PART. I. Pricing Theory and Risk Management

Transcription

1 PART. I Pricing Theory and Risk Managemen

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3 CHAPTER. 1 Pricing Theory Pricing heory for derivaive securiies is a highly echnical opic in finance; is foundaions res on rading pracices and is heory relies on advanced mehods from sochasic calculus and numerical analysis. This chaper summarizes he main conceps while presening he essenial heory and basic mahemaical ools for which he modeling and pricing of financial derivaives can be achieved. Financial asses are subdivided ino several classes, some being quie basic while ohers are srucured as complex conracs referring o more elemenary asses. Examples of elemenary asse classes include socks, which are ownership righs o a corporae eniy; bonds, which are promises by one pary o make cash paymens o anoher in he fuure; commodiies, which are asses, such as whea, meals, and oil ha can be consumed; and real esae asses, which have a convenience yield deriving from heir use. A more general example of an asse is ha of a conracual coningen claim associaed wih he obligaion of one pary o ener a sream of more elemenary financial ransacions, such as cash paymens or deliveries of shares, wih anoher pary a fuure daes. The value of an individual ransacion is called a pay-off or payou. Mahemaically, a pay-off can be modeled by means of a payoff funcion in erms of he prices of oher, more elemenary asses. There are numerous examples of coningen claims. Insurance policies, for insance, are srucured as conracs ha envision a paymen by he insurer o he insured in case a specific even happens, such as a car acciden or an illness, and whose pay-off is ypically linked o he damage suffered by he insured pary. Derivaive asses are claims ha disinguish hemselves by he propery ha he payoff funcion is expressed in erms of he price of an underlying asse. In finance jargon, one ofen refers o underlying asses simply as underlyings. To some exen, here is an overlap beween insurance policies and derivaive asses, excep he nomenclaure differs because he firs are markeed by insurance companies while he laer are raded by banks. A rading sraegy consiss of a se of rules indicaing wha posiions o ake in response o changing marke condiions. For insance, a rule could say ha one has o adjus he posiion in a given sock or bond on a daily basis o a level given by evaluaing a cerain funcion. The implemenaion of a rading sraegy resuls in pay-offs ha are ypically random. A major difference ha disinguishes derivaive insrumens from insurance conracs 3

4 4 C H A P T E R 1. Pricing heory is ha mos raded derivaives are srucured in such a way ha i is possible o implemen rading sraegies in he underlying asses ha generae sreams of pay-offs ha replicae he pay-offs of he derivaive claim. In his sense, rading sraegies are subsiues for derivaive claims. One of he driving forces behind derivaives markes is ha some marke paricipans, such as marke makers, have a compeiive advanage in implemening replicaion sraegies, while heir cliens are ineresed in aking cerain complex risk exposures synheically by enering ino a single conrac. A key propery of replicable derivaives is ha he corresponding payoff funcions depend only on prices of radable asses, such as socks and bonds, and are no affeced by evens, such as car accidens or individual healh condiions ha are no direcly linked o an asse price. In he laer case, risk can be reduced only by diversificaion and reinsurance. A relaed concep is ha of porfolio immunizaion, which is defined as a rade inended o offse he risk of a porfolio over a leas a shor ime horizon. A perfec replicaion sraegy for a given claim is one for which a posiion in he sraegy combined wih an offseing posiion in he claim are perfecly immunized, i.e., risk free. The posiion in an asse ha immunizes a given porfolio agains a cerain risk is radiionally called hedge raio. 1 An immunizing rade is called a hedge. One disinguishes beween saic and dynamic hedging, depending on wheher he hedge rades can be execued only once or insead are carried over ime while making adjusmens o respond o new informaion. The asses raded o execue a replicaion sraegy are called hedging insrumens. A se of hedging insrumens in a financial model is complee if all derivaive asses can be replicaed by means of a rading sraegy involving only posiions in ha se. In he following, we shall define he mahemaical noion of financial models by lising a se of hedging insrumens and assuming ha here are no redundancies, in he sense ha no hedging insrumen can be replicaed by means of a sraegy in he oher ones. Anoher very common expression is ha of risk facor: The risk facors underlying a given financial model wih a complee basis of hedging insrumens are given by he prices of he hedging insrumens hemselves or funcions hereof; as hese prices change, risk facor values also change and he prices of all oher derivaive asses change accordingly. The saisical analysis of risk facors allows one o assess he risk of financial holdings. Transacion coss are impedimens o he execuion of replicaion sraegies and correspond o coss associaed wih adjusing a posiion in he hedging insrumens. The marke for a given asse is perfecly liquid if unlimied amouns of he asse can be raded wihou affecing he asse price. An imporan noion in finance is ha of arbirage: If an asse is replicable by a rading sraegy and if he price of he asse is differen from ha of he replicaing sraegy, he opporuniy for riskless gains/profis arises. Pracical limiaions o he size of possible gains are, however, placed by he inaccuracy of replicaion sraegies due o eiher marke incompleeness or lack of liquidiy. In such siuaions, eiher riskless replicaion sraegies are no possible or prices move in response o posing large rades. For hese reasons, arbirage opporuniies are ypically shor lived in real markes. Mos financial models in pricing heory accoun for finie liquidiy indirecly, by posulaing ha prices are arbirage free. Also, marke incompleeness is accouned for indirecly and is refleced in correcions o he probabiliy disribuions in he price processes. In his sylized mahemaical framework, each asse has a unique price. 2 1 Noice ha he erm hedge raio is par of he finance jargon. As we shall see, in cerain siuaions hedge raios are compued as mahemaical raios or limis hereof, such as derivaives. In oher cases, expressions are more complicaed. 2 To avoid he percepion of a linguisic ambiguiy, when in he following we sae ha a given asse is worh a cerain amoun, we mean ha amoun is he asse price.

