Why we have always used the Black-Scholes-Merton option pricing formula

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1 Why we have always used he Black-Scholes-Meron opion pricing formula Charles J. Corrado Deakin Universiy Melbourne, Ausralia April 4, 9 Absrac Derman and aleb (he Issusions of Dynamic Hedging, 5 uncover a seeming anomaly in opion pricing heory which suggess ha saic hedging based on pu-call pariy provides sufficien heoreical suppor o jusify risk-neural opion pricing. From his hey sugges ha dynamic hedging as a heoreical basis for he celebraed opion pricing model of Black and Scholes (973 and Meron (973, while correc, is redundan [see also Haug and aleb (Why We Have Never Used he Black-Scholes- Meron Opion Pricing Formula, 9]. his paper examines he anomaly and finds ha pu-call pariy does no provide a basis for risk-neural opion pricing. Key words: Opion pricing, pu-call pariy, dynamic hedging, saic hedging, Black-Scholes-Meron JEL codes: G, G3 Commens and suggesions welcome. Conac informaion: Charles J. Corrado, School of Accouning, Economics, and Finance, Deakin Universiy, Burwood Highway, Burwood, VIC 35, Ausralia. corrado@deakin.edu.au. Elecronic copy available a: hp://ssrn.com/absrac=3575

2 Why we have always used he Black-Scholes-Meron opion pricing formula Absrac Derman and aleb (he Issusions of Dynamic Hedging, 5 uncover a seeming anomaly in opion pricing heory which suggess ha saic hedging based on pu-call pariy provides sufficien heoreical suppor o jusify risk-neural opion pricing. From his hey sugges ha dynamic hedging as a heoreical basis for he celebraed opion pricing model of Black and Scholes (973 and Meron (973, while correc, is redundan [see also Haug and aleb (Why We Have Never Used he Black-Scholes- Meron Opion Pricing Formula, 9]. his paper examines he anomaly and finds ha pu-call pariy does no provide a basis for risk-neural opion pricing.. Inroducion Since incepion, he opion pricing model of Black and Scholes (973 and Meron (973 has been aply celebraed as a landmark developmen in financial economics. heir conribuion is normally represened by heir famous formula, hough i is ofen hough o be misrepresened since similar formulas exised before 973. In his view, he rue achievemen of Black-Scholes-Meron lies in heir arbirage-free, risk-neural opion pricing model based on dynamically hedging an opion agains is underlying securiy. However, even for algebraically idenical formulas from he pre-black- Scholes-Meron era he resemblance is superficial. he Black-Scholes-Meron formula is disinguished by is origin wihin heir dynamic hedging paradigm. Recen debae criicizes heir dynamic hedging paradigm. In paricular, Derman and aleb (5 and Haug and aleb (9 make wo disinc argumens in a criique of dynamic hedging à la Black-Scholes-Meron: dynamic hedging is no feasible wih sufficien precision o offer realisic empirical suppor for Black-Scholes-Meron opion pricing heory, and dynamic hedging is heoreically redundan since pucall pariy already provides a heoreical basis for risk-neural opion pricing. he former argumen is no addressed in his paper. here is already a large lieraure on his opic. he laer argumen regarding pu-call pariy as a heoreical basis for riskneural opion pricing is he focal issue addressed here. his paper proceeds as follows: we firs review he argumen in Derman and aleb (5 ha pu-call pariy provides sufficien suppor o jusify risk-neural opion pricing, hereby making dynamic hedging redundan. Nex, we dissec he underlying srucure of pu-call pariy o expose wo consiuen pariy condiions ha reveal a See, for example, Boness (964, who assumes for convenience ha invesors in pus and calls are indifferen o risk. A parial sampling migh include Bakshi, Cao and Chen (997, Bossaers and Hillion (997, Boyle and Vors (99, Çein e al. (6, Consaninides and Zariphopoulou (999, Dumas, Fleming and Whaley (998, Figlewski (989, Galai (983, Leland (985, and Li and Pearson (7. Elecronic copy available a: hp://ssrn.com/absrac=3575

