Option Pricing: A Simplified Approach

Transcription

1 Otio Pricig: A Simlified Aroach Joh C. Cox Massachusetts Istitute of Techology ad Staford Uiversity Stehe A. Ross Yale Uiversity Mark Rubistei Uiversity of Califoria, Berkeley March 1979 revised July 1979 ublished uder the same title i Joural of Fiacial Ecoomics Setember 1979 [1978 wier of the Pomeraze Prize of the Chicago Board Otios Exchage] [rerited i Dyamic Hedgig: A Guide to Portfolio Isurace, edited by Do Luski Joh Wiley ad Sos 1988] [rerited i The Hadbook of Fiacial Egieerig, edited by Cliff Smith ad Charles Smithso Harer ad Row 1990] [rerited i Readigs i Futures Markets ublished by the Chicago Board of Trade, Vol. VI 1991] [rerited i Vasicek ad Beyod: Aroaches to Buildig ad Alyig Iterest Rate Models, edited by Risk Publicatios, Ala Brace 1996] [rerited i The Debt Market, edited by Stehe Ross ad Fraco Modigliai Edward Lear Publishig 000] [rerited i The Iteratioal Library of Critical Writigs i Fiacial Ecoomics: Otios Markets edited by G.M. Costatiides ad A..G. Malliaris Edward Lear Publishig 000] Abstract This aer resets a simle discrete-time model for valuig otios. The fudametal ecoomic riciles of otio ricig by arbitrage methods are articularly clear i this settig. Its develomet requires oly elemetary mathematics, yet it cotais as a secial limitig case the celebrated Black-Scholes model, which has reviously bee derived oly by much more difficult methods. The basic model readily leds itself to geeralizatio i may ways. Moreover, by its very costructio, it gives rise to a simle ad efficiet umerical rocedure for valuig otios for which remature exercise may be otimal. Our best thaks go to William Share, who first suggested to us the advatages of the discrete-time aroach to otio ricig develoed here. We are also grateful to our studets over the ast several years. Their favorable reactios to this way of resetig thigs ecouraged us to write this article. We have received suort from the Natioal Sciece Foudatio uder Grats Nos. SOC ad SOC

2 1. Itroductio A otio is a security that gives its ower the right to trade i a fixed umber of shares of a secified commo stock at a fixed rice at ay time o or before a give date. The act of makig this trasactio is referred to as exercisig the otio. The fixed rice is termed the strike rice, ad the give date, the exiratio date. A call otio gives the right to buy the shares; a ut otio gives the right to sell the shares. Otios have bee traded for ceturies, but they remaied relatively obscure fiacial istrumets util the itroductio of a listed otios exchage i Sice the, otios tradig has eoyed a exasio urecedeted i America securities markets. Otio ricig theory has a log ad illustrious history, but it also uderwet a revolutioary chage i At that time, Fischer Black ad Myro Scholes reseted the first comletely satisfactory equilibrium otio ricig model. I the same year, Robert Merto exteded their model i several imortat ways. These ath-breakig articles have formed the basis for may subsequet academic studies. As these studies have show, otio ricig theory is relevat to almost every area of fiace. For examle, virtually all cororate securities ca be iterreted as ortfolios of uts ad calls o the assets of the firm. 1 Ideed, the theory alies to a very geeral class of ecoomic roblems the valuatio of cotracts where the outcome to each arty deeds o a quatifiable ucertai future evet. Ufortuately, the mathematical tools emloyed i the Black-Scholes ad Merto articles are quite advaced ad have teded to obscure the uderlyig ecoomics. However, thaks to a suggestio by William Share, it is ossible to derive the same results usig oly elemetary mathematics. I this article we will reset a simle discrete-time otio ricig formula. The fudametal ecoomic riciles of otio valuatio by arbitrage methods are articularly clear i this settig. Sectios ad 3 illustrate ad develo this model for a call otio o a stock that ays o divideds. Sectio 4 shows exactly how the model ca be used to lock i ure arbitrage rofits if the market rice of a otio differs from the value give by the model. I sectio 5, we will show that our aroach icludes the Black-Scholes model as a secial limitig case. By takig the limits i a differet way, we will also obtai the Cox-Ross 1975 um rocess model as aother secial case. 1 To take a elemetary case, cosider a firm with a sigle liability of a homogeeous class of ure discout bods. The stockholders the have a call o the assets of the firm which they ca choose to exercise at the maturity date of the debt by ayig its ricial to the bodholders. I tur, the bods ca be iterreted as a ortfolio cotaiig a default-free loa with the same face value as the bods ad a short ositio i a ut o the assets of the firm. Share 1978 has artially develoed this aroach to otio ricig i his excellet ew book, Ivestmets. Redlema ad Bartter 1978 have recetly ideedetly discovered a similar formulatio of the otio ricig roblem.

3 Other more geeral otio ricig roblems ofte seem immue to reductio to a simle formula. Istead, umerical rocedures must be emloyed to value these more comlex otios. Michael Brea ad Eduardo Schwartz 1977 have rovided may iterestig results alog these lies. However, their techiques are rather comlicated ad are ot directly related to the ecoomic structure of the roblem. Our formulatio, by its very costructio, leads to a alterative umerical rocedure that is both simler, ad for may uroses, comutatioally more efficiet. Sectio 6 itroduces these umerical rocedures ad exteds the model to iclude uts ad calls o stocks that ay divideds. Sectio 7 cocludes the aer by showig how the model ca be geeralized i other imortat ways ad discussig its essetial role i valuatio by arbitrage methods.. The Basic Idea Suose the curret rice of a stock is S $50, ad at the ed of a eriod of time, its rice must be either S* $5 or S* $100. A call o the stock is available with a strike rice of K $50, exirig at the ed of the eriod. 3 It is also ossible to borrow ad led at a 5% rate of iterest. The oe iece of iformatio left ufurished is the curret value of the call, C. However, if riskless rofitable arbitrage is ot ossible, we ca deduce from the give iformatio aloe what the value of the call must be Cosider the followig levered hedge: 1 write 3 calls at C each, buy shares at $50 each, ad 3 borrow $40 at 5%, to be aid back at the ed of the eriod. Table 1 gives the retur from this hedge for each ossible level of the stock rice at exiratio. Regardless of the outcome, the hedge exactly breaks eve o the exiratio date. Therefore, to revet rofitable riskless arbitrage, its curret cost must be zero; that is, The curret value of the call must the be C $0. 3C To kee matters simle, assume for ow that the stock will ay o cash divideds durig the life of the call. We also igore trasactio costs, margi requiremets ad taxes. 3