5 Pricing Theory 5 Mos financial models are buil upon he perfec-markes hypohesis, according o which: There are no rading impedimens such as ransacion coss. The se of basic hedging insrumens is complee. Liquidiy is infinie. No arbirage opporuniies are presen. These hypoheses are robus in several ways. If liquidiy is no perfec, hen arbirage opporuniies are shor lived because of he acions of arbirageurs. The lack of compleeness and he presence of ransacion coss impacs prices in a way ha is uniform across classes of derivaive asses and can safely be accouned for implicily by adjusing he process probabiliies. The exisence of replicaion sraegies, combined wih he perfec-markes hypohesis, makes i possible o apply more sophisicaed pricing mehodologies o financial derivaives han is generally possible o devise for insurance claims and more basic asses, such as socks. The key o finding derivaive prices is o consruc mahemaical models for he underlying asse price processes and he replicaion sraegies. Oher sources of informaion, such as a counry s domesic produc or a akeover announcemen, alhough possibly relevan o he underlying prices, affec derivaive prices only indirecly. This firs chaper inroduces he reader o he mahemaical framework of pricing heory in parallel wih he relevan noions of probabiliy, sochasic calculus, and sochasic conrol heory. The dynamic evoluion of he risk facors underlying derivaive prices is random, i.e., no deerminisic, and is subjec o uncerainy. Mahemaically, one uses sochasic processes, defined as random variables wih probabiliy disribuions on ses of pahs. Replicaing and hedging sraegies are formulaed as ses of rules o be followed in response o changing price levels. The key principle of pricing heory is ha if a given payoff sream can be replicaed by means of a dynamic rading sraegy, hen he cos of execuing he sraegy mus equal he price of a conracual claim o he payoff sream iself. Oherwise, arbirage opporuniies would ensue. Hence pricing can be reduced o a mahemaical opimizaion problem: o replicae a cerain payoff funcion while minimizing a he same ime replicaion coss and replicaion risks. In perfec markes one can show ha one can achieve perfec replicaion a a finie cos, while if here are imperfecions one will have o find he righ rade-off beween risk and cos. The fundamenal heorem of asse pricing is a far-reaching mahemaical resul ha saes; The soluion of his opimizaion problem can be expressed in erms of a discouned expecaion of fuure pay-offs under a pricing (or probabiliy) measure. This represenaion is unique (wih respec o a given discouning) as long as markes are complee. Discouning can be achieved in various ways: using a bond, using he money marke accoun, or in general using a reference numeraire asse whose price is posiive. This is because pricing asses is a relaive, as opposed o an absolue, concep: One values an asse by compuing is worh as compared o ha of anoher asse. A key poin is ha expecaions used in pricing heory are compued under a probabiliy measure ailored o he numeraire asse. In his chaper, we sar he discussion wih a simple single-period model, where rades can be carried ou only a one poin in ime and gains or losses are observed a a laer ime, a fixed dae in he fuure. In his conex, we discuss saic hedging sraegies. We hen briefly review some of he relevan and mos basic elemens of probabiliy heory in he

6 6 C H A P T E R 1. Pricing heory conex of mulivariae coninuous random variables. Brownian moion and maringales are hen discussed as an inroducion o sochasic processes. We hen move on o furher discuss coninuous-ime sochasic processes and review he basic framework of sochasic (Iô) calculus. Geomeric Brownian moion is hen presened, wih some preliminary derivaions of Black Scholes formulas for single-asse and muliasse price models. We hen proceed o inroduce a more general mahemaical framework for dynamic hedging and derive he fundamenal heorem of asse pricing (FTAP) for coninuous-sae-space and coninuousime-diffusion processes. We hen apply he FTAP o European-syle opions. Namely, by he use of change of numeraire and sochasic calculus echniques, we show how exac pricing formulas based on geomeric Brownian moions for he underlying asses are obained for a variey of siuaions, ranging from elemenary sock opions o foreign exchange and quano opions. The parial differenial equaion approach for opion pricing is hen presened. We hen discuss pricing heory for early-exercise or American-syle opions. 1.1 Single-Period Finie Financial Models The simples framework in pricing heory is given by single-period financial models, in which calendar ime is resriced o ake only wo values, curren ime = and a fuure dae = T >. Such models are appropriae for analyzing siuaions where rades can be made only a curren ime =. Revenues (i.e., profis or losses) can be realized only a he laer dae T, while rades a inermediae imes are no allowed. In his secion, we focus on he paricular case in which only a finie number of scenarios 1 m can occur. Scenario is a common erm for an oucome or even. The scenario se = 1 m is also called he probabiliy space. Aprobabiliy measure P is given by a se of numbers p i i= 1 m, in he inerval 1 ha sum up o 1; i.e., m p i = 1 p i 1 (1.1) i=1 p i is he probabiliy ha scenario (even) i occurs, i.e., ha he ih sae is aained. Scenario i is possible if i can occur wih sricly posiive probabiliy p i >. Neglecing scenarios ha canno possibly occur, he probabiliies p i will henceforh be assumed o be sricly posiive; i.e., p i >. A random variable is a funcion on he scenario se, f, whose values f i represen observables. As we discuss laer in more deail, examples of random variables one encouners in finance include he price of an asse or an ineres rae a some poin in he fuure or he pay-off of a derivaive conrac. The expecaion of he random variable f is defined as he sum m E P f = p i f i (1.2) i=1 Asse prices and oher financial observables, such as ineres raes, are modeled by sochasic processes. In a single-period model, a sochasic process is given by a value f a curren ime = and by a random variable f T ha models possible values a ime T. In finance, probabiliies are obained wih wo basically differen procedures: They can eiher be inferred from hisorical daa by esimaing a saisical model, or hey can be implied from curren asse valuaions by calibraing a pricing model. The former are called hisorical, saisical, or, beer, real-world probabiliies. The laer are called implied probabiliies. The calibraion procedure involves using he fundamenal heorem of asse pricing o represen prices as discouned expecaions of fuure pay-offs and represens one of he cenral opics o be discussed in he res of his chaper.

