foreign scenario. Wysup () considers he example of, modelling he exchange rae of EUR/ USD. In New York, he cash-or-nohing digial call opion coss (,,,, σ,,, φ=+ ) USD and hence v( x, K, T,, σ, r, r, ) v x K T r r usd eur x S eur usd φ=+ EUR. If i ends in-he-money, he holder receives USD. For a Frankfur-based holder of he same opion, receiving one USD means receiving asse-or-nohing, where he uses reciprocal values for spo and srike, and for him, domesic currency is he one ha is foreign o he New Yorker; and vice versa. Since S and have he same volailiy, S he New York value and he Frankfur value mus agree. 5.5 Remarks on Binary Opions The following remark is made by Ingersoll (). The pricing of oher European derivaives wih piecewise linear and pah-independen payoffs only requires valuing digial opions and shares for evens of he ype L< ST < H. Any oher pahindependen even can be described as he union of such evens. These digials can be compued as he difference beween wo unlimied range binaries given in ( T,,,, σ,,, < T < ) = ( T,,,, σ,,, < T) v( S K T σ r q H S ) v S K T r q L S H v S K T r q L S T,,,,,,,. T 94

202 This can be seen by considering he following. A pure European syle opion is one wih a single payoff received on a mauriy dae known a he conrac s incepion. The level of he payoff depends on he evens ha occur beween he issuance dae and he mauriy dae. The payoffs on mos European conracs are piecewise linear in he underlying sock price on he mauriy dae. The value of any such conrac can be represened as av i ( ST, K, T,, σ, r, q, ξ i) + bjw( ST, K, T,, σ, r, q, ξj) i j. Here v( S, K, T,, σ, r, q, ) receiving T ξ is he value of a cash-or-nohing binary a ime of X = a ime T if, and only if, he even ξ occurs; w( S, K, T,, σ, r, q, ξ ) is he value of an asse-or-nohing binary a ime of receiving one share of he sock a im e T, if, and only if, ξ occurs. In general, he probabiliy of sock price being eiher above or below he srike price K. T ξ depends on he This binary porfolio pricing mehod is illusraed by he following examples. A sandard European pu opion can be represened as ( σ φ = ) ( σ φ = ) Xv S, K, T,,, r, q, w S, K, T,,, r, q,. T T Tha is, a fixed amoun of X is received and one share of sock is given up when he sock price, a mauriy, is below he srike price so ha he opion expires in-he-money. A down-and-ou call opion can be represened as ( T T min(, T) ) T T min(, T) ws, KT,,, σ, rqs,, > K S > H Xv S, KT,,, σ, rqs,, > K S > H. For his opion, a fixed amoun of X is paid o receive a share of he sock if he opion is in-he-money a mauriy, and he sock price never fell below he knock-ou price H. 95

203 Relaionship Beween Cash, Asse and Vanilla The simple equaion of payoffs, ( wt KvT ) ( ST K) φ = max φ,, leads o he formula (,,,,,,, ) (,,,,,,, ) (,,,,,,, ) vanilla x K T σ r q φ =φ w x K T σ r q φ Kv x K T σ r q φ. 5.6 Sensiiviies Firs, he sensiiviies are given direcly from Wysup (999) for he binary opions, where ( T,,,,,, ) (,,,,,, ) v S K T σ, r q φ = V = Xe rt cash or nohing ( ) ( ) ( φd ) qt w ST K T σ, r q φ = Wasse or nohing = STe N φd N and for a call opion, φ= for a pu opion. Dela v S T ( ) n( d ) T w ( ) qt n d rt ( ) =φ e + e N( φd S σ T T =φxe rt S σ T ) Gamma rt = φxe T T ( ) n( d) d T ( ) n( d) d v S S σ T w qt = φxe S S σ T T 96

204 Thea where v rt φn d = Xe rn ( φ d ) + d ( T ) d ( T ) w qt φn d = Se T qn ( φ d ) + S ln σ r q T K + d = σ T d = d σ T. Vega v = φ Xe σ w = φ Se T σ ( ) rt ( ) qt n d nd d σ d σ Rho ( ) φ ( ) qt = Se T v rt ( ) φn d T = Xe ( T ) N ( φ d ) + r σ w r n d T σ 97

205 Figure 44: The value, dela, gamma, vega, hea and rho for an cash-or-nohing call opion wih he parameers: S T =, K = 95, r = 5%, q = 3%, T = 365, σ = %. 98

206 Figure 45: The value, dela, gamma, vega, hea and rho for an asse-or-nohing pu opion wih he parameers: S T =, K = 5, r = 5%, q = 3%, T = 365, σ = %. 99