4 Table 1 Arbitrage Table Illustratig the Formatio of a Riskless Hedge exiratio date reset date S* $5 S* $100 write 3 calls 3C 150 buy shares borrow total If the call were ot riced at $0, a sure rofit would be ossible. I articular, if C $5, the above hedge would yield a curret cash iflow of $15 ad would exeriece o further gai or loss i the future. O the other had, if C $15, the the same thig could be accomlished by buyig 3 calls, sellig short shares, ad ledig $40. Table 1 ca be iterreted as demostratig that a aroriately levered ositio i stock will relicate the future returs of a call. That is, if we buy shares ad borrow agaist them i the right roortio, we ca, i effect, dulicate a ure ositio i calls. I view of this, it should seem less surrisig that all we eeded to determie the exact value of the call was its strike rice, uderlyig stock rice, rage of movemet i the uderlyig stock rice, ad the rate of iterest. What may seem more icredible is what we do ot eed to kow: amog other thigs, we do ot eed to kow the robability that the stock rice will rise or fall. Bulls ad bears must agree o the value of the call, relative to its uderlyig stock rice This examle is very simle, but it shows several essetial features of otio ricig. Ad we will soo see that it is ot as urealistic as it seems. 3. The Biomial Otio Pricig Formula I this sectio, we will develo the framework illustrated i the examle ito a comlete valuatio method. We begi by assumig that the stock rice follows a multilicative biomial rocess over discrete eriods. The rate of retur o the stock over each eriod ca have two ossible values: u 1 with robability q, or d 1 with robability 1 q. Thus, if the curret stock rice is S, the stock rice at the ed of the eriod will be either us or ds. We ca rereset this movemet with the followig diagram: us with robability q S ds with robability 1 q We also assume that the iterest rate is costat. Idividuals may borrow or led as much as they wish at this rate. To focus o the basic issues, we will cotiue to assume that there are o 4

5 taxes, trasactio costs, or margi requiremets. Hece, idividuals are allowed to sell short ay security ad receive full use of the roceeds. 4 Lettig r deote oe lus the riskless iterest rate over oe eriod, we require u > r > d. If these iequalities did ot hold, there would be rofitable riskless arbitrage oortuities ivolvig oly the stock ad riskless borrowig ad ledig. 5 To see how to value a call o this stock, we start with the simlest situatio: the exiratio date is ust oe eriod away. Let C be the curret value of the call, C u be its value at the ed of the eriod if the stock rice goes to us ad C d be its value at the ed of the eriod if the stock rice goes to ds. Sice there is ow oly oe eriod remaiig i the life of the call, we kow that the terms of its cotract ad a ratioal exercise olicy imly that C u max[0, us K] ad C d max[0, ds K]. Therefore, C C u max[0, us K] with robability q C d max[0, ds K] with robability 1 q Suose we form a ortfolio cotaiig Δ shares of stock ad the dollar amout B i riskless bods. 6 This will cost ΔS + B. At the ed of the eriod, the value of this ortfolio will be ΔuS + rb with robability q ΔS + B ΔdS + rb with robability 1 q Sice we ca select Δ ad B i ay way we wish, suose we choose them to equate the edof-eriod values of the ortfolio ad the call for each ossible outcome. This requires that Solvig these equatios, we fid ΔuS + rb C u ΔdS + rb C d Cu Cd ucd dcu, B 1 u d S u d r 4 Of course, restitutio is required for ayouts made to securities held short. 5 We will igore the uiterestig secial case where q is zero or oe ad u d r. 6 Buyig bods is the same as ledig; sellig them is the same as borrowig. 5

6 With Δ ad B chose i this way, we will call this the hedgig ortfolio. If there are to be o riskless arbitrage oortuities, the curret value of the call, C, caot be less tha the curret value of the hedgig ortfolio, ΔS + B. If it were, we could make a riskless rofit with o et ivestmet by buyig the call ad sellig the ortfolio. It is temtig to say that it also caot be worth more, sice the we would have a riskless arbitrage oortuity by reversig our rocedure ad sellig the call ad buyig the ortfolio. But this overlooks the fact that the erso who bought the call we sold has the right to exercise it immediately. Suose that ΔS + B < S K. If we try to make a arbitrage rofit by sellig calls for more tha ΔS + B, but less tha S K, the we will soo fid that we are the source of arbitrage rofits rather tha the reciiet. Ayoe could make a arbitrage rofit by buyig our calls ad exercisig them immediately. We might hoe that we will be sared this embarrassmet because everyoe will somehow fid it advatageous to hold the calls for oe more eriod as a ivestmet rather tha take a quick rofit by exercisig them immediately. But each erso will reaso i the followig way. If I do ot exercise ow, I will receive the same ayoff as a ortfolio with ΔS i stock ad B i bods. If I do exercise ow, I ca take the roceeds, S K, buy this same ortfolio ad some extra bods as well, ad have a higher ayoff i every ossible circumstace. Cosequetly, o oe would be willig to hold the calls for oe more eriod. Summig u all of this, we coclude that if there are to be o riskless arbitrage oortuities, it must be true that Cu Cd ucd dcu &, r - d, u - r # C S + B + Cu Cd / r u d u d r $ * + * u d u d % if this value is greater tha S K, ad if ot, C S K. 7 Equatio ca be simlified by defiig so that we ca write r d ad u d 1 u r u d C [C u + 1 C d ]/r 3 It is easy to see that i the reset case, with o divideds, this will always be greater tha S K as log as the iterest rate is ositive. To avoid sedig time o the uimortat situatios where the iterest rate is less tha or equal to zero, we will ow assume that r is always greater 7 I some alicatios of the theory to other areas, it is useful to cosider otios that ca be exercised oly o the exiratio date. These are usually termed Euroea otios. Those that ca be exercised at ay earlier time as well, such as we have bee examiig here, are the referred to as America otios. Our discussio could be easily modified to iclude Euroea calls. Sice immediate exercise is the recluded, their values would always be give by, eve if this is less tha S K. 6