7 1.1 Single-Period Finie Financial Models 7 Definiion 1.1. Financial Model A finie, single-period financial model = is given by a finie scenario se = 1 m and n basic asse price processes for hedging insrumens: = A 1 An = T (1.3) Here, A i models he curren price of he ih asse a curren (or iniial) ime = and Ai T is a random variable such ha he price a ime T> of he ih asse in case scenario j occurs is given by A i T j. The basic asse prices A i, i = 1 n, are assumed real and posiive. Definiion 1.2. Porfolio and Asse Le = be a financial model. A porfolio is given by a vecor wih componens i i = 1 n, represening he posiions or holdings in he he family of basic asses wih prices A 1 An. The worh of he porfolio a erminal ime T is given by n i=1 i A i T given he sae or scenario, whereas he curren price is n i=1 i A i. A porfolio is nonnegaive if i gives rise o nonnegaive pay-offs under all scenarios, i.e., n i=1 i A i T j j = 1 m. An asse price process A = A (a generic one, no necessarily ha of a hedging insrumen) is a process of he form n A = i A i (1.4) for some porfolio n. i=1 The modeling assumpion behind his definiion is ha marke liquidiy is infinie, meaning ha asse prices don vary as a consequence of agens rading hem. As we discussed a he sar of his chaper, his hypohesis is valid only in case rades are relaively small, for large rades cause marke prices o change. In addiion, a financial model wih infinie liquidiy is mahemaically consisen only if here are no arbirage opporuniies. Definiion 1.3. Arbirage: Single-Period Discree Case An arbirage opporuniy or arbirage porfolio is a porfolio = 1 n such ha eiher of he following condiions holds: A1. The curren price of is negaive, n i=1 ia i <, and he pay-off a erminal ime T is nonnegaive, i.e., n i=1 ia i T j for all j saes. A2. The curren price of is zero, i.e., n i=1 ia i =, and he pay-off a erminal ime T in a leas one scenario j is posiive, i.e., n i=1 ia i T j > for some jh sae, and he pay-off a erminal ime T is nonnegaive. Definiion 1.4. Marke Compleeness The financial model = is complee if for all random variables f, where f is a bounded payoff funcion, here exiss an asse price process or porfolio A in he basic asses conained in such ha A T = f T for all scenarios. This definiion essenially saes ha any pay-off (or sae-coningen claim) can be replicaed, i.e., is aainable by means of a porfolio consising of posiions in he se of basic asses. If an arbirage porfolio exiss, one says here is arbirage. The firs form of arbirage occurs whenever here exiss a rade of negaive iniial cos a ime = by means of which one can form a porfolio ha under all scenarios a fuure ime = T has a nonnegaive pay-off. The second form of arbirage occurs whenever one can perform a rade a zero cos a an iniial ime = and hen be assured of a sricly posiive payou a fuure ime T under

8 8 C H A P T E R 1. Pricing heory a leas one possible scenario, wih no possible downside. In realiy, in eiher case invesors would wan o perform arbirage rades and ake arbirarily large posiions in he arbirage porfolios. The exisence of hese rades, however, infringes on he modeling assumpion of infinie liquidiy, because marke prices would shif as a consequence of hese large rades having been placed. Le s sar by considering he simples case of a single-period economy consising of only wo hedging insrumens (i.e., n = 2 basic asses) wih price processes A 1 = B and A 2 = S. The scenario se, or sample space, is assumed o consis of only wo possible saes of he world: = +. S is he price of a risky asse, which can be hough of as a sock price. The riskless asse is a zero-coupon bond, defined as a process B ha is known o be worh he so-called nominal amoun B T = N a ime T while a ime = has worh B = 1 + rt 1 N (1.5) Here r> is called he ineres rae. As is discussed in more deail in Chaper 2, ineres raes can be defined wih a number of differen compounding rules; he definiion chosen here for r corresponds o selecing T iself as he compounding inerval, wih simple (or discree) compounding assumed. A curren ime =, he sock has known worh S. A a laer ime = T, wo scenarios are possible for he sock. If he scenario + occurs, hen here is an upward move and S T = S T + S + ; if he scenario occurs, here is a downward move and S T = S T S, where S + >S. Since he bond is riskless we have B T + = B T = B T. Assume ha he real-world probabiliies ha hese evens will occur are p + = p 1 and p = 1 p, respecively. Figure 1.1 illusraes his simple economy. In his siuaion, he hypohesis of arbirage freedom demands ha he following sric inequaliy be saisfied: S 1 + rt <S < S + (1.6) 1 + rt In fac, if, for insance, one had S < S, hen one could make unbounded riskless profis by 1+rT iniially borrowing an arbirary amoun of money and buying an arbirary number of shares in he sock a price S a ime =, followed by selling he sock a ime = T a a higher reurn level han r. Inequaliy (1.6) is an example of a resricion resuling from he condiion of absence of arbirage, which is defined in more deail laer. A derivaive asse, of worh A a ime, is a claim whose pay-off is coningen on fuure values of risky underlying asses. In his simple economy he underlying asse is he sock. An example is a derivaive ha pays f + dollars if he sock is worh S +, and f oherwise, a final ime T: A T = A T + = f + if S T = S + and A T = A T = f if S T = S. Assuming one can ake fracional posiions, his payou can be saically replicaed by means of a porfolio p + S + S p S FIGURE 1.1 A single-period model wih wo possible fuure prices for an asse S.

9 1.1 Single-Period Finie Financial Models 9 consising of a shares of he sock and b bonds such ha he following replicaion condiions under he wo scenarios are saisfied: The soluion o his sysem is as + bn = f (1.7) as + + bn = f + (1.8) a = f + f S + S b= f S + f + S (1.9) N S + S The price of he replicaing porfolio, wih pay-off idenical o ha of he derivaive, mus be he price of he derivaive asse; oherwise here would be an arbirage opporuniy. Tha is, one could make unlimied riskless profis by buying (or selling) he derivaive asse and, a he same ime, aking a shor (or long) posiion in he porfolio a ime =. A ime =, he arbirage-free price of he derivaive asse, A, is hen A = as + b 1 + rt 1 N ( ) S 1 + rt 1 S = f S + S + + ( 1 + rt 1 S + S S + S ) f (1.1) Dimensional consideraions are ofen useful o undersand he srucure of pricing formulas and deec errors. I is imporan o remember ha prices a differen momens in calendar ime are no equivalen and ha hey are relaed by discoun facors. The hedge raios a and b in equaion (1.9) are dimensionless because hey are expressed in erms of raios of prices a ime T. In equaion (1.1) he variables f ± and S + S are measured in dollars a ime T, so heir raio is dimensionless. Boh S and he discouned prices 1 + rt 1 S ± are measured in dollars a ime, as is also he derivaive price A. Rewriing his las equaion as A = 1 + rt 1 [( 1 + rt S S S + S ) f + + ( ] S+ 1 + rt S )f S + S (1.11) shows ha price A can be inerpreed as he discouned expeced pay-off. However, he probabiliy measure is no he real-world one (i.e., no he physical measure P) wih probabiliies p ± for up and down moves in he sock price. Raher, curren price A is he discouned expecaion of fuure prices A T, in he following sense: A = 1 + rt 1 E Q A T = 1 + rt 1 q + A T + + q A T (1.12) under he measure Q wih probabiliies (sricly beween and 1) q + = 1 + rt S S S + S q = S rt S S + S (1.13) q + + q = 1. The measure Q is called he pricing measure. Pricing measures also have oher, more specific names. In he paricular case a hand, since we are discouning wih a consan ineres rae wihin he ime inerval T, Q is commonly named he risk-neural or risk-adjused probabiliy measure, where q ± are so-called risk-neural (or risk-adjused) probabiliies. Laer we shall see ha his measure is also he forward measure, where he bond price B is used as numeraire asse. In paricular, by expressing all asse prices relaive

