Introduction to Arbitrage Pricing

Transcription

1 Inroducion o Arbirage Pricing Marek Musiela 1 School of Mahemaics, Universiy of New Souh Wales, 252 Sydney, Ausralia Marek Rukowski 2 Insiue of Mahemaics, Poliechnika Warszawska, -661 Warszawa, Poland 1 musiela@alpha.mahs.unsw.edu.au 2 markru@alpha.im.pw.edu.pl

2 Conens I Spo and Fuures Markes 5 1 An Inroducion o Financial Derivaives Derivaive Markes Opions Fuures Conracs and Opions Forward Conracs Call and Pu Spo Opions One-period Spo Marke Replicaing Porfolios Maringale Measure for a Spo Marke Absence of Arbirage Opimaliy of Replicaion Pu Opion Fuures Call and Pu Opions Fuures Conracs and Fuures Prices One-period Fuures Marke Maringale Measure for a Fuures Marke Absence of Arbirage One-period Spo/Fuures Marke Forward Conracs Opions of American Syle General No-arbirage Inequaliies The Cox-Ross-Rubinsein Model The CRR Model of a Sock Price The CRR Opion Pricing Formula The Black-Scholes Opion Pricing Formula Probabilisic Approach Condiional Expecaions Maringale Measure Risk-neural Valuaion Formula Valuaion of American Opions American Pu Opion Opions on a Dividend-paying Sock The Black-Scholes Model Iô Sochasic Calculus Iô s Lemma Predicable Represenaion Propery Girsanov s heorem The Black-Scholes Opion Valuaion Formula

3 2 CONTENTS Sock Price Self-financing Sraegies Maringale Measure for he Spo Marke The Pu-Call Pariy for Spo Opions The Black-Scholes PDE Sensiiviy Analysis Opion on a Dividend-paying Sock Hisorical Volailiy Implied Volailiy Numerical Mehods Fuures Marke Self-financing Sraegies Maringale Measure for he Fuures Marke The Black Fuures Opion Formula Opions on Forward Conracs Foreign Marke Derivaives Cross-currency Marke Model Domesic Maringale Measure Foreign Maringale Measure Currency Forward Conracs and Opions Forward Exchange Rae Currency Opion Valuaion Formula Foreign Equiy Forward Conracs Forward Price of a Foreign Sock Quano Forward Conracs Foreign Equiy Opions Opions Sruck in a Foreign Currency Opions Sruck in Domesic Currency Quano Opions Equiy-linked Foreign Exchange Opions American Opions Valuaion of American Claims American Call and Pu Opions Early Exercise Represenaion of an American Pu Free Boundary Problem Approximaions of he American Pu Price Opion on a Dividend-paying Sock Exoic Opions Packages Collars Break Forwards Range Forwards Forward-sar Opions Chooser Opions Compound Opions Digial Opions Barrier Opions Asian Opions Baske Opions Lookback Opions

4 CONTENTS 3 II Fixed-income Markes Ineres Raes and Relaed Conracs Zero-coupon Bonds Term Srucure of Ineres Raes Forward Ineres Raes Shor-erm Ineres Rae Coupon-bearing Bonds Ineres Rae Fuures Treasury Bond Fuures Bond Opions Treasury Bill Fuures Eurodollar Fuures Ineres Rae Swaps Models of he Shor-erm Rae Arbirage-free Family of Bond Prices Expecaions Hypoheses Case of Iô Processes Single-facor Models American Bond Opions Opions on Coupon-bearing Bonds Models of Forward Raes HJMModel Absence of Arbirage Forward Measure Approach Forward Price Forward Maringale Measure Gaussian HJM Model Model of LIBOR Raes Discree-enor Case Coninuous-enor Case Model of Forward Swap Raes Opion Valuaion in Gaussian Models European Spo Opions Bond Opions Sock Opions Opion on a Coupon-bearing Bond Pricing of General Coningen Claims Replicaion of Opions Fuures Prices Fuures Opions PDE Approach o Ineres Rae Derivaives PDEs for Spo Derivaives PDEs for Fuures Derivaives Valuaion of Swap Derivaives Ineres Rae Swaps Gaussian Model Forward Caps and Floors Capions

5 4 CONTENTS Swapions Model of Forward LIBOR Raes Caps Swapions Model of Forward Swap Raes Bibliography 187

6 Par I Spo and Fuures Markes 5

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8 Chaper 1 An Inroducion o Financial Derivaives We shall firs review briefly he mos imporan kinds of financial conracs, raded eiher on exchanges or over-he-couner OTC, beween financial insiuions and heir cliens. For a deailed accoun of he fundamenal feaures of spo i.e., cash and fuures financial markes he reader is referred, for insance, o Cox and Rubinsein 1985, Richken 1987, Chance 1989, Duffie 1989, Merrick 199, Kolb 1991, Dubofsky 1992, Edwards and Ma 1992, Sucliffe 1993, Hull 1994, 1997 or Redhead Derivaive Markes We sar his secion by describing he spo opions marke, afer which we shall focus on fuures conracs and fuures opions. Finally, he basic feaures of forward conracs will be discussed Opions Opions are examples of exchange-raded derivaive securiies ha is, securiies whose value depends on he prices of oher more basic securiies so called primary securiies or asses such as socks or bonds. By socks we mean common socks ha is, shares in he ne asse value no bearing fixed ineres. They give he righ o dividends according o profis, afer paymens on preferred socks. The preferred socks give some special righs o he sockholder, ypically a guaraneed fixed dividend. A bond is a cerificae issued by a governmen or a public company promising o repay borrowed money a a fixed rae of ineres a a specified ime. Basically, a call opion a pu opion, respecively is he righ o buy o sell, respecively he opion s underlying asse a some fuure dae for a prespecified price. Opions in paricular, warrans 1 have been raded for cenuries; unprecedened expansion of he opions marke sared, however, quie recenly wih he inroducion in 1973 of exchange-raded opions on socks in he Unied Saes. I should be emphasized ha mos of he raded opions are of American ype ha is, he holder has he righ o exercise an opion a any insan before he opion s expiry. When an invesor noifies his broker of he inenion o exercise an opion, he broker in urn noifies he OCC 2 member who clears he invesor s rade. This member hen places an exercise order wih he OCC. The OCC randomly selecs a member wih an ousanding shor posiion in he same 1 A warran is a call opion issued by a company or a financial insiuion. Warrans are frequenly issued by companies on heir own socks; new shares are issued when warrans are exercised. In some cases,he warrans are subsequenly raded on an exchange. Warrans are bough and sold in much he same way as socks. 2 OCC sands for he Opions Clearing Corporaion. The OCC keeps he record of all long and shor posiions. The OCC guaranees ha he opion wrier will fulfil obligaions under he erms of he opion conrac. The OCC has a number of members, and all opion rades mus be cleared hrough a member. 7