7 tha oe. Hece, 3 is the exact formula for the value of a call oe eriod rior to the exiratio i terms of S, K, u, d, ad r. To cofirm this, ote that if us K, the S < K ad C 0, so C > S K. Also, if ds K, the C S K/r > S K. The remaiig ossibility is us > K > ds. I this case, C us K/r. This is greater tha S K if 1 ds > rk, which is certaily true as log as r > 1. This formula has a umber of otable features. First, the robability q does ot aear i the formula. This meas, surrisigly, that eve if differet ivestors have differet subective robabilities about a uward or dowward movemet i the stock, they could still agree o the relatioshi of C to S, u, d, ad r. Secod, the value of the call does ot deed o ivestors attitudes toward risk. I costructig the formula, the oly assumtio we made about a idividual s behavior was that he refers more wealth to less wealth ad therefore has a icetive to take advatage of rofitable riskless arbitrage oortuities. We would obtai the same formula whether ivestors are risk-averse or risk-referrig. Third, the oly radom variable o which the call value deeds is the stock rice itself. I articular, it does ot deed o the radom rices of other securities or ortfolios, such as the market ortfolio cotaiig all securities i the ecoomy. If aother ricig formula ivolvig other variables was submitted as givig equilibrium market rices, we could immediately show that it was icorrect by usig our formula to make riskless arbitrage rofits while tradig at those rices. It is easier to uderstad these features if it is remembered that the formula is oly a relative ricig relatioshi givig C i terms of S, u, d, ad r. Ivestors attitudes toward risk ad the characteristics of other assets may ideed ifluece call values idirectly, through their effect o these variables, but they will ot be searate determiats of call value. Fially, observe that r d/u d is always greater tha zero ad less tha oe, so it has the roerties of a robability. I fact, is the value q would have i equilibrium if ivestors were risk-eutral. To see this, ote that the exected rate of retur o the stock would the be the riskless iterest rate, so qus + 1 qds rs ad q r d/u d Hece, the value of the call ca be iterreted as the exectatio of its discouted future value i a risk-eutral world. I light of our earlier observatios, this is ot surrisig. Sice the formula does ot ivolve q or ay measure of attitudes toward risk, the it must be the same for ay set of refereces, icludig risk eutrality. It is imortat to ote that this does ot imly that the equilibrium exected rate of retur o the call is the riskless iterest rate. Ideed, our argumet has show that, i equilibrium, holdig the call over the eriod is exactly equivalet to holdig the hedgig ortfolio. Cosequetly, the risk 7

8 ad exected rate of retur of the call must be the same as that of the hedgig ortfolio. It ca be show that Δ 0 ad B 0, so the hedgig ortfolio is equivalet to a articular levered log ositio i the stock. I equilibrium, the same is true for the call. Of course, if the call is curretly misriced, its risk ad exected retur over the eriod will differ from that of the hedgig ortfolio. Now we ca cosider the ext simlest situatio: a call with two eriods remaiig before its exiratio date. I keeig with the biomial rocess, the stock ca take o three ossible values after two eriods, us u S S dus ds d S Similarly, for the call, C uu max[0, u S K] C u C C du max[0, dus K] C d C dd max[0, d S K] C uu stads for the value of a call two eriods from the curret time if the stock rice moves uward each eriod; C du ad C dd have aalogous defiitios. At the ed of the curret eriod there will be oe eriod left i the life of the call, ad we will be faced with a roblem idetical to the oe we ust solved. Thus, from our revious aalysis, we kow that whe there are two eriods left, C u [C uu + 1 C ud ]/r ad 4 C d [C du + 1 C dd ]/r Agai, we ca select a ortfolio with ΔS i stock ad B i bods whose ed-of-eriod value will be C u if the stock rice goes to us ad C d if the stock rice goes to ds. Ideed, the 8

9 fuctioal form of Δ ad B remais uchaged. To get the ew values of Δ ad B, we simly use equatio 1 with the ew values of C u ad C d. Ca we ow say, as before, that a oortuity for rofitable riskless arbitrage will be available if the curret rice of the call is ot equal to the ew value of this ortfolio or S K, whichever is greater? Yes, but there is a imortat differece. With oe eriod to go, we could la to lock i a riskless rofit by sellig a overriced call ad usig art of the roceeds to buy the hedgig ortfolio. At the ed of the eriod, we kew that the market rice of the call must be equal to the value of the ortfolio, so the etire ositio could be safely liquidated at that oit. But this was true oly because the ed of the eriod was the exiratio date. Now we have o such guaratee. At the ed of the curret eriod, whe there is still oe eriod left, the market rice of the call could still be i disequilibrium ad be greater tha the value of the hedgig ortfolio. If we closed out the ositio the, sellig the ortfolio ad reurchasig the call, we could suffer a loss that would more tha offset our origial rofit. However, we could always avoid this loss by maitaiig the ortfolio for oe more eriod. The value of the ortfolio at the ed of the curret eriod will always be exactly sufficiet to urchase the ortfolio we would wat to hold over the last eriod. I effect, we would have to readust the roortios i the hedgig ortfolio, but we would ot have to ut u ay more moey. Cosequetly, we coclude that eve with two eriods to go, there is a strategy we could follow which would guaratee riskless rofits with o et ivestmet if the curret market rice of a call differs from the maximum of ΔS + B ad S K. Hece, the larger of these is the curret value of the call. Sice Δ ad B have the same fuctioal form i each eriod, the curret value of the call i terms of C u ad C d will agai be C [C u + 1 C d ]/r if this is greater tha S K, ad C S K otherwise. By substitutig from equatio 4 ito the former exressio, ad otig that C du C ud, we obtai C [ C uu + 1 C ud + 1 C dd ]/r 5 [ max[0, u S K] + 1 max[0, dus K] + 1 max[0, d S K]]/r A little algebra shows that this is always greater tha S K if, as assumed, r is always greater tha oe, so this exressio gives the exact value of the call. 8 All of the observatios made about formula 3 also aly to formula 5, excet that the umber of eriods remaiig util exiratio,, ow emerges clearly as a additioal determiat of the call value. For formula 5,. That is, the full list of variables determiig C is S, K,, u, d, ad r. 8 I the curret situatio, with o divideds, we ca show by a simle direct argumet that if there are o arbitrage oortuities, the the call value must always be greater tha S K before the exiratio date. Suose that the call is sellig for S K. The there would be a easy arbitrage strategy that would require o iitial ivestmet ad would always have a ositive retur. All we would have to do is buy the call, short the stock, ad ivest K dollars i bods. See Merto I the geeral case, with divideds, such a argumet is o loger valid, ad we must use the rocedure of checkig every eriod. 9