10 1 C H A P T E R 1. Pricing heory o (i.e., in unis of) he bond price A i /B, wih B T = N, regardless of he scenario and B /B T = 1 + rt 1, we can hence recas he foregoing expecaion as: A = B E Q A T /B T. Hence Q corresponds o he forward measure. We can also use as numeraire a discreely compounded money-marke accoun having value 1 + r (or 1 + r N ). By expressing all asse prices relaive o his quaniy, i is rivially seen ha he corresponding measure is he same as he forward measure in his simple model. As discussed laer, he name risk-neural measure shall, however, refer o he case in which he money-marke accoun (o be defined more generally laer in his chaper) is used as numeraire, and his measure generally differs from he forward measure for more complex financial models. Laer in his chaper, when we cover pricing in coninuous ime, we will be more specific in defining he erminology needed for pricing under general choices of numeraire asse. We will also see ha wha we jus unveiled in his paricularly simple case is a general and far-reaching propery: Arbirage-free prices can be expressed as discouned expecaions of fuure pay-offs. More generally, we will demonsrae ha asse prices can be expressed in erms of expecaions of relaive asse price processes. A pricing measure is hen a maringale measure, under which all relaive asse price processes (i.e., relaive o a given choice of numeraire asse) are so-called maringales. Since our primary focus is on coninuous-ime pricing models, as inroduced laer in his chaper, we shall begin o explicily cover some of he essenial elemens of maringales in he conex of sochasic calculus and coninuousime pricing. For a more complee and elaborae mahemaical consrucion of he maringale framework in he case of discree-ime finie financial models, however, we refer he reader o oher lieraure (for example, see [Pli97, MM3]). We now exend he pricing formula of equaion (1.12) o he case of n asses and m possible scenarios. Definiion 1.5. Pricing Measure A probabiliy measure Q = q 1 q m, <q j < 1, for he scenario se = 1 m is a pricing measure if asse prices can be expressed as follows: A i m = EQ A i T = q j A i T j (1.14) for all i = 1 nand some real number >. The consan is called he discoun facor. Theorem 1.1. Fundamenal Theorem of Asse Pricing (Discree, single-period case) Assume ha all scenarios in are possible. Then he following saemens hold rue: There is no arbirage if and only if here is a pricing measure for which all scenarios are possible. The financial model is complee, wih no arbirage if and only if he pricing measure is unique. Proof. Firs, we prove ha if a pricing measure Q = q 1 q m exiss and prices A i = E Q A i T for all i = 1 n, hen here is no arbirage. If i i A i T j, for all j, hen from equaion (1.14) we mus have i i A i. If i i A i =, hen from equaion (1.14) we canno saisfy he payoff condiions in (A2) of Definiion 1.3. Hence here is no arbirage, for any choice of porfolio n. On he oher hand, assume ha here is no arbirage. The possible price-payoff m + 1 uples { ( } n n n = i A i i A i T 1 i A i T m ) n (1.15) i=1 i=1 i=1 j=1

11 1.1 Single-Period Finie Financial Models 11 make up a plane in m. Since here is no arbirage, he plane inersecs he ocan + m + made up of vecors of nonnegaive coordinaes only in he origin. Le be he se of all vecors 1 m normal o he plane and normalized so ha >. Vecors in saisfy he normaliy condiion ( n ) m n ) i A i + j( i A i T j = (1.16) i=1 j=1 i=1 for all porfolios. Nex we obain wo Lemmas o complee he proof. Lemma 1.1. Suppose he financial model on he scenario se and wih insrumens A 1 A n is arbirage free and le m be he dimension of he linear space. If he marix rank dim <m, hen one can define l = m dim price-payoff uples B k Bk T k = 1 l, so ha he exended financial model wih basic asses A 1 A n B 1 B l and scenario se is complee and arbirage free. Proof. The price-payoff uples B k Bk T 1 BT k l can be found ieraively. Suppose ha l = m dim >. Then he complemen o he linear space has dimension l Le X = X k Xk T and Y = Y k Yk T be wo vecors orhogonal o each oher and orhogonal o. Then here is an angle such ha he vecor B 1 = cos X + sin Y has a leas one sricly posiive coordinae and one sricly negaive coordinae, i.e., B 1 +. Hence he financial model wih insrumens A 1 A n B 1 is arbirage free. Ieraing he argumen, one can complee he marke while reaining arbirage freedom. Lemma 1.2. If markes are complee, he space orhogonal o is spanned by a vecor 1 m lying in he main ocan = + m + of vecors wih sricly posiive coordinaes. Proof. In fac if =, hen conains he line x and all posiive payous would be possible, even for an empy porfolio, which is absurd. I is also absurd ha j =, j. In fac, in his case, since markes are complee, here is an insrumen paying one dollar in case he scenario j occurs and zero oherwise, and since j =, he price of his insrumen a ime = is zero, which is absurd. If markes are no complee, one can sill conclude ha he se conains a vecor 1 m wih sricly posiive coordinaes. In fac, hanks o Lemma 1.1, one can complee i while preserving arbirage freedom by inroducing auxiliary asses and he normal vecor can be chosen o have posiive coordinaes. Hence, in all cases of i values, according o equaion (1.16) we have m A i = EQ A i T = q j A i T j (1.17) where Q is he measure wih probabiliies j=1 q j = j m j=1 j (1.18) and discoun facor = 1 m j=1 j (1.19)

