Mathematical Modeling of Financial Derivative Pricing

Transcription

1 Unversty of Connectcut Honors Scolar Teses Honors Scolar Program Sprng Matematcal Modelng of Fnancal Dervatve Prcng Kelly L. Cosgrove Unversty of Connectcut, Follow ts and addtonal works at: ttp://dgtalcommons.uconn.edu/sronors_teses Part of te Fnance and Fnancal Management Commons, and te Oter Appled Matematcs Commons Recommended Ctaton Cosgrove, Kelly L., "Matematcal Modelng of Fnancal Dervatve Prcng" (207). Honors Scolar Teses. 55. ttp://dgtalcommons.uconn.edu/sronors_teses/55

2 Matematcal Modelng of Fnancal Dervatve Prcng Kelly Cosgrove May 6, 207

3 Abstract Te bnomal asset-prcng model s used to prce fnancal dervatve securtes. Ts text wll begn by gong over te probablty concepts necessary to understand ts dscrete-tme model. It ten develops te teory bend te bnomal model and dfferent propertes tat arse. It sows ow to use te bnomal model to predct future stock prces, and ten uses ts nformaton to prce dervatve securtes. It ntally focuses on te European call opton, but goes on to provde a prcng metod for te Amercan put opton. However, many of te teorems developed are applcable to all dervatve securtes. Te text wraps up by consderng a dfferent metod used n prcng dervatve securtes, te Black-Scoles model, wc s based on contnuous-tme concepts.

4 Contents Probablty Teory 3. Fnte Probablty Spaces Random Varables Martngales and Markov Processes Te Bnomal Model 7 2. Structure Prcng Dervatves Propertes Applcaton to Amercan Dervatve Securtes 5 3. Introducton Pat-Independent Stoppng Tmes Pat-Dependent Te Black-Scoles Model Random Walk Te Black-Scoles Model... 28

5 Introducton Te bnomal asset-prcng model s used to prce fnancal dervatve securtes. In ts text, we wll mostly use te example of te European call opton to llustrate te functon te bnomal model serves. Ts type of dervatve s one tat allows ts owner te rgt (but not te oblgaton) to buy astockataspecfedstrkeprceonaspecfedexpratondate. AsmlardervatvesteEuropean put opton wc gves te owner te rgt to sell stock at a specfc prce on a specfc date. We begn by revewng general probablty concepts needed to develop te bnomal asset-prcng model n Capter. Topcs nclude fnte probablty spaces, random varables, and propertes of condtonal expectatons. Capter ten goes on to develop te propertes of certan adapted stocastc processes. Namely, we sall look at martngales and Markov processes. We develop te dea of te bnomal asset-prcng model and ow to use t to prce European dervatve securtes n Capter 2. In order to accompls ts, we wll need to fgure out ow to replcate te dervatve securty n te stock and money markets. Once we are able to do ts, we can determne a far prce for te dervatve securty by settng ts value at tme n equal to tat of our replcated portfolo. In Capter 3 we wll explore Amercan dervatve securtes wc are defned smlarly to ter European counterparts except for te fact tat ts owner as te rgt to exercse te opton at any pont up to or on te expraton date. Ts complcates te process of buldng a replcatng portfolo because tere wll exst an optmal exercse date on wc te owner of te dervatve securty sould exercse t (wc we must fnd). Te concept of stoppng tmes wll be developed n ts capter and wll be crucal to fgurng out te optmal exercse date of an Amercan dervatve securty. Capter 4 wraps up te paper by ntroducng an alternate metod for prcng dervatve securtes known as te Black-Scoles model. It begns by descrbng te random walk, wose contnuous-tme counterpart, Brownan moton, s an underlyng assumpton of te Black-Scoles model. It ten provdes te general dea bend Black-Scoles and ow t s related to te bnomal model. Several sources were referenced n order to complete ts paper, but Steven E. Sreve s Stocastc Calculus for Fnance I Te Bnomal Asset Prcng Model [Sreve, 2004] wastemostpromnently referenced, as te source of te materal n Capters, 2, and 3, and Secton 4.. Furter, Rosstsa Yalamova s Smple eurstc approac to ntroducton of te Black-Scoles model [Yalamova, 200] was referenced to complete Secton 4.2, and Noga Alon and Joel H. Spencer s Te Probablstc Metod [Alon and Spencer, 205] was referenced for Teorem 4.2. (Azuma s Inequalty). 2

6 Capter Probablty Teory. Fnte Probablty Spaces Because te bnomal asset-prcng model s an applcaton of probablty teory, t s crucal to ave a good understandng of general probablty concepts n order to understand te model tself. In ts secton we wll buld up our understandng of a fnte probablty space. We wll use te example of a con toss to llustrate te components of a fnte probablty space. Say we toss a con 3 tmes. Let H denote a eads flp and T denote a tals flp. Te sample space,, stefntesetofallpossbleoutcomestatcouldresultfromtossngtatcon,or ={HHH,HHT,HTH,THH,HTT,THT,TTH,TTT} (.) Sequences of outcomes are denoted by! =!! 2! 3. We call subsets of events. Forexample,teeventAtatwegetatleasttwotals: A = {HTT,THT,TTH,TTT} If we let #T(!! 2! 3 )bedefnedastenumberoftalstatappearnoursequenceof3contosses, ten we can rewrte A as: A = {! 2 ; #T (!! 2! 3 ) 2} (.2) Fnally, we must determne te lkelood of eac sequence occurrng. We can do ts by lettng p be te probablty of te con landng on eads and q = p be te probablty of tals. Ten, te probabltes of eac! occurrng are P(HHH)=p 3, P(HHT)=p 2 q, P(HTH)=p 2 q, P(THH)=p 2 q, P(HTT)=pq 2, P(THT)=pq 2, P(TTH)=pq 2, P(TTT)=q 3 (.3) Furter, we can fgure out te probablty of an event by addng up te ndvdual probabltes of eac possble sequence n te event: Note tat P!2 P(!) =,orp( ) =. P(A) =P(HTT)+P(THT)+P(TTH)+P(TTT) = pq 2 + pq 2 + pq 2 + q 3 (.4) Defnton... A fnte probablty space conssts of a sample space and a probablty measure P tat takes eac element! 2 and assgns t a value n te nterval [0,]. We denote t by (, P). 3

7 .2 Random Varables Te fnte probablty space models a stuaton n wc a random experment s conducted. Tese experments typcally produce numercal data wc can be wrtten as random varables. Defnton.2.. Let (, P) be a fnte probablty space. A random varable s a functon tat maps onto R. It s mportant to note tat random varables temselves do not depend on te probablty measure P. Te dstrbuton of a random varable uses P to determne te probabltes of te random varable takng dfferent values. We can keep our sample space and random varable functon consstent between two fnte probablty spaces, but f ter probablty measures dffer, te same random varable can ave two dfferent dstrbutons. Ts s llustrated n te example below. Example.2.2. Suppose we toss a con 3 tmes suc tat = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Now defne te random varable X to be te total number of eads and Y to be te total number of tals. Terefore, X(HHH)=3 X(HHT)=X(HTH)=X(THH)=2 X(HTT)=X(THT)=X(TTH)= X(TTT)=0 (.5) Y(HHH)=0 Y(HHT)=Y(HTH)=Y(THH)= (.6) Y(HTT)=Y(THT)=Y(TTH)=2 Y(TTT)=3 We ave not yet set te probablty measure P for our fnte probablty space and we were stll able to determne te varous values for our random varables X and Y. Now, let us specfy P suc tat te probablty of gettng eads s 2 (and terefore te probablty of gettng tals s 2 = 2 ). Ten, P (!) = = 8 for all! 2. Let P{X = } := P{! 2 ;X(!) =}. Te dstrbuton would be as follows: P {X =3} = P {HHH} = 8 P {X =2} = P {HHT,HTH,THH} = 3 8 (.7) P {X =} = P {HTT,THT,TTH} = 3 8 P {X =0} = P {TTT} = 8 Now let us use a dfferent P, sayp 2,suctatteprobabltyofgettngeadss 2 3 of gettng tals s 3. By smlar calculatons, we get te followng dstrbuton: P 2 {X =3} = P 2 {HHH} = 8 27 P 2 {X =2} = P 2 {HHT,HTH,THH} = 2 27 P 2 {X =} = P 2 {HTT,THT,TTH} = 6 27 P 2 {X =0} = P 2 {TTT} = 27 and te probablty (.8) 4