9 8 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES opion. The chosen member, in urn, selecs a paricular invesor who has wrien he opion such an invesor is said o be assigned. If he opion is a call, his invesor is required o sell sock a he so-called srike price or exercise price if i is a pu, he is required o buy sock a he srike price. When he opion is exercised, he open ineres ha is, he number of opions ousanding goes down by one. In addiion o opions on paricular socks, a large variey of oher opion conracs are raded nowadays on exchanges: foreign currency opions such as, e.g., Briish pound, German mark or Japanese yen opion conracs raded on he Philadelphia Exchange, index opions e.g., hose on S&P1 and S&P5 raded on he CBOE, and fuures opions e.g., he Treasury bond fuures opion raded on he Chicago Board of Trade CBOT. Ineres rae opions are also implici in several oher ineres rae insrumens, such as caps or floors hese are, however, over-he-couner raded conracs. Derivaive financial insrumens involving opions are also widely raded ouside he exchanges by financial insiuions and heir cliens. We may idenify here such conracs as swapions ha is, opions on an ineres rae swap, or a large variey of exoic opions. Finally, opions are implici in several financial insrumens, for example in some bond or sock issues callable bonds, savings bonds or converible bonds, o menion a few. One of he mos appealing feaures of opions apar from he obvious chance of making exraordinary reurns is he possibiliy of easy speculaion on he fuure behavior of a sock price. Usually his is done by means of so called combinaions ha is, combined posiions in several opions, and possibly he underlying asse. For insance, a bull spread is porfolio creaed by buying a call opion on a sock wih a cerain srike price and selling a call opion on he same sock wih a higher srike price boh opions have he same expiry dae. Equivalenly, bull spreads can be creaed by buying a pu wih a low srike price and selling a pu wih a high srike price. An invesor enering a bull spread is hoping ha he sock price will increase. Like a bull spread, a bear spread can be creaed by buying a call wih one srike price and selling a call wih anoher srike price. The srike price of he opion purchased is now greaer han he srike price of he opion sold, however. An invesor who eners a bear spread is hoping ha he sock price will decline. A buerfly spread involves posiions in opions wih hree differen srike prices. I can be creaed by buying a call opion wih a relaively low srike price, buying anoher call opion wih a relaively high srike price, and selling wo call opions wih a srike price halfway beween he oher wo srike prices. The buerfly spread leads o a profi if he sock price says close o he srike price of he call opions sold, bu gives rise o a small loss if here is a significan sock price move in eiher direcion. A porfolio creaed by selling a call opion wih a cerain srike price and buying a longer-mauriy call opion wih he same srike price is commonly known as a calendar spread. A sraddle involves buying a call and pu wih he same srike price and expiry dae. If he sock price is close o his srike price a expiry of he opion, he sraddle leads o a loss. A sraddle is appropriae when an invesor is expecing a large move in sock price bu does no know in which direcion he move will be. Relaed ypes of rading sraegies are commonly known as srips, sraps and srangles Fuures Conracs and Opions Anoher imporan class of exchange-raded derivaive securiies comprises fuures conracs, and opions on fuures conracs, commonly known as fuures opions. Fuures conracs apply o a wide range of commodiies e.g., sugar, wool, gold and financial asses e.g., currencies, bonds, sock indices; he larges exchanges on which fuures conracs are raded are he Chicago Board of Trade and he Chicago Mercanile Exchange CME. In wha follows, we resric our aenion o financial fuures as opposed o commodiy fuures. To make rading possible, he exchange specifies cerain sandardized feaures of he conrac. Fuures prices are regularly repored in he financial press. They are deermined on he floor in he same way as oher prices ha is, by he law of supply and demand. If more invesors wan o go long han o go shor, he price goes up; if he reverse is rue, he price falls. Posiions in fuures conracs are governed by a specific daily selemen procedure commonly referred o as marking o marke. An invesor s iniial deposi,

