The Binomial Model and Risk Neutrality: Some Important Details

Transcription

1 The Binomial Model and Risk Neuraliy: Some Imporan Deails Sanjay K. Nawalkha* Donald R. Chambers** Absrac This paper reexamines he relaionship beween invesors preferences and he binomial opion pricing model of Cox, Ross, and Rubinsein (CRR). I is shown ha he independence of he binomial opion pricing model from invesors preferences is a resul of a special choice of binomial parameers made by CRR. For a more general choice of binomial parameers, risk neuraliy canno be obained in discree ime. This analysis reveals he essenial difference beween he risk neural valuaion approach of Cox and Ross and he equivalen maringale approach of Harrison and Kreps in a discree ime framework. *Universiy of Massachuses, Amhers, MA 0100 **Lafayee College, Eason, PA 1804 This version December 1994 Elecronic copy available a: hp://ssrn.com/absrac=17177

2 Inroducion Over a decade ago, he seminal work of Cox, Ross, and Rubinsein (CRR) [5] allowed he use of elemenary mahemaics in discree ime for opion valuaion. Since hen, he binomial model has been applied and exended in many ways. In general, he binomial model has made hree imporan conribuions o he opion lieraure. Firs, he binomial model has a considerable pedagogical value in demonsraing he economic inuiion behind he formaion of an arbirage-free hedge porfolio for opion pricing. Second, he binomial model allows simple coninuous ime numerical approximaions of complex opion valuaion problems where no analyical closed form soluions exis. Finally, he binomial model demonsraes how opion pricing can be done wihou any knowledge of he subjecive preferences of he invesors. Thus, according o CRR, he discree ime binomial model is consisen wih he risk neuraliy argumen of Cox and Ross [4]. This paper reexamines he consisency of he binomial opion pricing model wih he risk neuraliy argumen of Cox and Ross [4]. I is shown ha risk-neuraliy in discree ime is a consequence of a specific choice of binomial parameers by CRR. For a more general choice of binomial parameers (such as Jarrow and Rudd [7]) he discree ime binomial model is consisen wih he equivalen maringale approach of Harrison and Kreps [8], bu no wih he risk-neuraliy approach of Cox and Ross [4]. (The wo approaches become consisen in he coninuous ime limi of he binomial model.) 3 The above observaion underscores he necessiy of imposing sronger resricions on he asse reurn disribuions in discree ime, in order o obain risk neuraliy. Elecronic copy available a: hp://ssrn.com/absrac=17177

3 Specifically, he binomial opion pricing approach is consisen wih many binomial sock price disribuions, and only one ou of hese many disribuions (he one given by CRR) is consisen wih risk neuraliy. Furher, he above observaion is also consisen wih he discree ime coningen claims valuaion models of Rubinsein [11], Brennan [3], and Sapleon and Subrahmanyam [1], which require ha sronger disribuion specific resricions mus be imposed on asse reurns in order o obain risk neuraliy in he discree ime. 4 A General Se of Binomial Parameers and Risk Neuraliy This secion begins wih a review of he discree ime binomial model of CRR as follows. A ime = 0, le a call opion on a sock have T periods o expiraion. Le T be divided ino N number of sub-inervals. Le he curren ime be = (T/N) (N- 1). In oher words, a he curren ime he call is one sub-inerval or (T/N) periods from he expiraion dae. A his ime le he sock price equal S. In he nex period only wo saes can occur wih he upward movemen for he sock given by US and he downward movemen given by ds (where u d). The probabiliy of upward and downward movemens are q and 1 - q, respecively. Le C be he curren price of a call opion wih exercise price E. As shown by CRR he call price a he curren ime can be given as: C = [ p u! C u + p d! C d ] / r h (1) where: p u!! = r h d h, pd = u r, u! d u! d u C = Max[0, us! E ]

4 d C = Max[0, ds! E ] and r = 1 + riskless rae over a single period. h = T/N periods Now consider he binomial parameers specified by CRR as follows: u = exp(! " (T / N ) ) () d = exp(!" #(T / N ) ) (3) q = (1/ )![1+ (µ / " )! (T / N ) ] (4) where µ is he preference parameer and σ is he volailiy parameer. Since by definiion 0 < q < 1, i is implied ha σ (T/N) < µ (T/N) < σ (T/N). Obviously since u and d do no conain he preference parameer µ, he risk-neural probabiliies p u and p d, he fuure call prices C u and C d, and he curren call price C, are all independen of preferences. However, as shown by Jarrow and Rudd (JR) [7] and ohers, he above choice of binomial parameers is no unique. JR specify he following choice of binomial parameers for he call opion price in equaion (1): u = exp[µ!(t / N ) + "! (T / N ) ] (5) d = exp[µ! (T / N ) " #! (T / N ) ] (6) q = (1/ ) (7) JR argue in favor of he above parameers because he firs hree momens for he sock's log-reurn implied by he above parameers are consisen wih he respecive