10 10 We ow have a recursive rocedure for fidig the value of a call with ay umber of eriods to go. By startig at the exiratio date ad workig backwards, we ca write dow the geeral valuatio formula for ay : r K S d u C / ] max[0, 1 0 # $ $ % & * +, -. 6 This gives us the comlete formula, but with a little additioal effort we ca exress it i a more coveiet way. Let a stad for the miimum umber of uward moves that the stock must make over the ext eriods for the call to fiish i-the-moey. Thus a will be the smallest o-egative iteger such that u a d -a S > K. By takig the atural logarithm of both sides of this iequality, we could write a as the smallest o-egative iteger greater tha logk/sd /logu/d. For all < a, max[0, u d - S K] 0 ad for all a, max[0, u d - S K] u d - S K Therefore, a r K S d u C / ] [ 1 # $ % & * +, -. Of course, if a >, the call will fiish out-of-the-moey eve if the stock moves uward every eriod, so its curret value must be zero. By breakig u C ito two terms, we ca write # $ $ % & * * +, - * +, a r d u S C 1 # $ % & * +, -. a Kr 1 Now, the latter bracketed exressio is the comlemetary biomial distributio fuctio φ[a;, ]. The first bracketed exressio ca also be iterreted as a comlemetary biomial distributio fuctio φ[a;, ], where u/r ad 1 d/r1 is a robability, sice 0 < < 1. To see this, ote that < r/u ad r d r u r d u # $ % & # $ % & * +,,

11 I summary: Biomial Otio Pricig Formula C Sφ[a;, ] Kr φ[a;, ] where r d/u d ad u/r a the smallest o-egative iteger greater tha logk/sd /logu/d If a >, the C 0. It is ow clear that all of the commets we made about the oe eriod valuatio formula are valid for ay umber of eriods. I articular, the value of a call should be the exectatio, i a riskeutral world, of the discouted value of the ayoff it will receive. I fact, that is exactly what equatio 6 says. Why, the, should we waste time with the recursive rocedure whe we ca write dow the aswer i oe direct ste? The reaso is that while this oe-ste aroach is always techically correct, it is really useful oly if we kow i advace the circumstaces i which a ratioal idividual would refer to exercise the call before the exiratio date. If we do ot kow this, we have o way to comute the required exectatio. I the reset examle, a call o a stock ayig o divideds, it haes that we ca determie this iformatio from other sources: the call should ever be exercised before the exiratio date. As we will see i sectio 6, with uts or with calls o stocks that ay divideds, we will ot be so lucky. Fidig the otimal exercise strategy will be a itegral art of the valuatio roblem. The full recursive rocedure will the be ecessary. For some readers, a alterative comlete markets iterretatio of our biomial aroach may be istructive. Suose that π u ad π d rereset the state-cotiget discout rates to states u ad d, resectively. Therefore, π u would be the curret rice of oe dollar received at the ed of the eriod, if ad oly if state u occurs. Each security a riskless bod, the stock, ad the otio must all have returs discouted to the reset by π u ad π d if o riskless arbitrage oortuities are available. Therefore, 1 π u r + π d r S π u us + π d ds C π u C u + π d C d The first two equatios, for the bod ad the stock, imly 11

12 & r d # 1 u $ ad % u d r d & u r # 1 $ % u d r Substitutig these equalities for the state-cotiget rices i the last equatio for the otio yields equatio 3. It is imortat to realize that we are ot assumig that the riskless bod ad the stock ad the otio are the oly three securities i the ecoomy, or that other securities must follow a biomial rocess. Rather, however these securities are riced i relatio to others i equilibrium, amog themselves they must coform to the above relatioshis. From either the hedgig or comlete markets aroaches, it should be clear that three-state or triomial stock rice movemets will ot lead to a otio ricig formula based solely o arbitrage cosideratios. Suose, for examle, that over each eriod the stock rice could move to us or ds or remai the same at S. A choice of Δ ad B that would equate the returs i two states could ot i the third. That is, a riskless arbitrage ositio could ot be take. Uder the comlete markets iterretatio, with three equatios i ow three ukow state-cotiget rices, we would lack the redudat equatio ecessary to rice oe security i terms of the other two. 4. Riskless Tradig Strategies The followig umerical examle illustrates how we could use the formula if the curret market rice M ever diverged from its formula value C. If M > C, we would hedge, ad if M < C, reverse hedge, to try ad lock i a rofit. Suose the values of the uderlyig variables are S 80, 3, K 80, u 1.5, d 0.5, r 1.1 I this case, r d/u d 0.6. The relevat values of the discout factor are r , r , r The aths the stock rice may follow ad their corresodig robabilities usig robability are, whe 3, with S 80, 1

13 whe, if S 10,

14 whe, if S 40, Usig the formula, the curret value of the call would be C 0.751[ ] Recall that to form a riskless hedge, for each call we sell, we buy ad subsequetly kee adusted a ortfolio with ΔS i stock ad B i bods, where Δ C u C d /u ds. The followig tree diagram gives the aths the call value may follow ad the corresodig values of Δ:

15 With this relimiary aalysis, we are reared to use the formula to take advatage of misricig i the market. Suose that whe 3, the market rice of the call is 36. Our formula tells us the call should be worth The otio is overriced, so we could la to sell it ad assure ourselves of a rofit equal to the misricig differetial. Here are the stes you could take for a tyical ath the stock might follow. Ste 1 3: Sell the call for 36. Take of this ad ivest it i a ortfolio cotaiig Δ shares of stock by borrowig Take the remaider, , ad ut it i the bak. Ste : Suose the stock goes to 10 so that the ew Δ is Buy more shares of stock at 10 er share for a total exediture of Borrow to ay the bill. With a iterest rate of 0.1, you already owe Thus, your total curret idebtedess is Ste 3 1: Suose the stock rice ow goes to 60. The ew Δ is Sell shares at 60 er share, takig i Use this to ay back art of your borrowig. Sice you ow owe , the reaymet will reduce this to Ste 4d 0: Suose the stock rice ow goes to 30. The call you sold has exired worthless. You ow shares of stock sellig at 30 er share, for a total value of Sell the stock ad reay the that you ow owe o the borrowig. Go back to the bak ad withdraw your origial deosit, which has ow grow to Ste 4u 0: Suose, istead, the stock rice goes to 90. The call you sold is i the moey at the exiratio date. Buy back the call, or buy oe share of stock ad let it be exercised, icurrig a loss of either way. Borrow to cover this, brigig your curret idebtedess to You ow shares of stock sellig at 90 er share, for a total value of Sell the stock ad reay the borrowig. Go back to the bak ad withdraw your origial deosit, which has ow grow to I summary, if we were correct i our origial aalysis about stock rice movemets which did ot ivolve the ueviable task of redictig whether the stock rice would go u or dow, ad if we faithfully adust our ortfolio as rescribed by the formula, the we ca be assured of walkig away i the clear at the exiratio date, while still keeig the origial differetial ad the iterest it has accumulated. It is true that closig out the ositio before the exiratio date, which ivolves buyig back the otio at its the curret market rice, might roduce a loss which would more tha offset our rofit, but this loss could always be avoided by waitig util the exiratio date. Moreover, if the market rice comes ito lie with the formula value before the exiratio date, we ca close out the ositio the with o loss ad be rid of the cocer of keeig the ortfolio adusted. It still might seem that we are deedig o ratioal behavior by the erso who bought the call we sold. If istead he behaves foolishly ad exercises at the wrog time, could he makes thigs worse for us as well as for himself? Fortuately, the aswer is o. Mistakes o his art ca oly 15

16 mea greater rofits for us. Suose that he exercises too soo. I that circumstace, the hedgig ortfolio will always be worth more tha S K, so we could close out the ositio the with a extra rofit. Suose, istead, that he fails to exercise whe it would be otimal to do so. Agai there is o roblem. Sice exercise is ow otimal, our hedgig ortfolio will be worth S K. 9 If he had exercised, this would be exactly sufficiet to meet the obligatio ad close out the ositio. Sice he did ot, the call will be held at least oe more eriod, so we calculate the ew values of C u ad C d ad revise our hedgig ortfolio accordigly. But ow the amout required for the ortfolio, ΔS + B, is less tha the amout we have available, S K. We ca withdraw these extra rofits ow ad still maitai the hedgig ortfolio. The loger the holder of the call goes o makig mistakes, the better off we will be. Cosequetly, we ca be cofidet that thigs will evetually work out right o matter what the other arty does. The retur o our total ositio, whe evaluated at revailig market rices at itermediate times, may be egative. But over a eriod edig o later tha the exiratio date, it will be ositive. I coductig the hedgig oeratio, the essetial thig was to maitai the roer roortioal relatioshi: for each call we are short, we hold Δ shares of stock ad the dollar amout B i bods i the hedgig ortfolio. To emhasize this, we will refer to the umber of shares held for each call as the hedge ratio. I our examle, we ket the umber of calls costat ad made adustmets by buyig or sellig stock ad bods. As a result, our rofit was ideedet of the market rice of the call betwee the time we iitiated the hedge ad the exiratio date. If thigs got worse before they got better, it did ot matter to us. Istead, we could have made the adustmets by keeig the umber of shares of stock costat ad buyig or sellig calls ad bods. However, this could be dagerous. Suose that after iitiatig the ositio, we eeded to icrease the hedge ratio to maitai the roer roortios. This ca be achieved i two ways: a buy more stock, or b buy back some of the calls. If we adust through the stock, there is o roblem. If we isist o adustig through the calls, ot oly is the hedge o loger riskless, but it could eve ed u losig moey This ca hae if the call has become eve more overriced. We would the be closig out art of our ositio i calls at a loss. To remai hedged, the umber of calls we would eed to buy back deeds o their value, ot their rice. Therefore, sice we are ucertai about their rice, we the become ucertai about the retur from the hedge. Worse yet, if the call rice gets high eough, the loss o the closed ortio of our ositio could throw the hedge oeratio ito a overall loss. 9 If we were reverse hedgig by buyig a udervalued call ad sellig the hedgig ortfolio, the we would ourselves wat to exercise at this oit. Sice we will receive S K from exercisig, this will be exactly eough moey to buy back the hedgig ortfolio. 16