10 1.1. DERIVATIVE MARKETS 9 known as he iniial margin, is adjused daily o reflec he gains or losses ha are due o he fuures price movemens. Le us consider, for insance, a pary assuming a long posiion he pary who agreed o buy. When here is a decrease in he fuures price, her margin accoun is reduced by an appropriae amoun of money, her broker has o pay his sum o he exchange and he exchange passes he money on o he broker of he pary who assumes he shor posiion. Similarly, when he fuures price rises, brokers for paries wih shor posiions pay money o he exchange, and brokers of paries wih long posiions receive money from he exchange. This way, he rade is marked o marke a he close of each rading day. Finally, if he delivery period is reached and delivery is made by a pary wih a shor posiion, he price received is generally he fuures price a he ime he conrac was las marked o marke. In a fuures opion, he underlying asse is a fuures conrac. The fuures conrac normally maures shorly afer he expiry of he opion. When he holder of a call fuures opion exercises he opion, she acquires from he wrier a long posiion in he underlying fuures conrac plus a cash amoun equal o he excess of he curren fuures price over he opion s srike price. Since fuures conracs have zero value and can be closed ou immediaely, he payoff from a fuures opion is he same as he payoff from a sock opion, wih he sock price replaced by he fuures price. Fuures opions are now available for mos of he insrumens on which fuures conracs are raded. The mos acively raded fuures opion is he Treasury bond fuures opion raded on he Chicago Board of Trade. On some markes for insance, on he Ausralian marke, fuures opions have he same feaures as fuures conracs hemselves ha is, hey are no paid up-fron as classic opions, bu are raded a he margin. Unless oherwise saed, by a fuures opion we mean here a sandard opion wrien on a fuures conrac Forward Conracs A forward conrac is an agreemen o buy or sell an asse a a cerain fuure ime for a cerain price. One of he paries o a forward conrac assumes a long posiion and agrees o buy he underlying asse on a cerain specified fuure dae for a delivery price; he oher pary assumes a shor posiion and agrees o sell he asse on he same dae for he same price. A he ime he conrac is enered ino, he delivery price is deermined so ha he value of he forward conrac o boh paries is zero. I is hus clear ha some feaures of forward conracs resemble hose of fuures conracs. However, unlike fuures conracs, forward conracs do no rade on exchanges. Also, a forward conrac is seled only once, a he mauriy dae. The holder of he shor posiion delivers he asse o he holder of he long posiion in reurn for a cash amoun equal o he delivery price. The following lis summarizes he main differences beween forward and fuures conracs. 1. Conrac specificaion and delivery Fuures conracs. The conrac precisely specifies he underlying insrumen and price. Delivery daes and delivery procedures are sandardized o a limied number of specific daes per year, a approved locaions. Delivery is no, however, he objecive of he ransacion, and less han 2% are delivered. Forward conracs. There is an almos unlimied range of insrumens, wih individually negoiaed prices. Delivery can ake place on any individual negoiaed dae and locaion. Delivery is he objec of he ransacion, wih over 9% of forward conracs seled by delivery. 2. Prices Fuures conracs. The price is he same for all paricipans, regardless of ransacion size. Typically, here is a daily price limi alhough, for insance, on he FT-SE 1 index, fuures prices are unlimied. Trading is usually by open oucry aucion on he rading floor of he exchange. Prices are disseminaed publicly. Each ransacion is conduced a he bes price available a he ime. Forward conracs. The price varies wih he size of he ransacion, he credi risk, ec. There are no daily price limis. Trading akes place by elephone and fax beween individual buyers and sellers. Prices are no disseminaed publicly. Hence, here is no guaranee ha he price is he bes available.

11 1 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES 3. Markeplace and rading hours Fuures conracs. Trading is cenralized on he exchange floor, wih worldwide communicaions, during hours fixed by he exchange. Forward conracs. Trading akes place by elephone and elex beween individual buyers and sellers. Trading is over-he-couner world-wide, 24 hours per day, wih elephone and elex access. 4. Securiy deposi and margin Fuures conracs. The exchange rules require an iniial margin and he daily selemen of variaion margins. A cenral clearing house is associaed wih each exchange o handle he daily revaluaion of open posiions, cash paymens and delivery procedures. The clearing house assumes he credi risk. Forward conracs. The collaeral level is negoiable, wih no adjusmen for daily price flucuaions. There is no separae clearing house funcion. Thus, he marke paricipan bears he risk of he couner-pary defauling. 5. Volume and marke liquidiy Fuures conracs. Volume and open ineres informaion is published. There is very high liquidiy and ease of offse wih any oher marke paricipan due o sandardized conracs. Forward conracs. Volume informaion is no available. The limied liquidiy and offse is due o he variable conrac erms. Offse is usually wih he original couner-pary. 1.2 Call and Pu Spo Opions Le us firs describe briefly he se of general assumpions imposed on our models of financial markes. We consider hroughou, unless explicily saed oherwise, he case of a so-called fricionless marke, meaning ha: all invesors are price-akers, all paries have he same access o he relevan informaion, here are no ransacion coss or commissions, and all asses are assumed o be perfecly divisible and liquid. There is no resricion whasoever on he size of a bank credi, and he lending and borrowing raes are equal. Finally, individuals are allowed o sell shor any securiy and receive full use of he proceeds of course, resiuion is required for payoffs made o securiies held shor. Unless oherwise specified, by an opion we shall mean hroughou a European opion, giving he righ o exercise he opion only a he expiry dae. In mahemaical erms, he problem of pricing of American opions is closely relaed o opimal sopping problems. Unforunaely, closed-form expressions for he prices of American opions are rarely available; for insance, no closed-form soluion is available for he price of an American pu opion in he now classic framework of he Black-Scholes opion pricing model. A European call opion wrien on a common sockis a financial securiy ha gives is holder he righ bu no he obligaion o buy he underlying sock on a prespecified dae and for a prespecified price. The ac of making his ransacion is referred o as exercising he opion. If an opion is no exercised, we say i is abandoned. Anoher class of opions comprises so-called American opions. These may be exercised a any ime on or before he prespecified dae. The prespecified fixed price, say K, is ermed he srike or exercise price; he erminal dae, denoed by T in wha follows, is called he expiry dae or mauriy. I should be emphasized ha an opion gives he holder he righ o do somehing; however, he holder is no obliged o exercise his righ. In order o purchase an opion conrac, an invesor needs o pay an opion s price or premium o a second pary a he iniial dae when he conrac is enered ino. Le us denoe by S T he sock price a he erminal dae T. I is naural o assume ha S T is no known a ime, hence S T gives rise o uncerainy in our model. We argue ha from he perspecive of he opion holder, he payoff g a expiry dae T from a European call opion is given by he formula gs T =S T K + def = max {S T K, }, 1.1 ha is o say gs T = { ST K if S T >K opion is exercised, if S T K opion is abandoned.