8 Clearly tese dstrbutons dffer, even toug we are measurng te same random varable. Knowng te probablty dstrbuton of a random varable, we can calculate a sngle value of wat we expect te result of our random experment to be f we were to actually conduct t. Defnton.2.3. Let X be a random varable defned on a fnte probablty space (, P). Te expected value, E, of X s defned as EX = X!2 X(!)P(!) (.9) Te varance of X s Var(X) =E(X EX 2 ) (.0) Usng our sample space (, P 2 ), we can expect te number of eads tat sow up after tree con tosses to be EX = X!2 X(!)P 2 (!) =X(HHH)P 2 {HHH} + X(HHT)P 2 {HHT} + X(HTH)P 2 {HTH}+ X(THH)P 2 {THH} + X(HTT)P 2 {HTT} + X(THT)P 2 {THT}+ X(TTH)P 2 {TTH} + X(TTT)P 2 {TTT} = =2 (.) Ts agrees wt our ntuton tat f we ave a 2 3 probablty of gettng eads, ten every 3 tosses, we sould expect to see 2 eads. Ts s te best estmate f we are provded wt no addtonal nformaton. However, say we cange te scenaro so tat we ave already flpped te con two tmes, and bot tmes ave turned up tals! Surely, gettng two eads by our trd toss would be mpossble. We can adjust our expected value usng te nformaton about te frst two tosses tat we now ave. Ts s called te condtonal expectaton of X based on te nformaton we ave. Defnton.2.4. Gven n con tosses suc tat apple n apple N, tere are 2 N n possble contnuatons! n+...! N of te sequence!...! n. Let p be te probablty of gettng eads and q = - p be te probablty of gettng tals. Let #H(! n+...! N ) be te number of eads n! n+...! N and #T (! n+...! N ) be te number of tals n! n+...! N. Ten, our condtonal expectaton of X based on te nformaton at tme n s E n [X](!...! n ) = X! n+...! N p #H(!n+...!N ) q #T (!n+...!n ) X(!...! n! n+...! N ) (.2) Teorem.2.5 (Fundamental propertes of condtonal expectatons). Let N be a postve nteger and let X, Y be random varables dependng on te frst N con tosses. Let 0 apple n apple N be gven. Ten, te followng propertes old. () Lnearty of condtonal expectatons. For all constants c and c 2, E n [c X + c 2 Y ]=c E n [X]+c 2 E n [Y ] (.3) () Takng out wat s known. If X actually depends on te frst n con tosses, ten E n [XY ]=X E n [Y ] (.4) 5

9 () Iterated condtonng. If0apple n apple m apple N, ten E n [E m [X]] = E n [X] (.5) (v) Independence. If X depends only on tosses n + troug N, ten E n [X] =E[X] (.6) Te proof of ts teorem results largely from te defnton of condtonal expectaton and wll be left to te reader to work troug f desred..3 Martngales and Markov Processes Defnton.3.. Consder te con toss scenaro. Let M 0,M,...,M N be a sequence of random varables suc tat eac M n,0applenapplen, depends only on te frst n con tosses. Ten, we call ts sequence an adapted stocastc process. We can furter classfy adapted stocastc processes by ow we expect tem to cange from one con toss to te next. Gven M n,andcalculatngm n+ usng our defnton of condtonal expectaton, we wll fnd tat certan adapted stocastc processes can be expected to rse, oters can be expected to fall, and oters can be expected to reman constant. Suc classfcatons are defned formally below. Defnton.3.2. Let te sequence of random varables M 0,M,...,M N be an adapted stocastc process. If M n = E n [M n+ ],n=0,,...,n If M n apple E n [M n+ ],n=0,,...,n, ts process s a martngale., ts process s a submartngale. If M n E n [M n+ ],n=0,,...,n, ts process s a submartngale. Toug te martngale property s "one-step-aead" by only specfyng te expectaton of te mmedate next varable n te sequence, we can extend te property for any varable appearng after M n usng te terated condtonng property of condtonal expectaton. For example, f te sequence M 0,M,...,M N s a martngale and n apple N 2, weknowtat Applyng te terated condtonng property, we see tat And snce M n = E n [M n+ ], M n+ = E n+ [M n+2 ]. (.7) E n [M n+ ]=E n [E n+ [M n+2 ]] = E n [M n+2 ]. (.8) M n = E n [M n+2 ]. (.9) Te noton of condtonal expectatons gve rse to many algortms tat serve as powerful predctve tools n dfferent scenaros. But, f an adapted stocastc process s pat ndependent; namely, f a varable n te sequence only depends on te mmedate former varable, often tmes tese algortms can be greatly smplfed. Ts type of process s known as a Markov process. Defnton.3.3. Let X 0,X,...,X N be an adapted stocastc process. If for 0 apple n apple N exsts a functon g(x) for every f(x) dependng on n and f suc tat We call X 0,X,...,X N a Markov process. tere E n [f(x n+ )] = g(x n ) (.20) Te martngale s a specal case of (.20) wtf(x) =x and g(x) =x. Butsnceteremustexst a g(x) for every f(x) suc tat equaton (.20) oldsnorderforaprocesstobemarkov,notevery martngale s a Markov process. Of course, snce g(x) does not ave to equal x, noteverymarkov process s martngale. 6

10 Capter 2 Te Bnomal Model 2. Structure S 3 (HHH)=u 3 S 0 S 2 (HH)=u 2 S 0 S (H) =us 0 S 3 (HHT)=S 3 (HTH) = S 3 (THH)=u 2 ds 0 S 0 S 2 (HT)=S 2 (TH)=udS 0 S (T )=ds 0 S 3 (HTT)=S 3 (THT) = S 3 (TTH)=ud 2 S 0 S 2 (TT)=d 2 S 0 S 3 (TTT)=d 3 S 0 Atree-perodbnomalmodel. Te bnomal model above dsplays te possble pats of a stock prce, wt S 0 > 0 beng te ntal prce per sare at tme t =0. Te model s broken up nto perods, wt tme t =beng te end of te frst perod, t =2te end of te second, and t =3te end of te trd. We extend our con toss scenaro onto ts model and let eads represent te stock prce ncreasng and tals represent te stock prce decreasng. At te end of eac perod te stock can take on two possble values, one of wc s greater tan ts prevous value and one of wc s less. Here we can ntroduce two new varables, u and d, tat serve as te two possble ratos of te new stock prce to te former stock prce. If we ave a total of N perods, ten for eac n, <napple N and for every possble!...! n, u = S n(!...! n H) S n (!...! n ), d = S n(!...! n T ) S n (!...! n ) (2.) 7