12 12 C H A P T E R 1. Pricing heory The firs projec of Par II of his book is a sudy on single-period arbirage. We refer he ineresed reader o ha projec for a more deailed and pracical exposiion of he foregoing heory. In paricular, he projec provides an explici discussion of a numerical linear algebra implemenaion for deecing arbirage in single-period, finie financial models. Problems Problem 1. Consider he simple example in Figure 1.1 and assume he ineres rae is r. Under wha condiion is here no arbirage in he model? Problem 2. Compue E Q S T wihin he single-period wo-sae model. Explain your resul. Problem 3. Le pi denoe he curren price A i of he ih securiy and denoe by D ij = A i T j he marix elemens of he n m dividend marix wih i = 1 n, j = 1 m. Using equaion (1.14) wih = 1+rT 1 show ha he risk-neural expeced reurn on any securiy A i is given by he risk-free ineres rae [ ] A i E Q T A i A i = m j=1 q j ( Dij p i ) 1 = rt (1.2) where q j are he risk-neural probabiliies. Problem 4. Sae he explici marix condiion for marke compleeness in he single-period wo-sae model wih he wo basic asses as he riskless bond and he sock. Under wha condiion is his marke complee? Problem 5. Arrow Debreu securiies are claims wih uni pay-offs in only one sae of he world. Assuming a single-period wo-sae economy, hese claims are denoed by E ± and defined by 1 if = + if = + E + = E = if = 1 if = (a) Find exac replicaing porfolios + = a + b + and = a b for E + and E, respecively. The coefficiens a and b are posiions in he sock and he riskless bond, respecively. (b) Leing F T represen an arbirary pay-off, find he unique porfolio of Arrow Debreu securiies ha replicaes F T. 1.2 Coninuous Sae Spaces This secion, ogeher wih he nex secion, presens a review of basic elemens of probabiliy heory for random variables ha can ake on a coninuum of values while emphasizing some of he financial inerpreaion of mahemaical conceps. Modern probabiliy heory is based on measure heory. Referring he reader o exbook lieraure for more deailed and exhausive formal reamens, we will jus simply recall here ha measure heory deals wih he definiion of measurable ses D, probabiliy measures, and inegrable funcions f D for which one can evaluae expecaions as inegrals E f = f x dx (1.21) D

13 1.2 Coninuous Sae Spaces 13 In finance, one ypically deals wih siuaions where he measurable se D d, wih ineger d 1. Realizaions of he vecor variable x D correspond o scenarios for he risk facors or random variables in a financial model. Fuure asse prices are real-valued funcions of underlying risk facors f x defined for x D and hence hemselves define random variables. Probabiliy measures dx are ofen defined as dx = p x dx, where p x is a real-valued coninuous probabiliy disribuion funcion ha is nonnegaive and inegraes o 1; i.e., p x p x dx = 1 (1.22) The expecaion E P[ f ] of f under he probabiliy measure wih p as densiy is defined by he d-dimensional inegral E P[ f ] = f x p x dx (1.23) D The pair D dx is called a probabiliy space. In paricular, his formalism can also allow for he case of a finie scenario se of vecors D = x 1 x N, as was considered in he previous secion. In his case he probabiliy disribuion is a sum of Dirac dela funcions, p x = D N p i x x i (1.24) i=1 As furher discussed shorly, a dela funcion can be hough of as a singular funcion ha is posiive, inegraes o 1 over all space, and corresponds o he infinie limiing case of a sequence of inegrable funcions wih suppor only a he origin. Probabilisically, a disribuion, such as equaion (1.24), which is a sum of dela funcions, corresponds o a siuaion where only he scenarios x 1 N x can possibly occur, and hey do wih probabiliies p 1 p N. These probabiliies mus be posiive and add up o 1; i.e., N p i = 1 (1.25) i=1 In he case of a finie scenario se (i.e., a finie se of possible evens wih finie ineger N), he random variable f = f x is a funcion defined on he se of scenarios D, and is expecaion under he measure wih p as densiy is given by he finie sum E P f = N p i f x i (1.26) i=1 For an infiniely counable se of scenarios, hen, he preceding expressions mus be considered in he limi N. Hence in he case of a discree se of scenarios (as opposed o a coninuum) he probabiliy densiy funcion collapses ino he usual probabiliy mass funcion, as occurs in sandard probabiliy heory of discree-valued random variables. The Dirac dela funcion is no an ordinary funcion in d bu, raher, a so-called disribuion. Mahemaically, a disribuion is defined hrough is value when inegraed agains a smooh funcion. One can regard x x, x x d, as he limi of an infiniesimally narrow d-dimensional normal disribuion: f x x x 1 dx = lim d f x exp 2 d d ( x x ) dx = f x (1.27)

14 14 C H A P T E R 1. Pricing heory For example, in one dimension a represenaion of he dela funcion is x x 1 = lim 2 /2 2 e x x (1.28) 2 Evens are modeled as subses G D for which one can compue he inegral ha gives he expecaion E P 1 G. The funcion 1 G x denoes he random variable equal o 1 for x G and o zero oherwise; 1 G x is called he indicaor funcion of he se G. This expecaion is inerpreed as he probabiliy P(G) ha even G D will occur; i.e., P G = E P 1 G = 1 G x p x dx = p x dx (1.29) d G Examples of evens are subses, e.g., such as G = x D a<f x < b (1.3) wih b>a and where f is some funcion. An imporan concep associaed wih evens is ha of condiional expecaion. Given a random variable f, he expecaion of f condiioned o knowing ha even G will occur is E P[ f G ] = EP[ ] f 1 G (1.31) P G Two probabiliy measures dx = p x dx and dx = p x dx are said o be equivalen (or absoluely coninuous wih respec o one anoher) if hey share he same ses of null probabiliy; i.e., if he probabiliy condiion P G > implies P G >, where P G = E P 1 G = 1 G x p x dx = p x dx (1.32) d G wih E P denoing he expecaion wih respec o he measure. When compuing he expecaion of a real-valued random variable, say, of he general form of a funcion of a random vecor (such funcions are furher defined in he nex secion), f = f X d, i is someimes useful o swich from one choice of probabiliy measure o anoher, equivalen one. One can use he following change of measure (known as he Radon Nikodym heorem) for compuing expecaions: E P f = f x dx = f x d [ P x dx = E f d ] (1.33) D D d d The nonnegaive random variable denoed by d is called he Radon Nikodym derivaive of d wih respec o (or P w.r.. P). From his resul i also follows ha d = ( ) d 1 d d and E P d = 1. As will be seen laer in he chaper, a more general adapaion of his resul d for compuing cerain ypes of condiional expecaions involving maringales will urn ou o form one of he basic ools for pricing financial derivaives using changes of numeraire. Anoher paricular example of he use of his change-of-measure echnique is in he Mone Carlo esimaion of inegrals by so-called imporance-sampling mehods, as described in Chaper 4. Jus as inegrals are approximaed wih arbirary accuracy by finie inegral sums, coninuous probabiliy disribuions can be approximaed by discree ones. For insance, le D d be a bounded domain and p x be a coninuous probabiliy densiy on D and le G 1 G m