12 1.2. CALL AND PUT SPOT OPTIONS 11 In fac, if a he expiry dae T he sock price is lower han he srike price, he holder of he call opion can purchase an underlying sock direcly on a spo i.e., cash marke, paying less han K. In oher words, i would be irraional o exercise he opion, a leas for an invesor who prefers more wealh o less. On he oher hand, if a he expiry dae he sock price is greaer han K, an invesor should exercise his righ o buy he underlying sock a he srike price K. Indeed, by selling he sock immediaely a he spo marke, he holder of he call opion is able o realize an insananeous ne profi S T K noe ha ransacion coss and/or commissions are ignored here. In conras o a call opion, a pu opion gives is holder he righ o sell he underlying asse by a cerain dae for a prespecified price. Using he same noaion as above, we arrive a he following expression for he payoff h a mauriy T from a European pu opion or more explicily hs T =K S T + def = max {K S T, }, 1.2 { if ST K opion is abandoned, hs T = K S T if S T <K opion is exercised. I follows immediaely ha he payoffs of call and pu opions saisfy he following simple bu useful equaliy gs T hs T =S T K + K S T + = S T K. 1.3 The las equaliy can be used, in paricular, o derive he so-called pu-call pariy relaionship for opion prices. Basically, pu-call pariy means ha he price of a European pu opion is deermined by he price of a European call opion wih he same srike and expiry dae, he curren price of he underlying asse, and he properly discouned value of he srike price One-period Spo Marke Le us sar by considering an elemenary example of an opion conrac. Example Assume ha he curren sock price is $28, and afer hree monhs he sock price may eiher rise o $32or decline o $26. We shall find he raional price of a 3-monh European call opion wih srike price K = $28, provided ha he simple risk-free ineres rae r for 3-monh deposis and loansis r = 5%. Suppose ha he subjecive probabiliy of he price rise is.2, and ha of he fall is.8; hese assumpions correspond, loosely, o a so-called bear marke. Noe ha he word subjecive means ha we ake he poin of view of a paricular individual. Generally speaking, he wo paries involved in an opion conrac may have and usually do have differing assessmens of hese probabiliies. To model a bull marke one may assume, for example, ha he firs probabiliy is.8, so ha he second is.2. Le us focus firs on he bear marke case. The erminal sock price S T may be seen as a random variable on a probabiliy space Ω = {ω 1,ω 2 } wih a probabiliy measure P given by P{ω 1 } =.2 =1 P{ω 2 }. Formally, S T is a funcion S T :Ω R + given by he following formula { S S T ω = u = 32, if ω = ω 1, S d = 26, if ω = ω 2. Consequenly, he erminal opion s payoff X = C T =S T K + saisfies { C C T ω = u =4, if ω = ω 1, C d =, if ω = ω 2.

13 12 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES Noe ha he expeced value under P of he discouned opion s payoff equals E P 1 + r 1 C T = =7.62. I is clear ha he above expecaion depends on he choice of he probabiliy measure P; ha is, i depends on he invesor s assessmen of he marke. For a call opion, he expecaion corresponding o he case of a bull marke would be greaer han ha which assumes a bear marke. In our example, he expeced value of he discouned payoff from he opion under he bull marke hypohesis equals Sill, o consruc a reliable model of a financial marke, one has o guaranee he uniqueness of he price of any derivaive securiy. This can be done by applying he concep of he so-called replicaing porfolio, which we will now inroduce Replicaing Porfolios The wo-sae opion pricing model presened below was developed independenly by Sharpe 1978 and Rendleman and Barer 1979 a poin worh menioning is ha he ground-breaking papers of Black and Scholes 1973 and Meron 1973, who examined he arbirage pricing of opions in a coninuous-ime framework, were published five years earlier. The idea is o consruc a porfolio a ime which replicaes exacly he opion s erminal payoff a ime T. Le φ = φ =α,β R 2 denoe a porfolio of an invesor wih a shor posiion in one call opion. More precisely, le α sand for he number of shares of sock held a ime, and β be he amoun of money deposied on a bank accoun or borrowed from a bank. By V φ we denoe he wealh of his porfolio a daes = and = T ; ha is, he payoff from he porfolio φ a given daes. I should be emphasized ha once he porfolio is se up a ime, i remains fixed unil he erminal dae T. Therefore, for is wealh process V φ we have V φ =α S + β and V T φ =α S T + β 1 + r. 1.4 We say ha a porfolio φ replicaes he opion s erminal payoff whenever V T φ =C T, ha is, if { V V T φω = u φ =α S u +1+rβ = C u, if ω = ω 1, V d φ =α S d +1+rβ = C d, if ω = ω 2. For he daa of Example 1.2.1, he porfolio φ is deermined by he following sysem of linear equaions { 32 α +1.5 β =4, 26 α +1.5 β =, wih unique soluion α =2/3 and β = Observe ha for every call we are shor, we hold α of sock 3 and he dollar amoun β in riskless bonds in he hedging porfolio. Pu anoher way, by purchasing shares and borrowing agains hem in he righ proporion, we are able o replicae an opion posiion. Acually, one can easily check ha his propery holds for any coningen claim X which seles a ime T. I is naural o define he manufacuring cos C of a call opion as he iniial invesmen needed o consruc a replicaing porfolio, i.e., C = V φ =α S + β =2/ = I should be sressed ha in order o deermine he manufacuring cos of a call we did no need o know he probabiliy of he rise or fall of he sock price. In oher words, i appears ha he manufacuring cos is invarian wih respec o individual assessmens of marke behavior. In paricular, i is idenical under he bull and bear marke hypoheses. To deermine he raional price of a call we have used he opion s srike price, he curren value of he sock price, he range of flucuaions in he sock price ha is, he fuure levels of he sock price, and he risk-free rae of 3 We shall refer o he number of shares held for each call sold as he hedge raio. Basically,o hedge means o reduce risk by making ransacions ha reduce exposure o marke flucuaions.