11 And wen n =, u = S (H) S 0, d = S (T ) S 0 (2.2) Clearly, we wll ave u> and 0 <d<. It s mportant to note tat stock prces can never be negatve. Multplyng bot sdes by te denomnator wll result n te equatons gven n te bnomal model. Snce eac S n s a random varable tat depends on te frst n con tosses, S 0,S,...,S N s an adapted stocastc process. We stll use te probabltes p and q = p for tossng eads and tals, respectvely. We wll also ntroduce a varable r to be te money market nterest rate. If we nvest one dollar n te money market at tme t =0, we wll get + r dollars back at tme t =. If we borrow one dollar from te money market at tme t =0, we wll owe + r dollars at tme t =. We use te bnomal asset-prcng model to prce dervatves by replcatng tem n te stock and money markets, so we must assume tat 0 <d<+r<un order to avod arbtrage. Defnton 2... Arbtrage s a tradng strategy by wc partcpants start wt no money, ave a postve probablty of makng money, and ave zero probablty of losng money. We must be careful to avod arbtrage n our model because oterwse, analyzng t would result n nonsenscal conclusons. Toug money markets n real lfe sometmes ave arbtrage, t s always quckly dscovered and solved troug tradng. Assumng a no-arbtrage model results n te nequaltes 0 <d< + r<ufor te followng reasons: d<+r. Oterwse, one could start wt zero dollars, borrow from te money market, and guarantee tey make at least wat tey wll owe n nterest by nvestng n te stock. Tere would be no cance of losng money, ence tere would be arbtrage. +r<u. Oterwse, one could sell te stock sort and nvest te money n te money market. Ten, wen te securty expres, t wll cost at most wat was made from te money market nvestment to replace te stock. Agan, tere would be no cance of losng money, and tere would be arbtrage. It s common, but not necessary, to ave d and u suc tat d = u. 2.2 Prcng Dervatves In order to ensure no arbtrage wen prcng our dervatves, tere exsts an arbtrage prcng teory by replcatng t troug tradng n te stock and money markets. Frstly, one sould assgn a prce to te securty to prevent te possblty of arbtrage. Secondly, for an expraton tme N, atanytmen <N one sould be able to magne sellng te dervatve for a prce and nvest tat money n te stock and money markets suc tat te value of ts portfolo at tme N matces te payoff of te securty. Let s go troug te process usng te example of a European call opton tat expres at tme t =2. We defne K to be te strke prce at wc te owner may buy one stock of te sare at te expraton date. Terefore, te payoff of te deal wll be V 2 = (S 2 K) +, were (A) + means te maxmum value of te set (0,A). WearetryngtodetermneV 0 wc s te no-arbtrage prce for te opton at tme t =0. Now we suppose an agent sells te opton for V 0 at tme t =0and buys 0 sares of stock (prced at S 0 per sare). Se nvests V 0 0 S 0 dollars n te money market. Ts wll be a negatve quantty, taken to mean tat se as borrowed 0S 0 V 0 dollars from te money market to fnance er stock purcase. At tme t =,teagentasaportfolovaluedat X = 0 S +(+r)(v 0 0 S 0 ) (2.3) 8

12 But, because te value of S depends on te outcome of te frst con toss, we really ave two equatons: X (H) = 0 S (H)+(+r)(V 0 0 S 0 ) (2.4) X (T )= 0 S (T )+(+r)(v 0 0 S 0 ) (2.5) Te agent can adjust er portfolo based on te outcome of te frst con toss. Se now decdes to old sares of stock and nvests X S n te money market. Because we are replcatng te opton, we want te portfolo valued at V 2. In oter words, we want V 2 = S 2 +(+r)(x S ) (2.6) But, S 2 and V 2 depend on te frst two con tosses so we really ave four equatons: V 2 (HH)= (H)S 2 (HH)+(+r)(X (H) (H)S (HH)) (2.7) V 2 (HT)= (H)S 2 (HT)+(+r)(X (H) (H)S (HT)) (2.8) V 2 (TH)= (T )S 2 (TH)+(+r)(X (T ) (T )S (TH)) (2.9) V 2 (TT)= (T )S 2 (TT)+(+r)(X (T ) (T )S (TT)) (2.0) Now, we ave sx equatons and sx unknowns: V 0, 0, (H), (T ), X (H), X (T ). Let s frst fnd X (T ) and (T ) by lookng at equatons (2.9) and (2.0). We can easly solve for (T ) by subtractng (2.0) from (2.9) and solvng for t. Te soluton we obtan s called te delta-edgng formula. (T )= V 2(TH) V 2 (TT) (2.) S 2 (TH) S 2 (TT) Next we solve for X (T ). Snce S 2 (TH), S 2 (TT), S (TH), ands (TT) are random varables, we know tat f our frst con toss s tals, we ave some probablty p and q = p probablty of gettng tals. In oter words, we ave p probablty of equaton (2.9) and q probablty of equaton (2.0). In order to fnd te expected value of X (T ), wemultply(2.9)by p and (2.0) q and add tem togeter. We are also gong to dvde all terms by ( + r). pv 2 (TH)+ qv 2 (TT) = (T ) +r +r [ ps 2(TH)+ qs 2 (TT)] ( p q)s (T ) +( p q)x (T ) (2.2) Snce ( p + q) =,wecansmplfy(2.2)toget pv 2 (TH)+ qv 2 (TT) = (T ) +r +r [ ps 2(TH)+ qs 2 (TT)] S (T ) + X (T ) (2.3) If we coose p suc tat S (T )= +r [ ps 2(TH)+ qs 2 (TT)] (2.4) We can greatly smplfy (2.3). Use S (T )=ds 0, S 2 (TH)=duS 0,andS 2 (TT)=ddS 0,and q = p. Ten, (2.4) becomes ds 0 = +r [ pdus 0 +( p)dds 0 ] (2.5) Dvde bot sdes by ds 0 to get = [ pu +( p)d] (2.6) +r 9

13 And from tere we can easly solve for p. Smlarly,wecansolve(2.4)for q. p = +r d u d, q = u r u d Tese probabltes p and q are called rsk neutral probabltes. Normally, te average growt rate of a stock exceeds tat of te money market; oterwse, t would not make sense to rsk nvestng n stock. So, te actual probabltes p and q of a gven stock sould satsfy ( + r)s (T ) <ps 2 (TH)+qS 2 (TT) We nstead cose p and q to satsfy (2.4) to assst us n our calculatons. Upon smplfyng equaton (2.3) to reflect our strategc coce of p and q, wegottevaluete replcatng portfolo sould ave at tme t =f te stock prce goes down (our con toss results n tals): X (T )= pv 2 (TH)+ qv 2 (TT) (2.7) +r In terms of our stock opton, we defne ts to be te prce of te opton at tme t= f te frst con toss results n tal, denotedbyv (T ). It s an nstance of te rsk-neutral prcng formula, wc wll be formally defned later n ts capter. V (T )= pv 2 (TH)+ qv 2 (TT) +r (2.8) By gong troug a smlar procedure wt equatons (2.7) and (2.8), we come up wt te followng two equatons for (H) and V (H) =X (H): (H) = V 2(HH) S 2 (HH) V 2 (HT) S 2 (HT) V (H) = pv 2 (HH)+ qv 2 (HT) +r (2.9) (2.20) Te latter equaton s te prce of te opton at tme t= f te frst toss results n ead. Fnally, we can solve for V 0 and 0 by pluggng our values for X (H) =V (H) and X (T )=V (T ) nto equatons (2.4) and (2.5). Clearly, we ave tree stocastc processes: ( 0, ), (X 0,X,X 2 ),and(v 0,V,V 2 ).Werecursvely defned our portfolo, and f we specfy values for X 0, 0, (H), and (T ), wecanrecursvelydefne a replcatng portfolo wt any number of perods by te wealt equaton X n+ = n S n+ +(+r)(x n n S n ) (2.2) and we can prce our stock opton n a smlar manner as we dd for te two-perod model. Teorem 2.2. (Replcaton n te multperod bnomal model). Consder an N-perod bnomal asset-prcng model wt 0 < d < + r < u and wt p = +r d u d, q = u r u d (2.22) Let V N be a random varable (a dervatve securty payng off at tme N) dependng on te frst N con tosses!! 2...! N. Defne recursvely backward n tme te sequence of random varables V N,V N 2,...V 0,by V n (!! 2...! n )= +r [ pv n+(!! 2...! n H)+ qv n+ (!! 2...! n T )] (2.23) 0