17 To see how this could hae, let us reru the hedgig oeratio, where we adust the hedge ratio by buyig ad sellig calls. Ste 1 3: Same as before. Ste : Suose the stock goes to 10, so that the ew Δ The call rice has gotte further out of lie ad is ow sellig for 75. Sice its value is , it is ow overriced by With shares, you must buy back calls to roduce a hedge ratio of / This costs Borrow to ay the bill. With the iterest rate of 0.1, you already owe Thus, your total curret idebtedess is Ste 3 1: Suose the stock goes to 60 ad the call is sellig for Sice the call is ow fairly valued, o further excess rofits ca be made by cotiuig to hold the ositio. Therefore, liquidate by sellig your shares for ad close out the call ositio by buyig back calls for This ets Use this to ay back art of your borrowig. Sice you ow owe , after reaymet you owe.406. Go back to the bak ad withdraw your origial deosit, which has ow grow to Ufortuately, after usig this to reay your remaiig borrowig, you still owe Sice we adusted our ositio at Ste by buyig overriced calls, our rofit is reduced. Ideed, sice the calls were cosiderably overriced, we actually lost moey desite aaret rofitability of the ositio at Ste 1. We ca draw the followig adustmet rule from our exerimet: To adust a hedged ositio, ever buy a overriced otio or sell a uderriced otio. As a corollary, wheever we ca adust a hedged ositio by buyig more of a uderriced otio or sellig more of a overriced otio, our rofit will be ehaced if we do so. For examle, at Ste 3 i the origial hedgig illustratio, had the call still bee overriced, it would have bee better to adust the ositio by sellig more calls rather tha sellig stock. I summary, by choosig the right side of the ositio to adust at itermediate dates, at a miimum we ca be assured of earig the origial differetial ad its accumulated iterest, ad we may ear cosiderably more. 5. Limitig Cases I readig the revious sectios, there is a atural tedecy to associate with each eriod some articular legth of caledar time, erhas a day. With this i mid, you may have had two obectios. I the first lace, rices a day from ow may take o may more tha ust two ossible values. Furthermore, the market is ot oe for tradig oly oce a day, but, istead, tradig takes lace almost cotiuously. These obectios are certaily valid. Fortuately, our otio ricig aroach has the flexibility to meet them. Although it might have bee atural to thik of a eriod as oe day, there was othig that forced us to do so. We could have take it to be a much shorter iterval say a hour or eve a miute. By doig so, we have met both obectios simultaeously. Tradig 17

18 would take lace far more frequetly, ad the stock rice could take o hudreds of values by the ed of the day. However, if we do this, we have to make some other adustmets to kee the robability small that the stock rice will chage by a large amout over a miute. We do ot wat the stock to have the same ercetage u ad dow moves for oe miute as it did before for oe day. But agai there is o eed for us to have to use the same values. We could, for examle, thik of the rice as makig oly a very small ercetage chage over each miute. To make this more recise, suose that h reresets the elased time betwee successive stock rice chages. That is, if t is the fixed legth of caledar time to exiratio, ad is the umber of eriods of legth h rior to exiratio, the h t/ As tradig takes lace more ad more frequetly, h gets closer ad closer to zero. We must the adust the iterval-deedet variables r, u, ad d i such a way that we obtai emirically realistic results as h becomes smaller, or, equivaletly, as. Whe we were thikig of the eriods as havig a fixed legth, r rereseted both the iterest rate over a fixed legth of caledar time ad the iterest rate over oe eriod. Now we eed to make a distictio betwee these two meaigs. We will let r cotiue to mea oe lus the iterest rate over a fixed legth of caledar time. Whe we have occasio to refer to oe lus the iterest rate over a eriod tradig iterval of legth h, we will use the symbol rˆ. Clearly, the size of rˆ deeds o the umber of subitervals,, ito which t is divided. Over the eriods util exiratio, the total retur is rˆ, where t/h. Now ot oly do we wat rˆ to deed o, but we wat it to deed o i a articular way so that as chages the total retur rˆ over the fixed time t remais the same. This is because the iterest rate obtaiable over some fixed legth of caledar time should have othig to do with how we choose to thik of the legth of the time iterval h. If r without the hat deotes oe lus the rate of iterest over a fixed uit of caledar time, the over elased time t, r t is the total retur. 10 Observe that this measure of total retur does ot deed o. As we have argued, we wat to choose the deedece of rˆ o, so that t / r ˆ r for ay choice of. Therefore, rˆ r. This last equatio shows how rˆ must deed o for the total retur over elased time t to be ideedet of. We also eed to defie u ad d i terms of. At this oit, there are two sigificatly differet aths we ca take. Deedig o the defiitios we choose, as or, equivaletly, as h 0, we ca have either a cotiuous or a um stochastic rocess. I the first situatio, very t 10 The scale of this uit erhas a day, or a year is uimortat as log as r ad t are exressed i the same scale. 18

19 small radom chages i the stock rice will be occurrig i each very small time iterval. The stock rice will fluctuate icessatly, but its ath ca be draw without liftig e from aer. I cotrast, i the secod case, the stock rice will usually move i a smooth determiistic way, but will occasioally exeriece sudde discotiuous chages. Both ca be derived from our biomial rocess simly by choosig how u ad d deed o. We examie i detail oly the cotiuous rocess that leads to the otio ricig formula origially derived by Fischer Black ad Myro Scholes. Subsequetly, we idicate how to develo the um rocess formula origially derived by Joh Cox ad Stehe Ross. Recall that we suosed that over each eriod the stock rice would exeriece a oe lus rate of retur of u with robability q ad d with robability 1 q. It will be easier ad clearer to work, istead, with the atural logarithm of the oe lus rate of retur, log u or log d. This gives the cotiuously comouded rate of retur o the stock over each eriod. It is a radom variable which, i each eriod, will be equal to log u with robability q ad log d with robability 1 q. Cosider a tyical sequece of five moves, say u, d, u, u, d. The the fial stock rice will be S* uduuds; S*/S u 3 d, ad logs*/s 3 log u + log d. More geerally, over eriods, log S*/S log u + log d logu/d + log d where is the radom umber of uward moves occurrig durig the eriods to exiratio. Therefore, the exected value of logs*/s is ad its variace is E[logS*/S] logu/d E + log d Var[logS*/S] [logu/d] Var Each of the ossible uward moves has robability q. Thus, E q. Also sice the variace each eriod is q1 q + 1 q0 q q1 q, the Var q1 q. Combiig all of this, we have E [log S * / S] [ q log u / d + log d ] µˆ Var[log S * / S] q1 # q[log u / d ] ˆ Let us go back to our discussio. We were cosiderig dividig u our origial loger time eriod a day ito may shorter eriods a miute or eve less. Our rocedure calls for, over fixed legth of caledar time t, makig larger ad larger. Now if we held everythig else costat while we let become large, we would be faced with the roblem we talked about earlier. I fact, we would certaily ot reach a reasoable coclusio if either µˆ or ˆ wet to zero or ifiity as became large. Sice t is a fixed legth of time, i searchig for a realistic result, we must make the aroriate adustmets i u, d, ad q. I doig that, we would at least wat the mea ad variace of the cotiuously comouded rate of retur of the assumed stock rice movemet to coicide with that of the actual stock rice as. 19