14 1.2. CALL AND PUT SPOT OPTIONS 13 ineres. The invesor s ransacions and he corresponding cash flows may be summarized by he following wo exhibis one wrien call opion C, a ime = α shares purchased α S, amoun of cash borrowed β, and a ime = T payoff from he call opion C T, α shares sold α S T, loan paid back ˆrβ, where ˆr =1+r. I should be observed ha no ne iniial invesmen is needed o esablish he above porfolio; ha is, he porfolio is cosless. On he oher hand, for each possible level of sock price a ime T, he hedge exacly breaks even on he opion s expiry dae. Also, i is easy o verify ha if he call were no priced a $21.59, i would be possible for a sure profi o be gained, eiher by he opion s wrier if he opion s price were greaer han is manufacuring cos or by is buyer in he opposie case. Sill, he manufacuring cos canno be seen as a fair price of a claim X, unless he marke model is arbirage-free, in a sense examined below. Indeed, i may happen ha he manufacuring cos of a non-negaive claim is a sricly negaive number. Such a phenomenon conradics he usual assumpion ha i is no possible o make riskless profis Maringale Measure for a Spo Marke Alhough, as shown above, subjecive probabiliies are useless when pricing an opion, probabilisic mehods play an imporan role in coningen claims valuaion. They rely on he noion of a maringale, which is, inuiively, a probabilisic model of a fair game. In order o apply he so-called maringale mehod of derivaive pricing, one has o find firs a probabiliy measure P equivalen o P, and such ha he discouned or relaive sock price process S, which is defined by he formula S = S, S T =1+r 1 S T, follows a P -maringale; ha is, he equaliy S = E P ST holds. Such a probabiliy measure P is called a maringale measure for he discouned sock price process S. In he case of a wo-sae model, he probabiliy measure P is easily seen o be uniquely deermined provided i exiss by he following linear equaion S =1+r 1 p S u +1 p S d, 1.5 where p = P {ω 1 } and 1 p = P {ω 2 }. Solving his equaion for p yields P {ω 1 } = 1 + rs S d S u S d, P {ω 2 } = Su 1 + rs S u S d. 1.6 Le us now check ha he price C coincides wih C, where we wrie C o denoe he expeced value under P of an opion s discouned erminal payoff ha is C def = E P 1 + r 1 C T = E P 1 + r 1 S T K +. Indeed, using he daa of Example we find p =17/3, so ha C =1+r 1 p C u +1 p C d =21.59 = C. Remarks. Observe ha since he process S follows a P -maringale, we may say ha he discouned sock price process may be seen as a fair game model in a risk-neural economy ha is, in he sochasic economy in which he probabiliies of fuure sock price flucuaions are deermined by

15 14 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES he maringale measure P. I should be sressed, however, ha he fundamenal idea of arbirage pricing is based solely on he exisence of a porfolio ha hedges perfecly he risk exposure relaed o uncerain fuure prices of risky securiies. Therefore, he probabilisic properies of he model are no essenial. In paricular, we do no assume ha he real-world economy is acually risk-neural. On he conrary, he noion of a risk-neural economy should be seen raher as a echnical ool. The aim of inroducing he maringale measure is wofold: firsly, i simplifies he explici evaluaion of arbirage prices of derivaive securiies; secondly, i describes he arbirage-free propery of a given pricing model for primary securiies in erms of he behavior of relaive prices. This approach is frequenly referred o as he parial equilibrium approach, as opposed o he general equilibrium approach. Le us sress ha in he laer heory he invesors preferences, usually described in sochasic models by means of heir expeced uiliy funcions, play an imporan role. To summarize, he noion of an arbirage price for a derivaive securiy does no depend on he choice of a probabiliy measure in a paricular pricing model for primary securiies. More precisely, using sandard probabilisic erminology, his means ha he arbirage price depends on he suppor of a subjecive probabiliy measure P, bu is invarian wih respec o he choice of a paricular probabiliy measure from he class of muually equivalen probabiliy measures. In financial erminology, his can be resaed as follows: all invesors agree on he range of fuure price flucuaions of primary securiies; hey may have differen assessmens of he corresponding subjecive probabiliies, however Absence of Arbirage Le us consider a simple wo-sae, one-period, wo-securiy marke model defined on a probabiliy space Ω = {ω 1,ω 2 } equipped wih he σ-fields F = {, Ω}, F T =2 Ω i.e., F T conains all subses of Ω, and a probabiliy measure P on Ω, F T such ha P{ω 1 } and P{ω 2 } are sricly posiive numbers. The firs securiy is a sock whose price process is modelled as a sricly posiive discreeime process S =S {,T }. We assume ha he process S is F -adaped, i.e., ha he random variables S are F -measurable for {,T}. This means ha S is a real number, and { S u if ω = ω S T ω = 1, S d if ω = ω 2, where, wihou loss of generaliy, S u >S d. The second securiy is a riskless bond whose price process is B =1,B T =1+r for some real r. Le Φ sand for he linear space of all sock-bond porfolios φ = φ =α,β, where α and β are real numbers clearly, he class Φ may be hus idenified wih R 2. We shall consider he pricing of coningen claims in a securiy marke model M =S, B, Φ. We shall now check ha an arbirary coningen claim X which seles a ime T i.e., any F T -measurable real-valued random variable admis a unique replicaing porfolio in our marke model. In oher words, an arbirary coningen claim X is aainable in he marke model M. Indeed, if { X u if ω = ω Xω = 1, X d if ω = ω 2, hen he replicaing porfolio φ is deermined by he following sysem of linear equaions { α S u +1+rβ = X u, α S d +1+rβ = X d 1.7, which admis a unique soluion α = Xu X d S u S d, β = Xd S u X u S d 1 + rs u S d, 1.8 for arbirary values of X u and X d. Consequenly, an arbirary coningen claim X admis a unique manufacuring cos π X inm which is given by he formula π X def = V φ =α S + β = Xu X d S u S d S + Xd S u X u S d 1 + rs u S d. 1.9