14 suc tat eac V n depends on te frst n con tosses!! 2...! n, 0 apple n apple N n(!! 2...! n )= V n+(!! 2...! n H) V n+ (!! 2...! n T ) S n+ (!! 2...! n H) S n+ (!! 2...! n T ). Next, defne (2.24) were agan 0 apple n apple N. If we set X 0 = V 0 and defne recursvely forward n tme te portfolo values X,X 2,...,X N by te wealt equaton, ten we wll ave for all!! 2...! N. Proof. We wll use a proof of nducton on n tat X N (!! 2...! N )=V N (!! 2...! N ) (2.25) X n (!! 2...! n )=V n (!! 2...! n ) (2.26) for all!! 2...! n, 0 apple n apple N Base Case: n = 0 Te base case s gven by te defnton X 0 = V 0. Inductve Step Assume te teorem olds for some n<n.wewstosowtattalsooldsforn+. We know by te ypotess tat X n (!! 2...! n )=V n (!! 2...! n ). Now, consder!! 2...! n+ to be fxed and arbtrary. We do not know weter! n+ s H or T, so we must consder bot cases. However, te steps n bot cases are very smlar, so we wll only prove te case wen! n+ := H. From te wealt equaton, we know X n+ (!! 2...! n H)= n (!! 2...! n )us n (!! 2...! n )+ ( + r) X n (!! 2...! n ) n(!! 2...! n )S n (!! 2...! n ) For smplfcaton purposes, suppress te sequence!! 2...! n so ts equaton becomes X n+ (H) = n us n +(+r)(x n n S n ) (2.27) From equaton (2.24), and usng our smplfed notaton, we know tat We may rewrte ts as n = V n+(h) V n+ (T ) S n+ (H) S n+ (T ) (2.28) n = V n+(h) V n+ (T ) (u d)s n (2.29) Now, we can rewrte equaton (2.27) and substtute ts n for n, along wt usng our ypotess tat V n = X n : X n+ (H) =(+r)x n + n S n (u ( + r)) =(+r)v n + (V n+(h) V n+ (T ))(u ( + r) u d We can also use equaton (2.22) to substtute q nto te equaton: X n+ (H) =(+r)v n + qv n+ (H) qv n+ (T ) = pv n+ (H)+ qv n+ (T )+ qv n+ (H) qv n+ (T ) = V n+ (H) A smlar argument sows tat X n+ (T )=V n+ (T ). So, no matter wat! n+ s, we ave Te nducton step s complete. X n+ (!! 2...! n+ )=V n+ (!! 2...! n+ ) Ts teorem was prefaced wt te example of a European call opton for wc te payoff only depends on te fnal stock prce. Te teorem also apples to pat dependent optons wose payoff depends on te dfferent values te stock takes on between ts ntal value and ts fnal value. We wll explore te pat dependent example of Amercan dervatves n Capter 3.

15 2.3 Propertes In te prevous secton we came up wt rsk-neutral probabltes p and q. Wecandefnetese probabltes as te probablty measure P. Smlarly, our rsk-neutral expected value for random varable X under tese rsk-neutral probabltes s ẼX. Recalltat It s easy to ceck tat p = +r d u d, q = u r u d pu + qd +r = So, for all n and for every con toss sequence!...! n, we can wrte S n (!...! n )= ps n+ (!...! n H)+ qs n+ (!...! n T ) +r (2.30) But, usng our knowledge of condtonal expectaton from Defnton.2.4, we can rewrte te te above equaton as follows: S n = +r Ẽn[S n+ ] (2.3) If we dvde bot sdes of (2.3) by ( + r) n,wegetteequaton were S n (+r) n S n ( + r) n = S n+ Ẽn ( + r) n n + (2.32) s te dscounted stock prce at tme n. Tsequatonexpressesakeyfacttatunderte rsk-neutral probablty measure P, te best estmate of te dscounted stock prce at tme n + s te dscounted stock prce at tme n. In oter words, te dscounted stock prce process s a martngale. Teorem Consder te general bnomal model wt 0 < d < + r < u. Under te rsk neutral measure, te dscounted stock prce s a martngale,.e. equaton (2.32) olds at every tme n and for every sequence of con tosses. By defnton of P, our expected rate of growt of a stock s equal to te nterest rate, r, ofte money market (as was defned troug equaton (2.4)). Because of ts, no matter ow an agent dvdes up te wealt of ter replcatng portfolo between te stock and te money market, t wll ave an average rate of growt r. Ts s restated n te teorem below. Teorem Consder te bnomal model wt N perods. Let 0,,..., N be an adapted portfolo process, let X 0 be a real number, and let X,...,X N be generated recursvely by te wealt equaton, X n (+r) n,n=0,,...,n s a martngale under te rsk-neutral mea- Ten te dscounted wealt process sure. In oter words, X n+ = n S n+ +(+r)(x n n S n ) n =0,,...,N X n ( + r) n = Ẽn X n+ ( + r) n+, n =0,,...,N (2.33) 2

16 Proof. Ẽ n X n+ ( + r) n+ = Ẽn = Ẽn = Ẽn ns n+ +(+r)(x n n S n ) ( + r) n+ ns n+ ( + r) n+ + X n ns n ( + r) n ns n+ Xn ( + r) n+ + Ẽn = n Ẽ n S n+ ( + r) n+ n S n ( + r) n (Wealt equaton) (Lnearty) + X n ns n ( + r) n (Takng out wat s known) = n S n ( + r) n + X n ns n ( + r) n (Teorem 2.3.) S n S n = n ( + r) n n ( + r) n + X n ( + r) n = X n ( + r) n X n (+r) n Te expresson s called te P-martngale. Because of te terated condtonng property, we can nfer from Teorem tat Ẽ n X n ( + r) n = X 0, n =0,,...,N (2.34) A consequence of ts s known as te frst fundamental teorem of asset prcng. If we ave X 0 =0, ten Ẽn Xn (+r) =0and terefore ẼnX n n =0. Terefore, tere s no arbtrage n te asset-prcng model f we can fnd a rsk neutral measure n t. Anoter consequence of Teorem s te Rsk-Neutral Prcng Formula, statedbelowasateorem. Teorem (Rsk-Neutral Prcng Formula). Consder an N-perod bnomal asset-prcng model wt 0 <d<+r<uand wt rsk-neutral probablty measure P. Let V N be a random varable (a dervatve securty payng off at tme N dependng on con tosses. For 0 apple n apple N, te prce of te dervatve securty at tme n s gven by te rsk-neutral prcng formula V N V n = Ẽn ( + r) N n Furtermore, te dscounted prce of te dervatve securty s a martngale under P,.e. (2.35) V n ( + r) n = V n+ Ẽn ( + r) n+, n =0,,...,N (2.36) In addton to te dscounted prce of a dervatve securty beng a martngale, we can sow tat te prce process of a dervatve securty s Markov. Recall Defnton.3.3. If we can fnd a functon g(x) for every f(x) and for every n between 0 and N suc tat Ẽ[f(X n+)] = g(x n ),tenx 0,X,...,X N s a Markov process. We wll frst sow tat te stock prce process s Markov. Example (Stock Prce). We know we can compute te stock prce at tme n + usng te followng formula usn (! S n+ (!...! n! n+ )=...! n ) ds n (!...! n ) f! n+ = H f! n+ = T (2.37) 3