20 Suose we label the actual emirical values of µˆ ad ˆ as µt ad t, resectively. The we would wat to choose u, d, ad q so that [ q log u / d + log d ] µ t as q 1# q[log u / d] t A little algebra shows we ca accomlish this by lettig I this case, for ay, u e t /, d e t /, q µ / t / ˆ µ µt ad ˆ [ µ t / ] t Clearly, as, ˆ t while ˆ µ µ t for all values of. Alteratively, we could have chose u, d, ad q so that the mea ad variace of the future stock rice for the discrete biomial rocess aroach the resecified mea ad variace of the actual stock rice as. However, ust as we would exect, the same values will accomlish this as well. Sice this would ot chage our coclusios, ad it is comutatioally more coveiet to work with the cotiuously comouded rates of retur, we will roceed i that way. This satisfies our iitial requiremet that the limitig meas ad variaces coicide, but we still eed to verify that we are arrivig at a sesible limitig robability distributio of the cotiuously comouded rate of retur. The mea ad variace oly describe certai asects of that distributio. For our model, the radom cotiuously comouded rate of retur over a eriod of legth t is the sum of ideedet radom variables, each of which ca take the value log u with robability q ad log d with robably 1 q. We wish to kow about the distributio of this sum as becomes large ad q, u, ad d are chose i the way described. We eed to remember that as we chage, we are ot simly addig oe more radom variable to the revious sum, but istead are chagig the robabilities ad ossible outcomes for every member of the sum. At this oit, we ca rely o a form of the cetral limit theorem which, whe alied to our roblem, says that, as, if the q logu ˆ µ q logd ˆ µ # ˆ log S * / S / ˆ µ + $ Prob %, z N z ˆ &- 0 * # 0

21 where Nz is the stadard ormal distributio fuctio. Puttig this ito words, as the umber of eriods ito which the fixed legth of time to exiratio is divided aroaches ifiity, the robability that the stadardized cotiuously comouded rate of retur of the stock through the exiratio date is ot greater tha the umber z aroaches the robability uder a stadard ormal distributio. The iitial coditio says roughly that higher-order roerties of the distributio, such as how it is skewed, become less ad less imortat, relative to its stadard deviatio, as. We ca verify that the coditio is satisfied by makig the aroriate substitutios ad fidig 3 3 q logu ˆ µ + 1 q logd ˆ µ 1 q + q 3 ˆ q1 q 1 1 which goes to zero as sice q + µ / t /. Thus, the multilicative biomial model for stock rices icludes the logormal distributio as a limitig case. Black ad Scholes bega directly with cotiuous tradig ad the assumtio of a logormal distributio for stock rices. Their aroach relied o some quite advaced mathematics. However, sice our aroach cotais cotiuous tradig ad the logormal distributio as a limitig case, the two resultig formulas should the coicide. We will see shortly that this is ideed true, ad we will have the advatage of usig a much simler method. It is imortat to remember, however, that the ecoomic argumets we used to lik the otio value ad the stock rice are exactly the same as those advaced by Black ad Scholes 1973 ad Merto 1973, The formula derived by Black ad Scholes, rewritte i terms of our otatio, is Black-Scholes Otio Pricig Formula C t SN x Kr N x t where # t log S / Kr 1 x + t t We ow wish to cofirm that our biomial formula coverges to the Black-Scholes formula whe t is divided ito more ad more subitervals, ad rˆ, u, d, ad q are chose i the way we described that is, i a way such that the multilicative biomial robability distributio of stock rices goes to the logormal distributio. For easy referece, let us recall our biomial otio ricig formula: 1

22 C S [ a;, # ] Krˆ [ a;, ] The similarities are readily aaret. r ˆ is, of course, always equal to r -t. Therefore, to show the two formulas coverge, we eed oly show that as #[ a ;, ] N x ad [ a ;, ] $ N x # t We will cosider oly φ[a;, ], sice the argumet is exactly the same for φ[a;, ]. The comlemetary biomial distributio fuctio φ[a;, ] is the robability that the sum of radom variables, each of which ca take o the value 1 with the robability ad 0 with the robability 1, will be greater tha or equal to a. We kow that the radom value of this sum,, has mea ad stadard deviatio 1. Therefore, & 1 φ[a;, ] Prob[ a 1] Prob $ % a 1 # 1 1 Now we ca make a aalogy with our earlier discussio. If we cosider a stock which i each eriod will move to us with robability ad ds with robability 1, the logs*/s log u/d + log d. The mea ad variace of the cotiuously comouded rate of retur of this stock are µ ˆ log u / d + log d ad ˆ 1 [log u / d ] Usig these equalities, we fid that 1 log S * / S ˆ µ ˆ Recall from the biomial formula that a 1 log K / Sd / log u / d [log K / S log d ] / log u / d, where is a umber betwee zero ad oe. Usig this ad the defiitios of little algebra, we have Puttig these results together, a # 1 # log K / S # ˆ µ # log u / d 1 # ˆ µˆ ad ˆ, with a