3 clear disincion beween pu-call pariy and risk-neural opion pricing. In he following secion we draw on Margrabe (978 and generalize he discussion o include opions o exchange asses. We subsequenly discuss he relevance of he curren discussion o formulas for he expeced holding period reurn of an opion derived in Rubinsein (984. Finally, in he las secion we sae our conclusion.. Pu-call pariy and risk-neural opion pricing à la Derman and aleb (5 Derman and aleb (5 sugges ha a saic hedging sraegy based on pu-call pariy is sufficien o jusify risk-neural opion pricing. heir argumen challenges he imporance of he conribuion made by Black and Scholes (973 and Meron (973 and proceeds essenially as presened immediaely below. Le pu-call pariy be represened in he form shown in equaion (, where C( S, K, and P( S, K, denoe sandard European call and pu opions, respecively, wih srike price K, ime o expiraion, curren price of a nondividend paying securiy, and where r denoes he riskless ineres 3 rae. S (,, (,, C S K P S K S e K = ( Acuarial price formulas for European call and pu opions on a non-dividend paying securiy are saed in equaion (, where g is he expeced growh rae of he underlying securiy price and k is he rae used o discoun expiraion dae payoffs for boh opions. g ( σ k g (,, = ( σ C S K e S e N d KN d k (,, = ( + P S K e KN d Se N d ( ( + ( + σ ln S / K g / Subsiuing he acuarial call and pu opion prices from equaion ( ino he pu-call pariy condiion in equaion ( yields equaion (3. ( g k k ( σ k ( ( σ Se N d e KN d g k e KN d Se N d S e K + + = Equaion (3 reveals ha he requiremen of muual consisency beween he pu-call pariy condiion in equaion ( and he call and pu opion price formulas in equaion ( dicaes ha boh he discoun rae k and he growh rae g be equal o he riskless rae r. Wih hese equaliies, i.e., k = g = r, he call and pu prices in equaion ( are equivalen o he corresponding Black-Scholes-Meron formulas. On (3 3 Soll (969 is a widely cied reference on pu-call pariy.

4 his basis, Derman and aleb (5 conclude ha pu-call pariy is sufficien for riskneural opion pricing. heir argumen is remarkably simple. Bu is i correc? Ruffino and reussard (6 opine oherwise. hey sugges ha a single nonsochasic discoun rae boh for boh he call and he pu is inconsisen wih asse pricing heory. hey also poin ou ha allowing differen discoun facors for he call and he pu does no alleviae he inconsisency wih pu call pariy. 4 his poin is acue, as is easily demonsraed. Le kc and kp denoe separae discoun raes for he call opion and he pu opion. Inserion ino equaion (3 yields he expression shown immediaely below, which reveals ha allowing differen pu and call discoun raes exacerbaes he inconsisency wih pu-call pariy. ( C ( σ kp ( g kp ( σ g kc k S e N d e KN d e KN d Se N d S e K + + = he argumen in Derman and aleb (5 depends crucially on resricing he call and pu opion price formulas in equaion ( o a single discoun rae k. A firs glance his seems naural, so accusomed are we o hinking wihin he risk-neural paradigm. Of course, afer publicaion of Black and Scholes (973 and Meron (973 he use of only a riskless discoun rae o price opions is a direc consequence of heir dynamic hedging paradigm. Bu wihou he benefi of dynamic hedging à la Black-Scholes-Meron he assumpion of a single discoun rae is ad hoc. o see why, consider he experimen of modifying he call and pu opion pricing formulas in equaion ( so as o conain wo discoun raes, h and k, where he discoun rae k applies o he underlying securiy and he discoun rae h applies o he srike price. In his case, we obain he acuarial call and pu opion pricing formulas shown in equaion (4. (,, ( h ( σ h ( (,, = ( + σ g k C S K = S e N d e KN d P S K e KN d S e N d ( + ( + σ ln S / K g / ( g k (4 Subsiuing he call and pu opion prices specified in equaion (4 above ino he pucall pariy condiion in equaion ( yields equaion (5 immediaely below. 4 In paricular, he use of a non-sochasic discoun rae, k, common o boh he call and he pu opions is inconsisen wih modern equilibrium capial asse pricing heory. Correspondingly, he use of valid discoun facors sochasic and differen for he call and he pu would no allow Derman and aleb o combine heir acuarial formulas, mach he resuling expression wih he forward price and sill obain he Black Scholes formula. (Ruffino and reussard, 6, p.366 3