17 Terefore, E n [f(s n+ )](!...! n )=pf[us n (!...! n )] + qf[ds n (!...! n )] = g[s n (!...! n )] (2.38) Were g(x) s defned by g(x) =pf(ux)+qf(dx). Hence, we ave an algortm for fndng g(x) for every f(x) and te stock prce process s Markov. And, t s Markov under te rsk-neutral probablty measure or te actual probablty measure. We know tat te payoff of a dervatve securty at tme N s a functon v N of te stock prce at tme N. In oter words, V N = v N (S N ). If we take equaton (2.36) anddvdebotsdesby( + r) n, we get V n = +r Ẽn[V n+ ], n =0,,...,N (2.39) Snce te stock prce process s Markov, we can defne V N for some functon v N.Smlarly,wecandefneV N 2 : as V N = +r ẼN [v N (S N )] = v N (S N ) (2.40) V N 2 = +r ẼN 2[v N (S N )] = v N 2 (S N 2 ) (2.4) In general, we can defne recursvely backwards te functon v n suc tat V n = v n (S n ) by te algortm v n (s) = pv n+ (us)+ qv n+ (ds), n = N,N 2,...,0 (2.42) +r Ts sows tat te prce process of any dervatve securty s Markov under te rsk-neutral probablty measure. We can restate our fndngs as a teorem. Teorem Let X 0,X,...,X N be a Markov process under te rsk-neutral probablty measure P n te bnomal model. Let v N (x) be a functon of te dummy varable x, and consder a dervatve securty wose payoff at tme N s v N (X N ). Ten, for eac n between 0 and N, te prce of V n of ts dervatve securty s some functon v n of X n,.e. V n = v n (X n ), n =0,,...,N (2.43) Tere s a recursve algortm for computng v n wose exact formula depends on te underlyng Markov process X 0,X,...,X N. 4

18 Capter 3 Applcaton to Amercan Dervatve Securtes 3. Introducton So far, we ave only looked at European dervatve securtes of wc te owner can only coose to exercse on a gven expraton date. Ts capter dscusses Amercan dervatve securtes tat can be exercsed at any pont on or before te expraton date. Ts means tat tere wll exst an optmal exercse date, after wc te dervatve securty wll tend to lose value. As a result, te dscounted prce process of Amercan dervatve securtes s a supermartngale (unlke te European case, for wc te dscounted prce process s a martngale). We wll start by lookng at tose Amercan dervatve securtes tat are not pat dependent and ten move on to tose tat are pat dependent. We wll fnd tat we can stll come up wt a prcng algortm by replcatng te dervatve usng te stock and money markets muc lke we dd wt European dervatve securtes. 3.2 Pat-Independent Recall te prcng algortm for a European dervatve securty. We use an N-perod bnomal model wt up factor u, down factor d, andnterestrater suc tat 0 <d<+r<u. Let g(s N ) be te functon tat tells us te payoff of te dervatve securty at tme N. As was stated n Teorem 2.3.5, we can wrte te value V n of ts dervatve securty as a functon v n of te prce of te stock. Usng equaton (2.42) andusngtersk-neutralmeasure P, wecandefnev n recursvely backwards: v N (s) =[g(s)] +, v n (s) = pv n+ (us)+ qv n+ (ds), +r n = N,N 2,...,0 We also defne te replcatng portfolo usng equaton (2.24) to be (3.) n = v n+(us n ) v n+ (ds n ) (u d)s n, n =0,,...,N (3.2) For an Amercan dervatve securty, we wll stll ave a payoff functon g(s). But, snce te owner of te securty can coose to exercse t at any tme up to te expraton date and receve payment g(s n ),weneedtomakesureourreplcatngportfoloalways as a value X n of at least g(s n ) for n apple N,.e. X n g(s n ), n =0,,...,N (3.3) 5

19 Te functon g(s n ) tells us te ntrnsc value of te dervatve securty at tme n. Terefore, te dervatve securty wll always ave a value of at least g(s n ),andsncetevalueoftereplcatng portfolo must matc tat of te dervatve securty, t must also ave a value of at least g(s n ). Our defnton of v n s modfed n te case of te Amercan dervatve securty to account for ts. v N (s) =[g(s)] +, " v n (s) =max g(s), pv n+ (us)+ qv n+ (ds) #, n = N,N 2,...,0 +r (3.4) wt V n = v n (S n ) beng te dervatve securty prce at tme n. Example 3.2. (Amercan put opton). Let te followng bnomal model represent te possble stock prces of a gven stock exprng at tme t =2: S 2 (HH) = 6 S (H) =8 S 0 =4 S 2 (HT)=S 2 (TH)=4 S (T )=2 S 2 (TT)= Atwo-perodbnomalmodel. Consder an Amercan put opton. Let r =4,meanngourrsk-neutralprobabltesare p = q = 2. Also let te strke prce of te opton be 5, meanng te owner wll receve (5 S n ) f exercsed at tme n. So,g(s) =5 s, andweuseequaton(3.4) todefnetefunctonv n as v 2 (s) =[5 s] +, " v n (s) =max 5 s, v n+(2s)+ 2 v n+( # 2 s), n =, 0 (3.5) Pluggng n our values for te stock prce at tme n =2,, 0, tsgvesus v 2 (6) = [5 6] + =0, v 2 (4) = [5 4] + =, v 2 () = [5 ] + =4, v (8) = max 5 8, 2 5 (0 + ) = max( 3, 0.40) = 0.40 v (2) = max 5 2, 2 5 ( + 4) = max(3, 2) = 3 v 0 (4) = max 5 4, 2 5 ( ) = max(,.36) =.36 Had we calculated te values of v n for te European equvalent of ts put opton, we would get v (2) = 2 nstead of 3 and v 0 (4) = 2 5 ( ) = 0.96 nstead of.36. 6