16 1.2. CALL AND PUT SPOT OPTIONS 15 As already menioned, he manufacuring cos of a sricly posiive coningen claim may appear o be a negaive number, in general. If his were he case, here would be a profiable riskless rading sraegy so-called arbirage opporuniy involving only he sock and riskless borrowing and lending. To exclude such siuaions, which are clearly inconsisen wih any broad noion of a raional marke equilibrium as i is common o assume ha invesors are non-saiaed, meaning ha hey prefer more wealh o less, we have o impose furher essenial resricions on our marke model. Definiion We say ha a securiy pricing model M is arbirage-free if here is no porfolio φ Φ for which V φ =,V T φ and P{V T φ > } >. 1.1 A porfolio φ for which he se 1.1 of condiions is saisfied is called an arbirage opporuniy. A srong arbirage opporuniy is a porfolio φ for which V φ < and V T φ I is cusomary o ake eiher 1.1 or 1.11 as he definiion of an arbirage opporuniy. Noe, however, ha boh noions are no necessarily equivalen. We are in a posiion o inroduce he noion of an arbirage price; ha is, he price derived using he no-arbirage argumens. Definiion Suppose ha he securiy marke M is arbirage-free. Then he manufacuring cos π X is called he arbirage price of X a ime in securiy marke M. As he nex resul shows, under he absence of arbirage in a marke model, he manufacuring cos may be seen as he unique raional price of a given coningen claim ha is, he unique price compaible wih any raional marke equilibrium. Proposiion Suppose ha he spo marke M =S, B, Φ is arbirage-free. Le H sand for he raional price process of some aainable coningen claim X; more explicily, H R and H T = X. Le Φ H denoe he class of all porfolios in sock, bond and derivaive securiy H. Then he spo marke S, B, H, Φ H is arbirage-free if and only if H = π X. Proof. Since he proof is sraighforward, i is lef o he reader Opimaliy of Replicaion Le us show ha replicaion is, in a sense, an opimal way of hedging. Firsly, we say ha a porfolio φ perfecly hedges agains X if V T φ X, ha is, whenever { α S u +1+rβ X u, α S d +1+rβ X d The minimal iniial cos of a perfec hedging porfolio agains X is called he seller s price of X, and i is denoed by πx. s Le us check ha πx s =π X. By denoing c = V φ, we may rewrie 1.12 as follows { α S u S 1 + r + c1 + r X u, α S d S 1 + r + c1 + r X d I is rivial o check ha he minimal c R for which 1.13 holds is acually ha value of c for which inequaliies in 1.13 become equaliies. This means ha he replicaion appears o be he leas expensive way of perfec hedging for he seller of X. Le us now consider he oher pary of he conrac, i.e., he buyer of X. Since he buyer of X can be seen as he seller of X, he associaed problem is o minimize c R, subjec o he following consrains { α S u S 1 + r + c1 + r X u, α S d S 1 + r + c1 + r X d.

17 16 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES I is clear ha he soluion o his problem is π s X = πx =π X, so ha replicaion appears o be opimal for he buyer also. We conclude ha he leas price he seller is ready o accep for X equals he maximal amoun he buyer is ready o pay for i. If we define he buyer s price of X, denoed by π b X, by seing π b X = π s X, hen π s X =π b X =π X; ha is, all prices coincide. This shows ha in a wo-sae, arbirage-free model, he arbirage price of any coningen claim can be defined using he opimaliy crierion. I appears ha such an approach o arbirage pricing can be exended o oher models; we prefer, however, o define he arbirage price as ha value of he price which excludes arbirage opporuniies. Indeed, he fac ha observed marke prices are close o arbirage prices prediced by a suiable sochasic model should be explained by he presence of he raders known as arbirageurs 4 on financial markes, raher han by he clever invesmen decisions of mos marke paricipans. The nex proposiion explains he role of he so-called risk-neural economy in arbirage pricing of derivaive securiies. Observe ha he imporan role of risk preferences in classic equilibrium asse pricing heory is lef aside in he presen conex. Noice, however, ha he use of a maringale measure P in arbirage pricing corresponds o he assumpion ha all invesors are risk-neural, meaning ha hey do no differeniae beween all riskless and risky invesmens wih he same expeced rae of reurn. The arbirage valuaion of derivaive securiies is hus done as if an economy acually were risk-neural. Formula 1.14 shows ha he arbirage price of a coningen claim X can be found by firs modifying he model so ha he sock earns a he riskless rae, and hen compuing he expeced value of he discouned claim o he bes of our knowledge, his mehod of compuing he price was discovered in Cox and Ross Proposiion The spo marke M =S, B, Φ is arbirage-free if and only if he discouned sock price process S admis a maringale measure P equivalen o P. In his case, he arbirage price a ime of any coningen claim X which seles a ime T is given by he risk-neural valuaion formula π X =E P 1 + r 1 X, 1.14 or explicily π X = S 1 + r S d S u S d X u 1+r + Su S 1 + r S u S d X d 1+r Proof. We know already ha he maringale measure for S equivalen o P exiss if and only if he unique soluion p of equaion 1.5 saisfies <p < 1. Suppose here is no equivalen maringale measure for S ; for insance, assume ha p 1. Our aim is o consruc explicily an arbirage opporuniy in he marke model S, B, Φ. To his end, observe ha he inequaliy p 1 is equivalen o 1+rS S u recall ha S u is always greaer han S d. The porfolio φ = 1,S saisfies V φ = and { S V T φ = u +1+rS if ω = ω 1, S d +1+rS > if ω = ω 2, so ha φ is indeed an arbirage opporuniy. On he oher hand, if p, hen he inequaliy S d 1 + rs holds, and i is easily seen ha in his case he porfolio ψ =1, S = φ is an arbirage opporuniy. Finally, if <p < 1 for any porfolio φ saisfying V φ =, hen by virue of 1.9 and 1.6 we ge p V u φ+1 p V d φ = so ha V d φ < when V u φ > and V d φ > if V u φ <. This shows ha here are no arbirage opporuniies in M when <p < 1. To prove formula 1.14 i is enough o compare i 4 An arbirageur is ha marke paricipan who consisenly uses he price discrepancies o make almos risk-free profis. Arbirageurs are relaively few,bu hey are far more acive han mos long-erm invesors.