5 ( g k h ( σ h ( ( σ S e N d e KN d g k e KN d Se N d S e K + + = (5 Equaion (5 reveals ha he call and pu opion prices in equaion (4 are only consisen wih pu-call pariy in equaion ( when wo condiions are joinly saisfied, he discoun rae for he underlying securiy is equal o is growh rae, i.e., k = g, and he discoun rae for he srike price is equal o he growh rae of a riskless discoun bond, i.e., h= r. Imporanly, i is no a requiremen in equaion (5 ha he discoun rae k for he underlying securiy be equal o he riskless rae r for consisency wih pu-call pariy. Derman and aleb (5 make he implici assumpion a priori ha he discoun raes h and k are one in he same. his leads hem o conclude ha pu-call pariy provides sufficien suppor for risk-neural opion pricing, hereby appearing o render dynamic hedging redundan. Bu wih differen discoun raes h and k for he srike price and securiy price, respecively, heir argumen unravels. 3. Dissecing pu-call pariy o see why, we mus dissec pu-call pariy by drawing aenion o he fac ha sandard European call and pu opions each conain wo separae opions. Firs, he call opion C( S, K, conains: an asse-or-nohing binary call denoed by AC( S, K,, and a srike-or-nohing binary call denoed by KC( S, K,. A opion expiraion, a long posiion in he asse-or-nohing binary call has he payoff S I( S > K, where I ( x is a zero-one indicaor of he even x. Similarly, a shor posiion in he srike-or-nohing binary call has he payoff K I( S > K. In combinaion, hese long and shor posiions consiue a sandard European call opion as shown in equaion (6. (,, (,, (,, C S K = AC S K KC S K (6 P( S, K, Second, he pu opion conains: a srike-or-nohing binary pu denoed by KP( S, K,, and an asse-or-nohing binary pu denoed by AP( S, K,. A opion expiraion, a long posiion in he srike-or-nohing binary pu has he payoff K ( I( S > K and a shor posiion in he asse-or-nohing binary pu has he payoff S ( I( S > K. ogeher, hese long and shor posiions consiue a sandard European pu opion as shown in equaion (7. (,, (,, (,, P S K = KP S K AP S K (7 Subsiuing he asse-or-nohing and srike-or-nohing call and pu opions idenified immediaely above ino he pu-call pariy condiion in equaion ( reveals ha pu- 4

6 call pariy is acually a combinaion of wo disinc pariy condiions: asse-ornohing pu-call pariy, and srike-or-nohing pu-call pariy. hese wo disinc pariy condiions are saed in equaion (8 below. (,, (,, (,, (,, AC S K + AP S K = S KC S K + KP S K = e K (8 Expressed as an acuarial formula, he asse-or-nohing pu-call pariy condiion is, ( ( g k g k Se N d + Se N S ( + ( + σ ln S / K g / (9 he asse-or-nohing pariy condiion in equaion (9 requires ha he growh rae be equal o he discoun rae k, bu is agnosic regarding he discoun rae h. g Expressed as an acuarial formula, he srike-or-nohing pu-call pariy condiion is, ( σ ( σ h h e KN d + + e KN d = e K ( + ( + σ ln S / K g / he srike-or-nohing pariy condiion in equaion ( requires ha he discoun rae h be equal o he riskless rae r, bu is agnosic regarding he discoun rae k. he dissecion of pu-call pariy above reveals ha he discoun raes k and h corresponding o he securiy price and srike price, respecively, are deermined separaely by he underlying pariy condiions saed in equaions (9 and (. Nohing in he original pu-call pariy condiion imposes equaliy on he discoun raes k and h. Call and pu opion prices saisfying he asse-or-nohing and srike-or-nohing pariy condiions k = g and h= r, respecively, are saed in equaion ( below. hese are correc opion prices ouside he risk-neural realm of Black-Scholes-Meron. (,, = ( σ (,, = ( + σ ( C S K S N d e KN d ( + ( + σ ln S / K k / ( P S K e KN d SN d ( Equaliy of he discoun rae k wih he riskless rae r is no necessary for he call and pu opion prices in equaion ( o saisfy pu-call pariy. Risk-neural opion 5