20 Next, we construct te replcatng portfolo usng equaton (3.2), startng wt ntal captal v 0 = = v (us 0 ) v (ds 0 ) = v (8) v (2) = 0.43 (3.6) (u d)s We can verfy ts s correct by ceckng tat X (H) =v (S (H)) and X (T )=v (S (T )) usng te wealt equaton (2.2): X (H) = 0 S (H)+(+ 4 )(X 0 0S 0 ) X (T )= 0 S (T )+(+ 4 )(X 0 0S 0 ) =( 0.43)(8) ( ) = ( 0.43)(2) + 5 ( ) 4 =( 3.44) (3.08) = ( 0.86) (3.08) =0.40 = 3 = v (S (H)) = v (S (T )) (3.7) So, no matter wat te result of our con toss, te value of our replcatng portfolo at tme wll matc tat of te put opton f we were to sell t at tme. Te owner could coose to exercse te opton at tme t =, but let s assume oterwse. Frst, we consder te scenaro were te frst con toss results n tals. Te rsk-neutral prcng formula tells us tat at tme t =,wewantourportfolotobevaluedat X = V = Ẽ V2 +r = v 2(4) + 2 v 2() =2 Snce our portfolo s currently valued at 3 but we only need te value of our portfolo to be 2, we can consume a dollar and nvest te rest. We agan use equaton (3.2) tocalculatetevalueof (T ): (T )= v 2(uS (T )) v 2 (ds (T )) (u d)s (T ) = v 2(4) v 2 () 4 = We once agan verfy ts value of (T ) s correct by usng te wealt equaton to ceck f X 2 (TH)= v 2 (S 2 (TH)) and X 2 (TT)=v 2 (S 2 (TT)) X 2 (TH)= (T )S 2 (TH)+(+ 4 )(X (T ) (T )S (T )) =( )(4) + 5 (2 + 2) 4 =( 4) (4) = = v 2 (S 2 (TH)) X 2 (TT)= (T )S 2 (TT)+(+ 4 )(X (T ) (T )S (T )) (3.8) =( )() + 5 (2 + 2) 4 =( ) (4) =4 = v 2 (S 2 (TT)) 7

21 Fnally, we consder te scenaro were te frst con toss results n eads. Wt X (H) =0.40, we frst fnd (H): (H) = v 2(uS (H)) v 2 (ds (H)) (u d)s (H) = v 2(6) v 2 (4) 6 4 = 2 Ten we may easly verfy tat ts value for our replcatng portfolo s correct by smlar calculatons as tose n equaton (3.8). Lastly, we take a look at te dscounted Amercan put prces. ( 4 5 )2 v 2 (6) = v (8) = 0.32 v 0 (4) =.36 ( 4 5 )2 v 2 (4) = v (2) = 2.40 Dscounted Amercan put prces. ( 4 5 )2 v 2 () = 2.56 Clearly te dscounted Amercan put prce process s a supermartngale under te rsk-neutral probablty measure P because te average of te two rgtward brances of eac node s less tan or equal to te node tself. We can now formalze our fndngs from te prevous example wt a teorem. Teorem (Replcaton of pat-ndependent Amercan dervatves). Consder an N-perod bnomal asset-prcng model wt 0 < d < + r < u and wt p = +r d u d, q = u r u d Let a payoff functon g(s) be gven, and defne recursvely backward n tme te sequence of functons v N (s),v N (s),...,v 0 (s) by equaton (3.4). Next defne were n =0,,...,N X,X 2,...,X N by n = v n+(us n ) v n+ (ds n ), (3.9) (u d)s n C n = v n (S n ) pv n+ (us n )+ qv n+ (ds n ) (3.0) +r. If we set X 0 = v 0 (S 0 ) and defne recursvely forward te portfolo values X n+ = n S n+ +(+r)(x n C n n S n ) (3.) ten we wll ave X n (!...! n )=v n (S n (!...! n )) for all n and for all!...! n. X n g(s n ) for all n. Note tat equaton (3.0) s te wealt equaton wt te possblty of consumpton. In partcular, 8

22 3.3 Stoppng Tmes Wen consderng te general case of Amercan dervatve securtes, wc may be pat-dependent, we must develop te noton of stoppng tmes. Ts wll elp us determne wen te optmal tme for an owner to exercse te dervatve securty s based on te pat of te stock prce. If we look back at example 3.2., we can conclude tat f te frst con toss results n tals, te owner sould exercse te put opton at tme t =because te value of our replcatng portfolo was a dollar more tan wat te rsk-neutral prcng formula determned t sould be. If te frst con toss was eads, te owner would be out of te money snce S (H) =8and te strke prce was 5, so t would be wse for tem to wat. Ten, f te second con toss resulted n tals tey would be back n te money and tey could exercse te opton at tme t =2. But, f te second con toss resulted n eads te owner would stll be out of te money and sould not exercse te opton at all (.e. let t expre). We can use te random varable to rewrte our fndngs as follows: (HH)=, (HT)=2, (TH)=, (TT)= (3.2) And ts can be dsplayed usng te bnomal model: Don t exercse (HH)= Don t exercse Don t exercse Exercse (HT)=2 Exercse (TH)= (TT)= Exercse rule. Were takng on te value of means we allow te opton to expre. Te random varable s defned on te sample space =HH,HT,TH,TT and takes values n te set 0,, 2,. It s known as an exercse tme. Te values were determned based on te fact tat te owner dd not know wat te next con toss was to result n. If tey ad foreknowledge of te con tossng, ten tey surely would ave exercsed te opton at tme t =0f te frst con toss were eads. And, f te con toss sequence was TT, te owner would ave wated untl t =2to exercse nstead of exercsng te opton at tme t =.Wecandefneanoterrandomvarable,, wc s an exercse tme tat reflects ts. S 0 =4 (HH)= (HT)=0 S (T )=2 (TH)= Exercse rule. S 2 (TT)= (TT)=2 But, cannot be mplemented wtout nsder nformaton. So, wle t s an exercse tme, t s not a stoppng tme. Te random varable s a stoppng tme. We gve te formal defnton of stoppng 9

23 tme below. Defnton In an N-perod bnomal model, a stoppng tme s a random varable tat takes values 0,,...,N or and satsfes te condton tat f (!! 2...! n! n+...! N )=n, ten (!! 2...! n!0 n+...!0 N )=n for all!0 n+...!0 N. In oter words, te stoppng s based only on avalable nformaton. Wenever we ave a stocastc process and a stoppng tme, we can defne a stopped process. Forexample,wecanletY n be te process of te dscounted Amercan put prces from example Y 0 =.36, Y (H) =0.32, Y (T )=2.40, Y 2 (HH)=0, Y 2 (HT)=Y 2 (TH)=0.64, Y 2 (TT)=2.56 We can take our stoppng tme,, andletn ^ mean te mnmum of n and. We can let Y 0^ = Y 0 =.36 and Y ^ = Y because regardless of te con toss outcome, we wll always old onto our opton untl at least tme t =. 2 ^ wll depend on te con tossng: Y 2^ (HH)=Y 2 (HH)=0, Y 2^ (HT)=Y 2 (HT)=0.64, Y 2^ (TH)=Y (T )=2.40, Y 2^ (TT)=Y (T )=2.40 Note tat even f te stoppng tme s less tan 2, te process contnues untl t =2. Te value of te process merely freezes upon reacng te stoppng tme. Terefore, we may llustrate a stopped process usng a bnomal model, albet slgtly modfed from wat we are used to snce Y 2^ (TH) does not equal Y 2^ (HT). Y 0^ =.36 Y ^ (H) =0.32 Y ^ (T )=2.40 Astoppedprocess. Y 2^ (HH)=0 Y 2^ (HT)=0.64 Y 2^ (TH)=2.40 Y 2^ (TT)=2.40 Ts process s a martngale (wc s also a supermartngale by defnton). Ts fact s true for all stopped prce processes of an Amercan put opton under te rsk-neutral probablty measure P. Addtonally, EY n^ EY n. Teorem (Optonal Samplng Part I). A martngale stopped at a stoppng tme s a martngale. A supermartngale (or submartngale) stopped at a stoppng tme s a supermartngale (or submartngale, respectvely). Teorem (Optonal Samplng Part II). Let X n, n =0,,...,N be a supermartngale and let be a stoppng tme. Ten EX n^ EX n. If X n s a submartngale, ten EX n^ apple EX n. If X n s a martngale, ten EX n^ = EX n. Ts does not old for all exercse tmes, suc as. 20