15 1.2 Coninuous Sae Spaces 15 be a pariion of D made up of a family of noninersecing evens G i D whose union covers he enire sae space D and ha have he shape of hypercubes. Le p i be he probabiliy of even G i under he probabiliy measure wih densiy p(x). Then an approximaion for p x is p x = m p i x x i (1.34) i=1 where x i is he cener of he hypercube corresponding o even G i. Le be he volume of he larges hypercube among he cubes in he pariion G 1 G m and le f x be a random variable on D. In he limi, as he pariion becomes finer and finer, he number of evens m will diverge o. In his limi, we find m E P f = lim p i f x i (1.35) By using sums as approximaions o expecaions, which are essenially mulidimensional Riemann inegrals, one can exend he heorem in he previous secion o he case of coninuous probabiliy disribuions. Consider a single-period financial model wih curren (i.e., iniial) ime = and ime horizon = T and wih n basic asses whose curren prices are A i, i = 1 n. The prices of hese basic asses a ime T are indexed by a coninuous sae space represened by he domain d, and he values of he basic asses are random variables A i T x, wih x. Tha is, he asse prices Ai are random variables assumed o ake on real posiive values, i.e., A i +. Le s denoe by p x dx he real-world probabiliy measure in and assume ha he measure of all open subses of is sricly posiive. A porfolio is modeled by a vecor whose componens denoe posiions or holdings i, i = 1 n,in he basic asses. The definiion of arbirage exends as follows. Definiion 1.6. Nonnegaive Porfolio A porfolio is nonnegaive if i gives rise o nonnegaive expeced pay-offs under almos all evens G of nonzero probabiliy, i.e., such ha [ n ] E P i A i T x x G (1.36) i=1 i=1 Definiion 1.7. Arbirage: Single-Period Coninuous Case The marke admis arbirage if eiher of he following condiions holds: A1. There is a nonnegaive porfolio of negaive iniial price n i=1 ia i <. A2. There is a nonnegaive porfolio of zero iniial cos, n i=1 ia i =, for which he expeced payoff is sricly posiive, i.e., E [ P n i=1 ] ia i T > Definiion 1.8. Pricing Measure: Single-Period Coninuous Case 3 A probabiliy measure Q of densiy q x dx on D is a pricing measure if all asse prices a curren ime = can be expressed as follows: A i = EQ f i = f i x q x dx (1.37) for some real number >. The consan is called he discoun facor. The funcions f i x = A i T x are payoff funcions for a given sae or scenario x. 3 Laer we relae such pricing measures o he case of arbirary choices of numeraire asse wherein he pricing formula involves an expecaion of asse prices relaive o he chosen numeraire asse price. Changes in numeraire correspond o changes in he probabiliy measure.

16 16 C H A P T E R 1. Pricing heory Marke compleeness is defined in a manner similar o ha in he single-period discree case of he previous secion. From he foregoing definiions of arbirage and pricing measure we hen have he following resul, whose proof is lef as an exercise. Theorem 1.2. Fundamenal Theorem of Asse Pricing (Coninuous Single-Period Case) Assume ha all scenarios in are possible. Then he following saemens hold rue. There is no arbirage if and only if here is a pricing measure for which all scenarios are possible. If he linear span of he se of basic insrumens A i T, i = 1 n, is complee and here is no arbirage, hen here is a unique pricing measure Q consisen wih he prices A i of he reference asses a curren ime =. The single-period pricing formalism can also be exended o he case of a muliperiod discree-ime financial model, where rading is allowed o ake place a a finie number of inermediae daes. This feaure gives rise o dynamic rading sraegies, wih porfolios in he basic asses being rebalanced a discree poins in ime. The foregoing definiions and noions of arbirage and asse pricing mus hen be modified and exended subsanially. Raher han presen he heory for such discree-ime models, we shall insead inroduce more imporan heoreical ools in he following secions ha will allow us ulimaely o consider coninuous-ime financial models. Muliperiod discree-ime (coninuous-sae-space) models can hen be obained, if desired, as special cases of he coninuous models via a discreizaion of ime. A furher discreizaion of he sae space leads o discree-ime muliperiod finie financial models. 1.3 Mulivariae Coninuous Disribuions: Basic Tools Marginal probabiliy disribuions arise, for insance, when one is compuing expecaions on some reduced subspace of random variables. Consider, for example, a se of coninuous random variables ha can be separaed or grouped ino wo random vecor spaces X = X 1 X m and Y = Y 1 Y n m ha can ake on values x = x 1 x m m and y = y 1 y n m n m, respecively, wih 1 m<n, n 2. The funcion p x y is he join probabiliy densiy or probabiliy disribuion funcion (pdf) in he produc space n = m n m. The inegral p y y p x y dx (1.38) m defines a marginal densiy p y y. This funcion describes a probabiliy densiy in he subspace of random vecors Y n m and inegraes o uniy over n m. The condiional densiy funcion, denoed by p x Y = y p x y for he random vecor X, is defined on he subspace of m (for a given vecor value Y = y) and is defined by he raio of he join probabiliy densiy funcion and he marginal densiy funcion for he random vecor Y evaluaed a y: p x y = p x y (1.39) p y y assuming p y y. From he foregoing wo relaions i is simple o see ha, for any given y, he condiional densiy also inegraes o uniy over x m.