5 momens of he lognormal process over every lengh of discree sub-inerval (T/N). These momens can be given as he mean =µ (T/N), variance = (σ (T/N), and skewness = 0. However, he hree momens of he log-reurn for he binomial process using he CRR parameers are no consisen wih he corresponding momens of he lognormal process. This can be seen from he following definiions of he momens using CRR parameers: he mean =µ (T/N), variance =σ (T/N)- µ (T/N), and skewness =µ [µ (T/N) 3 - σ (T/N) ]. The CRR parameers imply a non-zero level of skewness for log-reurns in discree ime. The variance and skewness of he binomial process implied by CRR parameers converge o he variance and he skewness of he lognormal process only in he coninuous ime limi (since (T/N) and (T/N) 3 become insignifican in comparison o (T/N) as N ends o infiniy). I can be shown ha he JR parameers are no consisen wih risk neural approach o opion valuaion in discree ime. Subsiuing Jarrow and Rudd's choice of u and d (given in equaions (5) and (6)) in equaion (1) implies ha he risk-neural probabiliies p u, and p d, he fuure call prices C u and C d, and he curren call price C, all depend on he preference parameer µ. Hence, he resuling risk-neural probabiliies and he call price are preference dependen. Now consider a general form of binomial parameers given as follows: u = exp[m! (T / N ) + "! (T / N ) ] (8) d = exp[m! (T / N ) + "! (T / N ) ] (9) q = (1 / )![1+ ((µ " m) / # )! (T / N ) ] (10) where -σ (T/W) < (µ- m) (T/N) <σ (T/N) (since 0 < q < 1).

6 Given he above choice of binomial parameers, he firs hree momens of he logreurn for he binomial process over a discree sub-inerval (T/N) can be given as: mean = µ (T/N), variance = σ (T/N) - (µ-m) (T/N), and skewness = (µ-m) [(µ-m) (T/N) 3 -σ (T/N) ]. The erms (T/N) and (T/N) 3 become insignifican in comparison o (T/N) as N ends o infiniy, and herefore he variance and skewness of he binomial process implied by he above parameers converge o he variance and skewness of he lognormal process in he coninuous ime limi. The above choice of binomial parameers (see equaions (8) and (9)) implies ha he erms p u, p d, C u, C d, and C in equaion (1) depend upon he parameer m. To obain a unique opion price, invesors may disagree abou he preference parameer µ (and hus, disagree on he acual probabiliies), bu all invesors mus agree on he parameer m. For he CRR model, all invesors agree ha he parameer m equals zero. For he JR model, all invesors agree ha he parameer m equals µ. In general, if he parameer m canno be uniquely deermined, he resuling binomial opion prices are no uniquely defined in discree ime. To preclude arbirage, addiional resricions mus be imposed on he binomial parameers. Specifically, a unique equivalen probabiliy measure mus exis such ha he sock price discouned a he riskless rae is a maringale wih respec o his measure (see Harrison and Kreps [8]). To saisfy he above condiion, he risk neural probabiliies p u and p d mus be greaer han zero in equaion (1). This implies ha for he CRR choice of parameers exp(-σ (T/N) ) < r h < exp(σ (T/N) ), for he JR parameers exp[µ (T/N) -σ (T/N) ] < r h < exp[µ (T/N) + σ (T/N) ], and for he revealed preference parameers exp[m (T/N) -σ (T/N) ] < r h < exp[m (T/N) + σ (T/N) ].