23 1 φ[a;, ] Prob & log S * / S ˆ µ K S u d # log / ˆ µ * log / $ $ % ˆ ˆ We are ow i a ositio to aly the cetral limit theorem. First, we must check if the iitial coditio, 3 3 logu ˆ µ 1 log ˆ + d µ 1 + # ˆ 1 0 as, is satisfied. By first recallig that rˆ d / u d, ad the t / ad d e, it is ossible to show that as, rˆ t / t / r, u e, 1 & 1 $ logr 1 + $ $ % # t As a result, the iitial coditio holds, ad we are ustified i alyig the cetral limit theorem. To do so, we eed oly evaluate discussio for arameterizig q shows that as µ ˆ, ˆ ad logu/d as. 11 Examiatio of our 11 A surrisig feature of this evaluatio is that although q ad thus µ ˆ µ ˆ ad ˆ ˆ, oetheless ˆ 1 $ ad ˆ have the same limitig value as. By cotrast, sice µ log r % #, µ ˆ ad µˆ do ot. & This results from the way we eeded to secify u ad d to obtai covergece to a logormal distributio. Rewritig this as t log u, it is clear that the limitig value σ of the stadard deviatio does ot deed o or q, ad hece must be the same for either. However, at ay oit before the limit, sice & t # t ˆ $ µ t ad t % & #, 1. ˆ $. - * log r -. $ % + ˆ ad ˆ will geerally have differet values. & 1 # The fact that µ ˆ $ log r t ca also be derived from the roerty of the logormal distributio that % 1 log E[ S * / S] µ t + t where E ad µ are measured with resect to robability. Sice rˆ d / u d, it follows that r ˆ u + 1 d. For ideedetly distributed radom variables, the exectatio of a roduct equals the roduct of their exectatios. Therefore, t E [ S * / S] [ u + 1 d ] rˆ r Substitutig r t for E[S*/S] i the revious equatio, we have 3

24 4 t r # $ % & 1 log ˆ µ ad t ˆ Furthermore, logu/d 0 as. For this alicatio of the cetral limit theorem, the, sice t t r S K z d u S K µ # $ % & * 1 log / log ˆ / log ˆ / log we have # $ % & + t t S Kr N z N a t * 1 / log ], ; [ 1 The fial ste i the argumet is to use the symmetry roerty of the stadard ormal deviatio distributio that 1 Nz N z. Therefore, as 1 / log ], ; [ t x N t t Kr S N z N a t # $ % & # # * # Sice a similar argumet holds for φ[a;, ], this comletes our demostratio that the biomial otio ricig formula cotais the Black-Scholes formula as a limitig case. 1,13 1 log µ r 1 The oly differece is that, as, t r / / 1 log 1 1 # $ % & * +, Further, it ca be show that as, Δ Nx. Therefore, for the Black-Scholes model, ΔS SNx ad t x N Kr B t. 13 I our origial develomet, we obtaied the followig equatio somewhat rewritte relatig the call rices i successive eriods: 0 ˆ ˆ ˆ # $ % & + # $ % & rc C d u r u C d u d r d u By their more difficult methods, Black ad Scholes obtaied directly a artial differetial equatio aalogous to our discrete-time differece equatio. Their equatio is 0 log log 1 + C r t C S C S r S C S #. The value of the call, C, was the derived by solvig this equatio subect to the boudary coditio C* max[0, S* K].

25 As we have remarked, the seeds of both the Black-Scholes formula ad a cotiuous-time um rocess formula are both cotaied withi the biomial formulatio. At which ed oit we arrive deeds o how we take limits. Suose, i lace of our former corresodece for u, d, ad q, we istead set u u, d e ζt/, q λt/. This corresodece catures the essece of a ure um rocess i which each successive stock rice is almost always close to the revious rice S ds, but occasioally, with low but cotiuig robability, sigificatly differet S us. Observe that, as, the robability of a chage by d becomes larger ad larger, while the robability of a chage by u aroaches zero. With these secificatios, the iitial coditio of the cetral limit theorem we used is o loger satisfied, ad it ca be show the stock rice movemets coverge to a log-poisso rather tha a logormal distributio as. Let us defie %[ x; y] $ i x # y i e y i as the comlemetary Poisso distributio fuctio. The limitig otio ricig formula for the above secificatios of u, d ad q is the Jum Process Otio Pricig Formula C S [ x; y] Kr t [ x; y / u], where y log r # ut / u 1, ad x the smallest o-egative iteger greater tha logk/s ζt/log u. A very similar formula holds if we let u e ζt/, d d, ad 1 q λt/. Based o our revious aalysis, we would ow susect that, as, our differece equatio would aroach the Black-Scholes artial differetial equatio. This ca be cofirmed by substitutig our defiitios of rˆ, u, d i h terms of i the way described earlier, exadig C u, C d i a Taylor series aroud e S, t h h e S, t h, resectively, ad the exadig ad e h e h,, ad r h i a Taylor series, substitutig these i the equatio ad collectig terms. If we the divide by h ad let h 0, all terms of higher order tha h go to zero. This yields the Black-Scholes equatio. 5

26 6. Divideds ad Put Pricig So far we have bee assumig that the stock ays o divideds. It is easy to do away with this restrictio. We will illustrate this with a secific divided olicy: the stock maitais a costat yield, δ, o each ex-divided date. Suose there is oe eriod remaiig before exiratio ad the curret stock rice is S. If the ed of the eriod is a ex-divided date, the a idividual who owed the stock durig the eriod will receive at that time a divided of either δus or δds. Hece, the stock rice at the ed of the eriod will be either u1 δ v S or d1 δ v S, where v 1 if the ed of the eriod is a ex-divided date ad v 0 otherwise, Both δ ad v are assumed to be kow with certaity. Whe the call exires, its cotract ad a ratioal exercise olicy imly that its value must be either C u max[0, u1 δ v S K] or C d max[0, d1 δ v S K] Therefore, C u max[0, u1 δ v S K] C C d max[0, d1 δ v S K] Now we ca roceed exactly as before. Agai, we ca select a ortfolio of Δ shares of stock ad the dollar amout B i bods that will have the same ed-of-eriod value as the call. 14 By retractig our revious stes, we ca show that C [ C + 1 C u if this is greater tha S K ad C S K otherwise. Here, oce agai, rˆ d / u d ad C C / u d S. u d Thus far the oly chage is that 1 δ v S has relaced S i the values for C u ad C d. Now we come to the maor differece: early exercise may be otimal. To see this, suose that v 1 ad d1 δs > K. Sice u > d, the, also, u1 δs > K. I this case, C u u1 δs K ad C d d1 δs K. Therefore, sice u / rˆ + d / rˆ1 1, the d ]/ rˆ [ C + 1 C ] / rˆ 1 S K / rˆ u d 14 Remember that if we are log the ortfolio, we will receive the divided at the ed of the eriod; if we are short, we will have to make restitutio for the divided. 6