17 1.3 Mulivariae Coninuous Disribuions: Basic Tools 17 Condiional disribuions play an imporan role in finance and pricing heory. As we will see laer, derivaive insrumens can be priced by compuing condiional expecaions. Assuming a condiional disribuion, he condiional expecaion of a coninuous random variable g = g X Y, given Y = y, is defined by E g Y = y = g x y p x y dx (1.4) m Given any wo coninuous random variables X and Y, hen E X Y = y is a number while E X Y is iself a random variable as Y is random, i.e., has no been fixed. We hen have he following propery ha relaes uncondiional and condiional expecaions: E X = E [ E X Y ] = E X Y = y p y y dy (1.41) This propery is useful for compuing expecaions by condiioning. More generally, for a random variable given by he funcion g = g X Y we have he propery n m E g = g x y p x y dxdy m [ ] = g x y p x y dx p y y dy n m m = E g Y = y p y y dy = E [ E g Y ] (1.42) n m Funcions of random variables, such as g X Y, are of course also random variables. In general, he pdf of a random variable given by a mapping f = f X n is he funcion p f, ( ) P f X + p f = lim (1.43) defined on some open or closed inerval beween a and b. This inerval may be finie or infinie; some examples are 1,, and. The cumulaive disribuion funcion (cdf) C f for he random variable f is defined as C f z = z a p f d (1.44) and gives he probabiliy P a f z, wih C f b = 1. Le us consider anoher independen real-valued random variable g c d, where (c,d) is generally any oher inerval. We recall ha any wo random variables f and g are independen if he join pdf (or cdf) of f and g is given by he produc of he respecive marginal pdfs (or cdfs). The sum of wo independen random variables f and g is again a random variable h = f + g. The cumulaive disribuion funcion, denoed by C h, for he random variable h is given by he convoluion inegral C h = p f p g d d + b = a p f C g d = d c p g C f d (1.45)

18 18 C H A P T E R 1. Pricing heory where p g and C g are he densiy and cumulaive disribuion funcions, respecively, for he random variable g. By differeniaing he cumulaive disribuion funcion we find he densiy funcion for he variable h: p h = b a p f p g d = d c p g p f d (1.46) The preceding formulas are someimes useful because hey provide he cumulaive (or densiy) funcions for a sum of wo independen random variables as convoluion inegrals of he separae densiy and cumulaive funcions. The definiion for cumulaive disribuion funcions exends ino he mulivariae case in he obvious manner. Given a pdf p n for n -valued random vecors X = X 1 X n, he corresponding cdf is he funcion C p n defined by he join probabiliy C p x = P X 1 x 1 X n x n = xn x1 p x dx (1.47) We recall ha any wo random variables X i and X j (i j) are independen if he join probabiliy P X i a X j b = P X i a P X j b for all a b, i.e., if he evens X i a and X j b are independen. Hence, for wo independen random variables he join cdf and join pdf are equal o he produc of he marginal cdf and marginal pdf, respecively: p x i x j = p i x i p j x j and C p x i x j = C i x i C j x j. Anoher useful formula for mulivariae disribuions is he relaionship beween probabiliy densiies (wihin he same probabiliy measure, say, dx expressed on differen variable spaces or coordinae variables. Tha is, if p x and p X x represen probabiliy densiies on n-dimensional real-valued vecor spaces x and x, respecively and he wo spaces are relaed by a one-o-one coninuously differeniable mapping x = x x, hen p x = p X x d x dx (1.48) where d x is he Jacobian marix of he inverible ransformaion x x. The noaion M dx refers o he deerminan of a marix M. A probabiliy disribuion ha plays a disinguished role is he n-dimensional Gaussian (or normal) disribuion, wih mean (or average) vecor = 1 n, defined on x n as follows: p x C = ( 1 2 n C exp 1 ) 2 x C 1 x (1.49) The shorhand noaion x N n C is also used o denoe he values of an n-dimensional random vecor wih componens x 1 x n ha are obained by sampling wih disribuion p x C. C = C ij is called covariance marix and enjoys he propery of being posiive definie, i.e., is such ha he inner produc x Cx x Cx > for all real vecors x, and C ij = C ji. I follows ha he cdf of he n-dimensional mulivariae normal random vecor is defined by he n-dimensional Gaussian inegral n x C = xn x1 p x C dx (1.5) A paricularly imporan special case of equaion (1.5) for n = 1 is he univariae sandard normal cdf (i.e., 1 x 1 ), defined by N x 1 x e y2 /2 dy (1.51) 2

19 1.3 Mulivariae Coninuous Disribuions: Basic Tools 19 The mean of a random vecor X wih given pdf p x, is defined by he componens i = E [ X i = x i p x dx = xp i x dx (1.52) n and he covariance marix elemens are defined by he expecaions C ij Cov X i X j = E [ X i i X j j ] = x i i x j j p x dx (1.53) n for all i j = 1 n. The sandard deviaion of he random variable X i is defined as he square roo of he variance: i Var X i = E [ X i i 2 (1.54) and he correlaion beween wo random variables X i and X j is defined as follows: ij Corr X i X j = C ij i j (1.55) Since C ii = i, he correlaion marix has a uni diagonal, i.e., ii = 1. As well, hey obey he inequaliy ij 1 (see Problem 1 of his secion). For random variables ha may be posiively or negaively correlaed (e.g., as is he case for differen sock reurns) i follows ha 1 ij 1 (1.56) In he paricular case of a mulivariae normal disribuion wih posiive definie covariance marix as in equaion (1.49), he sric inequaliies 1 < ij < 1 hold. The main propery of normal disribuions is ha he convoluion of wo normal disribuions is also normal. A random variable ha is a sum of random normal variables is, herefore, also normally disribued (see Problem 2). Because of his propery, mulivariae normal disribuions can be regarded as affine ransformaions of sandard normal disribuions wih = n 1 and C = I n n (he ideniy marix). Consider he vecor = 1 n of independen sandard normal variables wih zero mean and uni covariance, i.e., wih probabiliy densiy p = n i=1 e 2 i /2 2 (1.57) If L = L ij,isann-dimensional marix, hen he random vecor X = + L is normally disribued wih mean and covariance C = LL, marix ranspose. Indeed, aking expecaions over he componens gives and E [ X i ] = E [ i + E [ X i i X j j ] [( n = E ] n L ij j = i (1.58) j=1 k=1 L ik k)( n l=1 L jl l )] = n L ik L jk = C ij (1.59) k=1