24 3.4 Pat-Dependent Usng our newfound knowledge of stoppng tmes, we may now work out ow to prce Amercan dervatve securtes tat are permtted to be pat-dependent. Frst, we must redefne te prce process V n to nclude stoppng tmes. We stll ave an N-perod bnomal model wt u and d as te up-factor and down-factor respectvely, along wt nterest rate r suc tat 0 < d < + r < u. Let us defne P n as te set of all stoppng tmes takng on values n te set n, n +,...,N,. Ten, P 0 would be te set of all stoppng tmes and P N can take on only te values N and. WealsoletG n be te ntrnsc value process of te dervatve securty. Obvously, we need to ave te ntrnsc value be a random varable unlke te pat-ndependent equvalent g(s) wc only depended on te stock prce S n at tme n. Defnton For eac n, 0 apple n apple N, letg n be a random varable dependng on te frst n con tosses. An Amercan dervatve securty wt ntrnsc value process G n s a contract tat can be exercsed at any tme pror to or at tme N. If exercsed at tme n, ts payoff s G n. We defne te prce process V n by te Amercan rsk-neutral prcng formula: V n = max Ẽ n I { applen} P n ( + r) n G, n =0,,...,N (3.3) In te above defnton, I { applen} tells us tat I { applen} (+r) smply equals (+r) n G =0wen =, and oterwse t G n.togetaclearerunderstandngoftsdefnton,supposeweareattmen. Te owner can coose weter or not to exercse te dervatve based on only te prevous pat (.e. not based on a later date). Ts means te exercse date s ndeed a stoppng tme. Snce te owner as not yet exercsed te securty, te stoppng tme must be n te set P n. If t s never exercsed, te owner would receve no money wc s wy we use te notaton I { applen}. Oterwse, te owner sould coose P n tat makes te expected value of (+r) G n as large as possble. We wll now develop some propertes of te Amercan dervatve securty prce process as defned n Defnton Teorem Te Amercan dervatve securty prce process gven by Defnton 3.4. as te followng propertes: () V n max{g n, 0} for all n () Te dscounted process (+r) n V n s a supermartngale () If Y n s anoter process satsfyng Y n max{g n, 0} for all n and for wc (+r) Y n n supermartngale, ten Y n V n for all n. Proof. () Let n be gven. Consder te stoppng tme ˆ n P n tat only takes on te value n. Ts means Ẽn n Gˆ = Ẽn G n n = G n. Snce V n s te maxmum value of I {ˆ applen} (+r)ˆ (+r) n Ẽ n I { applen} (+r) n G,tmustbeatleastequaltoG n.but,wealsoknowtatftesecurty s left to expre, te owner wll receve no money for t and we wll ave V n =0. Snce V n s at least equal to G n and at least equal to 0, we ave V n max{g n, 0}. () Let n be gven. Let attan te maxmum n te defnton of V n+,.e.. But P n. Usng ts fact as well as te terated V n+ = Ẽn+ I { applen} (+r) (n+) G s a 2

25 condtonng property of condtonal expectaton, we see tat V n Ẽ n I { applen} ( + r) n G " # = Ẽn Ẽ n+ I { applen} ( + r) n G " # = Ẽn +r Ẽn+ I { applen} ( + r) n G = Ẽn +r V n+ Fnally we dvde bot sdes by ( + r) n to see tat V n (+r) n (+r) n+ V n+. () To prove te last part of te teorem, let Y n be a process satsfyng condtons () and (). Let n N be gven and be a stoppng tme n P n.becauseofcondton(),weave I { applen} G apple I { applen} max{g, 0} apple I { applen} max{g N^, 0} + I { =} max{g N^, 0} = max{g N^, 0} apple Y N^ Next, we can use Teorem along wt condton (): Ẽ n I { applen} ( + r) G = Ẽn I { applen} ( + r) ^N G apple Ẽn ( + r) ^N Y ^N apple ( + r) ^n Y ^n = ( + r) n Y n wt te last step due to te fact tat P n s greater tan or equal to n on every possble pat. If we multply bot sdes by ( + r) n,wegetteequaton Ẽ n I { applen} ( + r) n G apple Y n (3.4) And, snce V n s te maxmum value of te left sde of equaton (3.4) forall P n,tmustbe tat V n apple Y n. Eac part of Teorem tells us sometng about buldng a replcatng portfolo of a patdependent Amercan dervatve securty. Property () tells us tat t s possble to construct a portfolo wose agent starts wt ntal captal V 0 and te value of te portfolo at eac tme t = n s V n. Property () tells us tat f te agent does ts, tey wll ave edged a sort poston n te securty, no matter te exercse tme. Terefore, propertes () and () togeter guarantee tat te prce of te dervatve securty s acceptable to te seller. In addton, property () sows tat te prce s te mnmum requred to be acceptable for te seller, and tus s a far prce for te buyer. We wll now provde a teorem for and prove te Amercan prcng algortm for pat-dependent dervatve securtes. 22

26 Teorem We ave te followng Amercan prcng algortm for te pat-dependent dervatve securty prce process gven by Defnton 3.4.: V N (!...! N )=max{g N (!...! N ), 0}, (3.5) n o V n (!...! n )=max G n (!...! n ), pv n+ (!...! n H)+ qv n+ (!...! n T ), (3.6) +r for n = N,...,0. Proof. In ts proof we wll sow tat ts process satsfes all tree propertes of Teorem and s terefore te same Amercan rsk-neutral prcng formula as defned n Defnton () It s clear tat V N satsfes property () of Teorem Now, we use backwards nducton. Suppose for some n between 0 and N, weavev n+ max{g n+, 0}. Fromequaton(3.6), t follows tat V n (!...! n ) max{g n (!...! n ), 0}. () From equaton (3.6), t follows tat V n (!...! n ) +r = Ẽn pv n+ (!...! n H)+ qv n+ (!...! n T ) +r V n+ Multplyng bot sdes of ts equaton by (+r) n (!...! n ) wll lead to property () beng satsfed. () It s clear tat V N s te smallest random varable satsfyng V N max{g N, 0} snce V N = max{g N, 0}. We must terefore sow by backwards nducton tat te same s true for all n. Suppose for some n between 0 and N, V n s te smallest random varable satsfyng V n+ max{g n+, 0}. We know from () tat V n (!...! n ) +r pv n+ (!...! n H)+ qv n+ (!...! n T ) because ts process s a supermartngale. We also know from () tat V n G n. Combnng tese two facts, we get n o V n (!...! n ) max G n (!...! n ), pv n+ (!...! n H)+ qv n+ (!...! n T ) (3.7) +r But, equaton (3.6) tellsustatbotsdesoftsnequaltyareequal. So,V n (!...! n ) s as small as possble. Fnally, we wll prove te process for replcatng pat-dependent Amercan dervatves n te stock and money markets. Te proof s anoter nducton on n, so we wll skp t. Teorem Consder an N-perod bnomal asset-prcng model wt 0 < d < + r < u and wt p = +r d u d, q = u r u d For eac n, 0 apple n apple N, letg n be a random varable dependng on te frst n con tosses. Wt V n, 0 apple n apple N gven by Defnton 3.4., we defne C n (!...! n )=V n (!...! n ) n(!...! n )= V n+(!...! n H) V n+ (!...! n T ) S n+ (!...! n H) S n+ (!...! n T ), (3.8) pv n+ (!...! n H)+ qv n+ (!...! n T ), (3.9) +r 23