18 1.2. CALL AND PUT SPOT OPTIONS 17 wih 1.9. Alernaively, we may observe ha for he unique porfolio φ =α,β which replicaes he claim X, we have E P 1 + r 1 X = E P 1 + r 1 V T φ = E P α ST + β = α S + β = V φ =π X, so ha we are done. Remarks. The choice of he bond price process as a discoun facor is no essenial. Suppose, on he conrary, ha we have chosen he sock price S as a numeraire. In oher words, we now consider he bond price B discouned by he sock price S B = B /S for {,T}. The maringale measure P for he process B is deermined by he equaliy B = E PB T, or explicily p 1+r 1+r + q Su S d = 1, 1.16 S where q =1 p. One finds ha P{ω 1 } = p = 1 S d rs S u S d S u S d 1.17 and 1 P{ω 2 } = q = S u 1 S u S d 1 + rs S d S u I is easy o show ha he properly modified version of he risk-neural valuaion formula has he following form π X =S E P S 1 T X, 1.19 where X is a coningen claim which seles a ime T. I appears ha in some circumsances he choice of he sock price as a numeraire is more convenien han ha of he savings accoun. Le us apply his approach o he call opion of Example One finds easily ha p =.62, and hus formula 1.19 gives as expeced. Ĉ = S E P S 1 T S T K + =21.59 = C, Pu Opion We refer once again o Example However, we shall now focus on a European pu opion insead of a call opion. Since he buyer of a pu opion has he righ o sell a sock a a given dae T, he erminal payoff from he opion is now P T =K S T +, i.e., { P P T ω = u =, if ω = ω 1, P d =2, if ω = ω 2, where we have aken, as before, K= $28. The porfolio φ =α,β which replicaes he European pu opion is hus deermined by he following sysem of linear equaions { 32 α +1.5 β =, 26 α +1.5 β =2, so ha α = 1/3 and β = Consequenly, he arbirage price P of he European pu opion equals P = 1/ =8.25.

19 18 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES Noice ha he number of shares in a replicaing porfolio is negaive. This means ha an opion wrier who wishes o hedge risk exposure should sell shor a ime he number α =1/3 shares of sock for each sold pu opion. The proceeds from he shor-selling of shares, as well as he opion s premium, are invesed in an ineres-earning accoun. To find he arbirage price of he pu opion we may alernaively apply Proposiion By virue of 1.14, wih X = P T, we ge P = E P 1 + r 1 P T =8.25. Finally, he pu opion value can also be found by applying he following relaionship beween he prices of call and pu opions. Corollary The following pu-call pariy relaionship is valid C P = S 1 + r 1 K. 1.2 Proof. The formula is an immediae consequence of equaliy 1.3 and he pricing formula 1.14 applied o he claim S T K. I is worhwhile o menion ha relaionship 1.2 is universal ha is, i does no depend on he choice of he model he only assumpion we need o make is he addiiviy of he price. Using he pu-call pariy, we can calculae once again he arbirage price of he pu opion. Formula 1.2 yields immediaely P = C S +1+r 1 K =8.25. For ease of furher reference, we shall wrie down explici formulae for he call and pu price in he one-period, wo-sae model. We assume, as usual, ha S u >K>S d. Then and C = S 1 + r S d S u S d P = Su S 1 + r S u S d S u K 1+r, 1.21 K S d 1+r Fuures Call and Pu Opions We will firs describe very succincly he main feaures of fuures conracs, which are refleced in sochasic models of fuures markes o be developed laer. As in he previous secion, we will focus mainly on he arbirage pricing of European call and pu opions; clearly, insead of he spo price of he underlying asse, we will now consider is fuures price. The model of fuures prices we adop here is quie similar o he one used o describe spo prices. Sill, due o he specific feaures of fuures conracs used o se up a replicaing sraegy, one has o modify significanly he way in which he payoff from a porfolio is defined Fuures Conracs and Fuures Prices A fuures conrac is an agreemen o buy or sell an asse a a cerain dae in he fuure for a cerain price. The imporan feaure of hese conracs is ha hey are raded on exchanges. Consequenly, he auhoriies need o define precisely all he characerisics of each fuures conrac in order o make rading possible. More imporanly, he fuures price he price a which a given fuures conrac is enered ino is deermined on a given fuures exchange by he usual law of demand and supply in a similar way as for spo prices of lised socks. Fuures prices are herefore seled daily and he quoaions are repored in he financial press. A fuures conrac is referred o by is delivery monh, however an exchange specifies he period wihin ha monh when delivery mus be made. The exchange specifies he amoun of he asse o be delivered for one conrac, as well as some addiional deails when necessary e.g., he qualiy of a given commodiy or he mauriy