7 Two limiaions of he discree ime binomial approach can now be summarized wih respec o he general binomial parameers given in equaions (8), (9), and (10), as follows: 1. If m = µ, hen he binomial model is preference dependen and is inconsisen wih he risk-neuraliy argumen of Cox and Ross [4] in discree ime.. If m µ, hen risk neuraliy can sill be obained since invesors are allowed o disagree abou he preference parameer µ. However, he difficuly here is in deermining a unique value for m, eiher heoreically or empirically. 6 Differen values of m will resul in differen call opion prices in he discree ime. Forunaely, boh he above limiaions of he binomial approach can be resolved in he coninuous ime limi. I can be shown ha he dependence of he discree ime binomial opion pricing model on he parameer m diminishes as he number of subinervals N becomes large. Wih an infiniely large N, he Black and Scholes [] opion formula is obained as a limiing case of he binomial opion formula (see he Appendix). Thus, he binomial model is independen of he parameer m, only in he coninuous ime limi. This underscores he necessiy of coninuous ime porfolio rebalancing o guaranee risk neuraliy as originally noed by Black and Scholes [] and subsequenly formalized by Cox and Ross [4]. Conclusions This paper shows ha he consisency of he discree ime binomial opion pricing model of CRR wih he risk neuraliy argumen of Cox and Ross [4] depends upon a specific choice of binomial parameers. For a more general choice of binomial

8 parameers, he resuling opion prices may be preference dependen. This preference dependence diminishes as he number of sub-inervals N becomes large and disappears compleely only in he coninuous ime limi as he binomial model converges o he Black and Scholes model. The implicaions of hese resuls perain o one of he cenral issues of modern finance: he risk neural pricing of coningen claims. In order o obain risk neuraliy, he CRR model mus assume very specific behavior regarding he price changes of underlying asses. However, boh heoreically and pracically speaking, alernaive price behavior models are reasonable. This paper demonsraes a binomial opion pricing model using an alernaive and reasonable specificaion of behavior in underlying asse price changes and demonsraes ha risk neuraliy does no obain in he model. Thus, his paper provides an addiional demonsraion of he failure of risk neuraliy o obain in discree ime in paricular cases.

9 Appendix This appendix shows ha he binomial opion pricing model converges o he Black-Scholes model for a general choice of binomial parameers given by equaions (8), (9), and (10), as he number of sub-inervals N becomes infiniely large. Though i is possible o demonsrae he acual convergence of he binomial call opion price o he Black-Scholes call opion price, a much easier and inuiive proof follows from Cox and Rubinsein [6], page 09. Using he alernaive approach, he Black-Scholes P.D.E. is derived from he binomial equaion for he general choice of binomial parameers. Reconsider he call opion defined in he second secion a ime (0 T). The call price a ime can be given as follows: C = [ p! C u + p! C d ] / r h (A1) u + h d + h where he parameers p u, p d, r h, and h are defined in equaion (1), and he general binomial parameers u and d are defined in equaions (8) and (9). Following Cox and Rubinsein, he call price a ime is assumed o be a coninuously differeniable funcion of he sock price a ime, and he ime remaining o he expiraion dae. Thus, C = C(S, T- ), C u +h = C(u S, T - ( + h)), C u +h = C(d S, T - ( + h)), where subscrip implies he ime price of a given securiy. By appropriae subsiuions equaion (Al) can be rewrien as:

10 h r " exp[ m# h "! #( h )] [ ]# C(exp[ m# h +! #( h )]# S, ( )) T " + h exp[ m# h +! #( h )]" exp[ m# h "! #( h )] h exp[ m# h +! #( h )]" r [ ] C(exp[ m h! ( h )] S, ( )) T h + # # " # # " + exp[ m# h +! #( h )] " exp[ m# h "! #( h )] h " r # C( S, T " ) = 0 (A) By Taylor series expansions of he expressions C m" h + " h " S T # + h and (exp[! ( )], ( )) C m" h # " h " S T # + h around (exp[! ( )], ( )) he poin (S, T-): " C C(exp[ m# h +! #( h )]# S, T $ ( + h)) = C + [exp[ m# h +! #( h )] $ 1] # S # " S 1 " C 1 " C " C + #[exp[ m# h +! #( h )] $ 1] # S # + #[exp[ m# h +! #( h )] $ 1] # S # + h # +... " S " S " (A3) and a similar expression for replaces! "( h ). C m" h # " h " S T # + h, excep (exp[! ( )], ( )) "! # ( h ) By Taylor series expansions of he expressions exp[ m h! ( h )] " + " and exp[ m h! ( h )] " # " : 1 " + " = + " + " + " " + " + + (A4) exp[ m h! ( h )] 1 [ m h! ( h )] [ m h! ( h )]... and a similar expression for exp[ m h! ( h ) " # ", excep "! #( h ) replaces! "( h ). Finally, by a Taylor series expansion of r h : 1 r h r h r h r h = exp(! log ) = 1+! log +!(! log ) (A5) By subsiuion of he appropriae values from equaions (A3), (A4), and (A5) ino equaion (A) and simplifying:

11 " C " C " C " S " S " S 1 #! # S # # h + (log r) # S # # h + # h $ (log r) # C # h + Z = 0 (A6) where Z conains all erms of higher orders of h (i.e. Z = [erms wih (h) 1 (h),, ]). Dividing equaion (A6) by h: " C " C " C " S " D " 1 #! # S # + (log r) # S # + $ (log r) # C + Z / h = 0 (A7) I can be seen from he above equaion ha he revealed preference parameer m (see equaion (A)) is conained only in he erm Z/h. For non-infiniesimal values of h, he magniude of Z/h will be significan, and he soluion o equaion (A7) will depend upon he revealed preference parameer. However, as he number of sub-inervals N ends o infiniy, he erm Z/h goes o zero (as h goes o zero), bu oher erms do no. Thus, in he coninuous ime limi, equaion (A7) converges o he Black-Scholes parial differenial equaion, which is independen of he revealed preference parameer m. 7 Q.E.D.

12 End Noes 1. Recenly, he economic inuiion behind he binomial approach has been exended o a mulivariae-mulinomial approach by He [9] using he dynamic complee marke framework. The argumens given in his paper apply o he mulivariae-mulinomial models, as he binomial model can be considered as a special case of hese models.. See Meron [10], pp , for he applicaion of he risk-neuraliy argumen of Cox and Ross [4] o he binomial model of CRR. 3. Recenly, Beck [1] demonsraed ha he radiional derivaion of Black-Scholes opion formula is mahemaically unsaisfacory. Specifically, Beck shows ha he hedge porfolio used in Black-Scholes is neiher riskless nor self-financing. Beck presens an alernaive derivaion of he Black-Scholes formula ha avoids hese inconsisencies. Though he issues analyzed in his paper are quie differen, i is similar o he Beck s paper in spiri. 4. For example, assume ha invesors preferences are given by CPRA uiliy. Then, i is only necessary ha he underlying asse and he marke porfolio be bivariae lognormally disribued o obain risk-neuraliy in he discree-ime. Of course, he above assumpion is no necessary in he coninuous-ime analog of he above model (i.e., Black and Scholes []). 5. Jarrow and Rudd [7], pp , show heir binomial parameers o be consisen wih risk-neuraliy in he coninuous ime limi. This is consisen wih he Appendix A of his paper, which obains a similar resul. 6. CRR do no provide any heoreical jusificaion or empirical evidence for choosing zero as he appropriae value for m. 7. A he firs glance, equaion (A7) looks slighly differen from he Black-Scholes parial differenial equaion. However, noe ha r equals 1 plus he discree-ime single period riskless rae (see equaion (1)). However, Black-Scholes use he coninuous-ime riskless rae, which can be defined as R. From he basic rules of compounding over ime, he relaionship beween r and R over any lengh of ime h is given as r h = exp(r h), which in urn implies logr = R.

13 References [1] Beck, Thomas M. Black-Scholes Revisied: Some Imporan Deails. The Financial Review 8(February 1993): [] Black, Fischer, and Myron S. Scholes. The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy (May-June 1973): [3] Brennan, Michael J. The Pricing of Coningen Claims in Discree Time Models. Journal of Finance 34.1(1979): [4] Cox, John C., and Sephen A. Ross. The Valuaion of Opions for Alernaive Sochasic Processes. Journal of Financial Economics (January-March 1976): [5] Cox, John C., Sephen A. Ross, and Mark Rubinsein. Opion Pricing: A Simplified Approach. Journal of Financial Economics (Sepember 1979): [6] Cox, John C., and Mark Rubinsein. Opion Markes. Englewood Cliffs, NJ: Prenice Hall, [7] Jarrow, Rober A., and Andrew Rudd. Opion Pricing. Homewood, IL: Richard D. Irwin, [8] Harrison, J. Michael, and David M. Kreps. Maringales and Arbirage in Muliperiod Securiies Markes. Journal of Economic Theory 0(1979): [9] He, Hua. Convergence from Discree o Coninuous-Time Coningen Claim Prices. The Review of Financial Sudies 3.4(1990): [10] Meron, Rober C. Coninuous-Time Finance. Cambridge, MA: Basil Blackwell, [11] Rubinsein, Mark. The Valuaion of Uncerain Income Sreams and he Pricing of Opions. Bell Journal of Economics 7(1976): [1] Sapleon, Richard C., and Mari G. Subrahmanyam. The Valuaion of Mulivariae Coningen Claims in Discree Time Models, Journal of Finance 39(1984): 07-8.