27 were 0 apple n apple N. We ave C n 0 for all n. If we set X 0 = V 0 and defne recursvely forward te portfolo values X,X 2,...X N by X n+ = n S n+ +(+r)(x n C n n S n ) (3.20) ten we wll ave X n (!...! n )=V n (!...! n ) for all n and all!...! n. In partcular, we ave X n G n for all n. Ts teorem sows gves us an algortm tat s acceptable for te seller. Now, we wll consder te buyer. We must frst establs anoter teorem regardng a seres of payments. Teorem (Cas flow valuaton). Consder an N-perod bnomal asset-prcng model wt 0 < d < + r < u and rsk-neutral probablty measure P. Let C 0,C,...,C N be a sequence of random varables suc tat eac C n depends only on!...! n. Te prce at tme n of te dervatve securty tat makes payments C n,...,c N at tmes n,..., N, respectvely,s " N # X C k V n = Ẽn ( + r) k n,n=0,,...,n (3.2) n and X n+ by Equa- Te prce process V n, n =0,,...,N satsfes Equaton (3.9) and we defne tons (3.8) and (3.20), respectvely, from Teorem k=n We can magne we are at a fxed tme n and tat te securty as not yet been exercsed. Denote P n as te stoppng tme tat attans te maxmum V n,.e. V n = Ẽn I { applen} ( + r) n G (3.22) Also defne C k = I { applek}g k, k = n, n +,...,N (3.23) as te cas flows tat te owner receves f te dervatve securty s exercsed accordng to. At most only one C k wll be nonzero, and te k wll correspond wt te exercse tme f te securty s exercsed at all. But, by our defnton of C k,weave V n = Ẽn N X k=n I { =k} ( + r) k n G k = Ẽn N X k=n C k ( + r) k n (3.24) And, ts s just te cas flows C n,...,c N receved at tme n,..., N, respectvely. Terefore, V n s acceptable to te seller as well. Now, we only need to defne a metod by wc to determne an optmal exercse tme. We let n =0so we can consder all possble stoppng tmes from te tme te securty s obtaned. Teorem (Optmal exercse). Te stoppng tme = mn{n V n = G n } (3.25) maxmzes te rgt and sde of Equaton (3.3) wen n = 0;.e. V 0 = Ẽ I { applen} ( + r) n G Proof. Frst, we need to sow tat te stopped process ( + r) n^ V n^ (3.26) s a martngale. Let te frst n con tosses result n te sequence!...! n 24

28 () If, along ts pat, n+, V n (!...! n ) G n (!...! n ) from our assumpton of.furter, as a consequence of Equaton (3.6), V n^ (!...! n )=V n (!...! n ) = pv n+ (!...! n H)+ qv n+ (!...! n T ) +r = pv (n+)^ (!...! n H)+ qv (n+)^ (!...! n T ) +r And tus te martngale property s satsfed. () If, along ts pat, apple n, V n^ (!...! n )=V (!...! n ) And agan, te martngale property s satsfed. Terefore, we can say = pv (!...! )+ qv (!...! ) = pv (n+)^ (!...! n H)+ qv (n+)^ (!...! n T ) V 0 = Ẽ ( + r) N^ V N^ = Ẽ I { applen} ( + r) G + Ẽ I { =} ( + r) N G N But f = ten t must be tat V n >G n for all n, ncludngforn = N. But,becauseofequaton (3.5), t must be true tat G N < 0 and V N =0.So,I { =}V N =0and we can smplfy te equaton for V 0 : V 0 = Ẽ I { applen} ( + r) G Wc s exactly Equaton (3.26). We ave now seen ow te bnomal model can be used to prce multple dfferent dervatve securtes, ncludng European and Amercan optons. Tere are, of course, dfferent means by wc one may prce suc securtes. Te next capter wll gve a bref overvew of one suc way: te Black-Scoles model. 25

29 Capter 4 Te Black-Scoles Model 4. Random Walk Te Black-Scoles model reles on Brownan moton. Toug we wll not explore ts concept, we wll develop ts dscrete-tme counterpart known as te random walk. We wll focus on te symmetrc random walk weren we repeatedly toss a far con (p = q = 2 ). If we cose probabltes oter tan ts, te resultng walk would be asymmetrc. Let te process M n be our random walk and M 0 =0. If!...! n s our sequence of con tosses, ten for eac! k tat s eads, M k = M k + and for eac! k tat s tals, M k = M k. M n s bot a martngale and a Markov process. 3 2 M 2 M M 3 2 M 0 M 4 M M 5 Asx-steprandomwalk. We can let m be te frst tme te random walk reaces some nteger m,.e. m = mn{n M n = m} (4.) We call m te frst passage tme of te random walk to level m. If te walk never reaces level m, m =. Wewouldlketofgureouttedstrbutonof m and we wll do so by usng te reflecton prncple. Let s say we toss a con an odd number of tmes. We can represent ts number as (2j ) for some postve nteger j. Outofalltepossblepatstatcouldresultfromdfferentsequencesoftscon 26

30 toss, some wll reac level and oters wll not. For example, f we ave a seres of tree con tosses, tere are 8 possble pats total. Fve of tese wll reac level (resultng from con toss sequences HHH, HHT, HTH, THH, HTT). Consder one of te pats tat reac level. Ts means te frst passage tme would be no more tan te total number of con tosses;.e. apple 2j. We can create a reflected pat tat dverges from te orgnal pat at step and mrrors all future steps. If te orgnal pat ends above, te reflected pat ends below. If te orgnal pat ends at, ten so does te reflected pat. But, ts covers all possble pats. And snce tere are te same amount of pats endng above as tere are reflected pats endng below, we can say: P( apple 2j ) = P(M 2j = ) + 2P(M 2j 3) Note tat snce we ave an odd number of con tosses, t s mpossble for te pat to end on level 2, terefore n order to ave ended above level t must be at least at level 3. But, snce tere s an equal probablty of gettng eads as tere s tals, Wc means P(M 2j 3) = P(M 2j apple 3) P( apple 2j ) = P(M 2j = ) + P(M 2j 3) + P(M 2j apple 3) Ts covers all cases except for wen M 2j =. So, P( apple 2j ) = P(M 2j = ) If M 2j =, tecorrespondngcontosssequencecontans(j ) eads and j tals. Te number of possble pats wose con toss fts ts crtera s: 2j (2j )! = j j!(j )! and eac of tese con toss sequences ave a probablty of 2j (2j )! P(M 2j = ) = 2 j!(j )! 2 2j. Terefore, We come to smlar conclusons for (2j 3) con tosses. In addton, t s clear tat for j 2, We can state ts as a teorem. P( =2j ) = P( apple 2j ) P( apple 2j 3) =[ P(M 2j = )] [ P(M 2j 3 = )] = P(M 2j 3 = ) P(M 2j = ) 2j 3 (2j 3)! = 2 (j )!(j 2)! 2 2j (2j 3)! = 2 j!(j )! 2j (2j 3)! = 2 j!(j )! 2j (2j 3)! = (2j 2) 2 j!(j )! 2j (2j 2)! = 2 j!(j )! 2j (2j )! j!(j )! 4j(j ) 2(j )(2j 2) 2j(2j 2) 2(j )(2j 2) Teorem 4... Let be te frst passage tme to level of a symmetrc random walk. Ten, 2j (2j 2)! P( =2j ) = j =, 2,... (4.2) 2 j!(j )! 27