20 1.3. FUTURES CALL AND PUT OPTIONS 19 of a bond. From our perspecive, he mos fundamenal feaure of a fuures conrac is he way he conrac is seled. The procedure of daily selemen of fuures conracs is called marking o marke. A fuures conrac is worh zero when i is enered ino; however, each invesor is required o deposi funds ino a margin accoun. The amoun ha should be deposied when he conrac is enered ino is known as he iniial margin. A he end of each rading day, he balance of he invesor s margin accoun is adjused in a way ha reflecs daily movemens of fuures prices. To be more specific, if an invesor assumes a long posiion, and on a given day he fuures price rises, he balance of he margin accoun will also increase. Conversely, he balance of he margin accoun of any pary wih a shor posiion in his fuures conrac will be properly reduced. Inuiively, i is hus possible o argue ha fuures conracs are acually closed ou afer each rading day, and hen sar afresh he nex rading day. Obviously, o offse a posiion in a fuures conrac, an invesor eners ino he opposie rade o he original one. Finally, if he delivery period is reached, he delivery is made by he pary wih a shor posiion One-period Fuures Marke I will be convenien o sar his secion wih a simple example which, in fac, is a sraighforward modificaion of Example o a fuures marke. Example Le f = f S, T be a one-period process which models he fuures price of a cerain asse S, for he selemen dae T T. We assume ha f = 28, and { f f T ω = u = 32, if ω = ω 1, f d = 26, if ω = ω 2, where T = 3 monhs. 5 We consider a 3-monh European fuures call opion wih srike price K = $28. As before, we assume ha he simple risk-free ineres rae for 3-monh deposis and loans is r = 5%. The payoff from he fuures call opion C f T =f T K + equals C f T ω = { C fu =4, if ω = ω 1, C fd =, if ω = ω 2. A porfolio φ which replicaes he opion is composed of α fuures conracs and β unis of cash invesed in riskless bonds or borrowed. The wealh process V f φ, {, T}, of his porfolio equals V f φ =β, since fuures conracs are worhless when hey are firs enered ino. Furhermore, he erminal wealh of φ is V f T φ =α f T f +1+rβ, 1.23 where he firs erm on he righ-hand side represens gains or losses from he fuures conrac, and he second corresponds o a savings accoun or loan. Noe ha 1.23 reflecs he fac ha fuures conracs are marked o marke daily ha is, afer each period in our model. A porfolio φ =α,β is said o replicae he opion when V f T = Cf T, or more explicily, if he equaliies V f T ω = { α f u f rβ = C fu, if ω = ω 1, α f d f +1+rβ = C fd, if ω = ω 2 are saisfied. For Example 1.3.1, his gives he following sysem of linear equaions { 4 α +1.5 β =4, 2 α +1.5 β =, 5 Noice ha in he presen conex,he knowledge of he selemen dae T of a fuures conrac is no essenial. I is sufficien o assume ha T T.

21 2 CHAPTER 1. AN INTRODUCTION TO FINANCIAL DERIVATIVES yielding α =2/3 and β =12.7. The manufacuring cos of a fuures call opion is hus C f = V f φ =β =12.7. Similarly, he unique porfolio replicaing a sold pu opion is deermined by he following condiions { 4 α +1.5 β =, 2 α +1.5 β =2, so ha α = 1/3 and β =12.7in his case. Consequenly, he manufacuring coss of pu and call fuures opions are equal in our example. As we shall see soon, his is no a pure coincidence; in fac, by virue of formula 1.29 below, he prices of call and pu fuures opions are equal when he opion s srike price coincides wih he iniial fuures price of he underlying asse. The above consideraions may be summarized by means of he following exhibis noe ha β is a posiive number one sold fuures opion C f, a ime = fuures conracs, cash deposied in a bank β = C f, and a ime = T where, as before, ˆr =1+r. opion s payoff C f T, profis/losses from fuures α f T f, cash wihdrawal ˆrβ, Maringale Measure for a Fuures Marke We are looking now for a probabiliy measure P which makes he fuures price process wih no discouning follow a P-maringale. A probabiliy P, if i exiss, is hus deermined by he equaliy I is easily seen ha f = E P ft = pf u +1 p f d P{ω 1 } = p = f f d f u f d, P{ω2 } =1 p = f u f f u f d Using he daa of Example 1.3.1, one finds easily ha p = 1/3. Consequenly, he expeced value under he probabiliy P of he discouned payoff from he fuures call opion equals C f = E P 1 + r 1 f T K + =12.7= C f. This illusraes he fac ha he maringale approach may be used also in he case of fuures markes, wih a suiable modificaion of he noion of a maringale measure. Remarks. Using he radiional erminology of mahemaical finance, we may conclude ha he risk-neural fuures economy is characerized by he fair-game propery of he process of a fuures price. Remember ha he risk-neural spo economy is he one in which he discouned sock price as opposed o he sock price iself models a fair game Absence of Arbirage In his subsecion, we shall sudy a general wo-sae, one-period model of a fuures price. We consider he filered probabiliy space Ω, F {,T }, P inroduced in Sec The firs process, which inends o model he dynamics of he fuures price of a cerain asse for he fixed selemen dae T T, is an adaped and sricly posiive process f = f S, T,=,T. More specifically, f is assumed o be a real number, and f T is he following random variable { f f T ω = u, if ω = ω 1, f d, if ω = ω 2,