7 pricing à la Black-Scholes-Meron replaces he discoun rae rae r. Pu-call pariy does no do his. k wih he riskless A his poin i is worh noing ha by seing k = g in equaion ( above we obain he opion prices shown in equaion ( immediaely below. k (,, = ( σ k (,, = ( + σ ( C S K S N d e KN d P S K e KN d S N d ( + ( + σ ln S / K k / he call and pu opion prices in equaion ( are no legiimae ouside he riskneural realm of Black-Scholes-Meron, where hey are inconsisen wih pu-call pariy. While hey are rescued wihin he risk-neural realm, hey do no drag pu-call pariy wih hem. Pu-call pariy holds boh inside and ouside he risk-neural realm. ( 4. Generalizaion o opions o exchange asses he discussion above gains srengh and clariy by generalizaion o include opions o exchange one asse for anoher. Sandard call and pu opions are special cases of opions wih fixed srikes wihin a broader framework of opions o exchange one asse for anoher. he discussion below draws heavily on Margrabe (978. Consider European call and pu opions o exchange he risky asses S and R a opion expiraion. he call grans he buyer he righ o deliver asse R in exchange for asse S a conrac expiraion. he pu grans he buyer he righ o deliver asse S in exchange for asse R a conrac expiraion. A porfolio aking a long posiion in he call and a shor posiion in he pu will have he expiraion dae payoff indicaed in equaion (3. max, S R max, R S = S R (3 Hence, he pu-call pariy condiion for hese call and pu exchange opions is C S, R, and expressed as shown in equaion (4 immediaely below, where ( P( S, R, represen call and pu opion prices based on curren securiy prices S, R and ime o opion expiraion. (,, (,, C S R P S R = S R (4 We shall assume ha securiy prices S and R follow dynamic processes specified in equaion (5, where gs and gr are securiy price growh raes, σ S and σ R are insananeous reurn sandard deviaions, and Z S and Z R are Brownian moions wih correlaion parameer ρ RS. 6

8 ( σ / ( σ / ds = g Sd + σ S ddz S S S S dr = gr R Rd + σrr ddzr (5 E dz dz = ρ d S R RS Given he discoun raes ks and kr for securiies S and R, respecively, he acuarial formulas for hese call and pu exchange opions are saed in equaion (6 below. (,, (,, ( gs ks gr kr ( ( ˆ σ C S R = S e N d R e N d ( ( gr kr gs ks σ + ( ˆ S R + σ P S R = R e N d + ˆ S e N d ln S / R g g / ˆ σ = σs + σs ρrsσsσ R ˆ Subsiuing he asse exchange opion price formulas in equaion (6 above ino he pu-call pariy condiion in equaion (4 reveals wo consiuen pariy condiions as shown in equaion (7. (6 ( ( gs ks gs ks + = Se N d Se N d S ( gr kr ( gr kr ( ˆ σ ( ˆ σ R e N d + + R e N d = R (7 hese consiuen pariy condiions require ha g S = k S and gr yields he refined pariy condiions in equaion (8 below. = k R, which in urn SN d + SN S ( ˆ σ ( ˆ σ RN d + + RN d = R + ( + ˆ σ ln S / R ks kr / ˆ (8 Refined call and pu exchange opion price formulas saisfying he pariy condiions in equaion (8 above are saed in equaion (9 immediaely below. hese are correc opion prices ouside he risk-neural realm of Black-Scholes-Meron. (,, = ( ˆ σ (,, = ( + ˆ σ ( C S R S N d R N d P S R R N d S N d + ( + ˆ σ ln S / R ks kr / ˆ (9 7

9 Equaliy of he discoun raes k and k is no necessary o saisfy pu-call pariy. Risk-neural opion pricing à la Black-Scholes-Meron replaces boh discoun raes and k R S wih he riskless rae r. Pu-call pariy does no do his. R As a final noe o his secion, again drawing on Margrabe (978, leing σ R in equaion (9 o yield r and R = e K we obain he pricing formulas for call kr = and pu opions wih fixed srikes given earlier in equaion (. k S 5. Expeced holding period reurn of an opion Call and pu opion prices based on differen discoun raes for he underlying sock and srike prices have pracical relevance when we are ineresed in opion price behaviour ouside he risk-neural realm. Rubinsein (984 provides an ineresing applicaion wih formulas for expeced fuure call and pu opion prices based on discoun raes k and r for he sock and srike prices, respecively. In a noaion adaped o he curren ex, hese formulas are shown in equaion ( below, which sae he expeced fuure values of call and pu opions a ime given opion expiraion a ime, where <. k ( ( σ ( ( σ ( E C = e S N d e KN d k E P = e KN d + e S N d ( ( + ( + ( + σ ln S / K k r k / hese expeced opion prices based on he discoun raes k and r for he underlying sock and srike prices, respecively, saisfy he following version of pu-call pariy: ( k E C E P e S e K = ( Rubinsein (984 shows ha he formulas for expeced fuure opion prices can provide expeced fuure reurns from invesmens in call and pu opions. In he curren noaion, he annualized expeced holding period reurns hrough ime for sandard European call and pu opions are given in equaion (. (, ( ( r ( σ k e SN d k, r e KN d k, r + HPRC = SN( d r, r e KN( d( r, r σ ( (, + σ (, (, + σ, ( + ( + ( + σ ( k e KN d k r e S N d k r + HPRP = e KN( d r r SN d( r r d b a ln S / K b a b / = ( 8