20 2 C H A P T E R 1. Pricing heory Here we have used E i = and E i j = ij, where ij is Kronecker s dela, wih value 1 if i = j and zero oherwise. Conversely, given a posiive definie marix C, one can show ha here is a lower riangular marix L = L ij wih L ij = ifj>i, such ha C = LL. The marix L can be evaluaed wih a procedure known as Cholesky facorizaion. As discussed laer in he book, his algorihm is a he basis of Mone Carlo mehods for generaing scenarios obeying a mulivariae normal disribuion wih a given covariance marix. A special case of a mulivariae normal is he bivariae disribuion defined for x = x 1 x 2 2 : p x 1 x = e [ x x ] 2 x 1 1 x The parameers i and i > are he mean and he sandard deviaion of X i, i = 1 2, respecively, and ( 1 < <1) is he correlaion beween X 1 and X 2, i.e., = 12 = C 12 / 1 2. In his case he covariance marix is ( ) 2 C = (1.6) and he lower Cholesky facorizaion of C is given by ( ) 1 L = (1.61) The correlaion marix is simply = ( ) 1 (1.62) 1 wih Cholesky facorizaion =, ( ) 1 = (1.63) 1 2 The covariance marix has inverse C 1 = ( 1/ 2 1 / 1 2 / 1 2 1/ 2 2 ) (1.64) Condiional and marginal densiies of he bivariae disribuion are readily obained by inegraing over one of he variables in he foregoing join densiy (see Problem 3). For mulivariae normal disribuions one has he following general resul, which we sae wihou proof. Proposiion. Consider he random vecor X n wih pariion X = X 1 X 2, X 1 m, X 2 n m wih 1 m<n, n 2. Le X N n C wih mean = 1 2 and n n covariance ( ) C11 C C = 12 C 21 C 22

21 1.3 Mulivariae Coninuous Disribuions: Basic Tools 21 wih nonzero deerminan C 22, where C 11 and C 22 are m m and n m n m covariance marices of X 1 and X 2, respecively, and C 12 = C 21 is he m n m crosscovariance marix of he wo subspace vecors. The condiional disribuion of X 1, given X 2 = x 2, is he m-dimensional normal densiy wih mean = 1 + C 12 C 1 22 x 2 2 and covariance C = C 11 C 12 C 1 22 C 21, i.e., x 1 N m C condiional on X 2 = x 2. A relaively simple proof of his resul follows by applicaion of known ideniies for pariioned marices. This resul is useful in manipulaing mulidimensional inegrals involving normal disribuions. In deriving analyical properies associaed wih expecaions or condiional expecaions of random variables, he concep of a characerisic funcion is useful. Given a pdf p n for a coninuous random vecor X = X 1 X n, he (join) characerisic funcion is he funcion X n defined by X u = E e iu X = e iu x p x dx (1.65) n where u = u 1 u n n, i 1. Since X is he Fourier ransform of p, hen from he heory of Fourier inegral ransforms we know ha he characerisic funcion gives a complee characerizaion of he probabiliic laws of X, equivalenly as p does. Tha is, any wo random variables having he same characerisic funcion are idenically disribued; i.e., he characerisic funcion uniquely deermines he disribuion. From he definiion we observe ha X is always a well-defined coninuous funcion, given ha p is a bonafide disribuion. Evaluaing a he origin gives X = E 1 = 1. The exisence of derivaives k X / u k i, k 1 is dependen upon he exisence of he respecive momens of he random variables X i. The kh momen of a single random variable X is defined by m k = E X k = while he kh cenered momen is defined by k = E X k = x k p x dx (1.66) x k p x dx (1.67) = E X, k 1. [Noe: for X = X i hen p p i is he ih marginal pdf, i = E X i, k k i = E X i i k, ec.] From hese inegrals we hus see ha he exisence of he momens depends on he decay behavior of p a he limis x ±. For insance, a disribuion ha exhibis asympoic decay a leas as fas as a decaying exponenial has finie momens o all orders. Obvious examples of hese include he disribuions of normal, exponenial, and uniform random variables. In conras, disribuions ha decay as some polynomial o a negaive power may, a mos, only possess a number of finie momens. A classic case is he Suden disribuion wih ineger d degrees of freedom, which can be shown o possess only momens up o order d. This disribuion is discussed in Chaper 4 wih respec o modeling risk-facor reurn disribuions. The momens can be obained from he derivaives of X a he origin. However, i is a lile more convenien o work direcly wih he momen-generaing funcion (mgf). The (join) momen-generaing funcion is given by M X u = E e u X = n e u x p x dx (1.68)

22 22 C H A P T E R 1. Pricing heory If he mgf exiss (which is no always rue), hen i is relaed o he characerisic funcion: M X u = X iu. I can be shown ha if E X r <, hen M X u (and X u ) has coninuous rh derivaive a u = wih momens given by m k = E X k = dk M X du k = i k dk X du k k= 1 r (1.69) Hence, a random variable X has finie momens of all orders when M X u (or X u ) is coninuously differeniable o any order wih m k = M k X = i k k X, k = 1. Given wo independen random variables X and Y, he characerisic funcion of he sum X + Y simplifies ino a produc of funcions: X+Y u = E e iu X+Y = E e iux E e iuy = X u Y u. Hence for Z = n i=1 X i we have Z u = n i=1 X i u if all X i are independen. Characerisic funcions or mgfs can be obained in analyically closed form for various common disribuions. Problems Problem 1. Make use of equaions (1.53) and (1.54) and he Schwarz inequaliy, ( ) 2 ( )( ) f x g x dx f x 2 dx g x 2 dx x n (1.7) n n n o demonsrae he inequaliy C ij i j, hence ij 1. Problem 2. Consider wo independen normal random variables X and Y wih probabiliy disribuions p x x = 1 x 1 x 2 /2 2 e x y y 2 /2 2 and p y y = e y (1.71) x 2 y 2 respecively. Use convoluion o show ha Z = X + Y is a normal random variable wih probabiliy disribuion p z z = 1 z z 2 /2 2 e z (1.72) z 2 where z 2 = 2 x + 2 y and z = x + y. Problem 3. Show ha he join densiy funcion for he bivariae normal has he form /2 2 p x y = e y [ exp and hereby obain he marginal and condiional disribuions: p Y Y = [ 1 exp ( x 1 ) 2 ] 1 y 2 (1.73) e Y 2 2 /2 2 2 (1.74) [ x 1 ] 2 ] 1 Y 2 (1.75) 2 1 p x Y = Verify ha his same resul follows as a special case of he foregoing proposiion.