10 I is sraighforward o show ha expeced holding period reurns for he asse-ornohing call and pu opions, i.e., HPR, HPR, are: AC AP (, ( (, ( ( + ( + ( + σ ( (, + σ σ k ( e N d k r e N d k r + HPRAC = + HPRAP = N d r, r N( d r, r + d b a ln S / K b a b / = ( Similarly, expeced holding period reurns for he srike-or-nohing call and pu opions, i.e., HPRKC, HPR KP, are: ( (, σ (, σ r e N d k r + HPRKC = N( d r r ( (, + σ (, + σ r e N d k r + HPRKP = N( d r r (3 6. Conclusion Saic hedging via pu-call pariy is no a sufficien heoreical basis for risk-neural opion pricing. We have always used he Black-Scholes-Meron opion pricing formula because only he Black-Scholes-Meron formula originaes wihin he dynamic hedging paradigm of risk-neural opion pricing. References Bakshi, G., Cao, C., and Chen, Z.W., 997, Empirical performance of alernaive opion pricing models, Journal of Finance 5 (December, Black, F. and Scholes, M., 973, he pricing of opions and corporae liabiliies, Journal of Poliical Economy 8, Boness, A.J., 964, Elemens of a heory of sock opion value, Journal of Poliical Economy 8(3, Bossaers, P. and Hillion, P., 997, Local parameric analysis of hedging in discree ime, Journal of Economerics 8, Boyle, P., and Vors,., 99, Opion replicaion in discree ime wih ransacions coss, Journal of Finance 47(,

11 Çein, U., Jarrow, R., Proer, P., and Warachka, M., 6, Pricing opions in an exended Black-Scholes economy wih illiquidiy: heory and empirical evidence, Review of Financial Sudies 9(, Consaninides, G., and Zariphopoulou,., 999, Bounds on prices of coningen claims in an ineremporal economy wih proporional ransacion coss and general preference, Finance and Sochasics 3, Derman, E. and aleb, N.N., 5, he illusions of dynamic replicaion, Quaniaive Finance 5(4, Dumas, B., Fleming, J. and Whaley, R.E., 998, Implied volailiy funcions: Empirical ess, Journal of Finance 53(6, Figlewski, S., 989, Opions arbirage in imperfec markes, Journal of Finance 44(5, Galai, D., 983, he componens of he reurn from hedging opions agains socks, Journal of Business 56(, Haug, E.G. and aleb, N.N., 9, Why we have never used he Black-Scholes- Meron opion pricing formula. Working paper available a: hp://ssrn.com/absrac=75 Leland, H., 985, Opion pricing and replicaion wih ransacion coss, Journal of Finance 4, Li, M.Q. and Pearson, N.D., 7, A Horse Race among compeing opion pricing models using S&P 5 Index Opions, Working paper available a SSRN: hp://ssrn.com/absrac=9577 Margrabe, W., 978, he value of an opion o exchange one asse for anoher, Journal of Finance 33(, Meron, R.C., 973, heory of raional opion pricing, Bell Journal of Economics and Managemen Science 4, Rubinsein, M., 984, A Simple Formula for he Expeced Rae of Reurn of an Opion over a Finie Holding Period, Journal of Finance 39(5, Ruffino, D. and reussard, J. Derman and aleb s he illusions of dynamic replicaion : A commen, Quaniaive Finance 6(5, Soll, H.R., 969, he relaionship beween pu and call prices, Journal of Finance 4(5, 8 84.