Skewness and Kurtosis Adjusted Black-Scholes Model: A Note on Hedging Performance

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1 Finance Leers, 003, (5), 6- Skewness and Kurosis Adjused Black-Scholes Model: A Noe on Hedging Performance Sami Vähämaa * Universiy of Vaasa, Finland Absrac his aricle invesigaes he dela hedging performance of he skewness and kurosis adjused Black-Scholes model of Corrado and Su (996) and Brown and Robinson (00). he empirical ess in he FSE 00 index opion marke show ha he more sophisicaed skewness and kurosis adjused model performs worse han he simplisic Black-Scholes model in erms of dela hedging. he hedging errors produced by he skewness and kurosis adjused model are consisenly larger han he Black-Scholes hedging errors, regardless of he moneyness and mauriy of he opions and he lengh of he hedging horizon. Keywords: Opion Hedging, Dela Hedging, Skewness, Kurosis JEL classificaion: G0, G3. INRODUCION he Black-Scholes (973) model assumes he asse price dynamics o be described by a geomeric Brownian moion, and hus, implies consan volailiy and Gaussian log-reurns. However, boh he assumpions of consan volailiy and Gaussian reurns are obviously violaed in financial markes. I has been recognized for a long ime ha asse reurn disribuions end o be lepokuric. More recenly, a vas lieraure has documened volailiy o be ime-varying. In addiion, here seems o be a endency for changes in sock prices o be negaively correlaed wih changes in volailiy, due o which sock reurn disribuions end o be negaively skewed. Given he possible misspecificaions of he Black-Scholes model, a subsanial lieraure has devoed o he developmen of opion pricing models which accoun for he observed empirical violaions, such as he volailiy smile. he Black-Scholes consan volailiy assumpion is relaxed in sochasic volailiy models such as Hull and Whie (987) and Heson (993a), in deerminisic volailiy models by Dupire (99), Derman and Kani (99), and Rubinsein (99), and in ARCH models of Engle and Musafa (99), Duan (995), and Heson and Nandi (000). he assumpion of lognormal erminal price disribuion is relaxed, e.g., in skewness and kurosis adjused models of Jarrow and Rudd (98), and Corrado and Su (996), log-gamma model of Heson (993b), lognormal mixure model by Melick and homas (997), and hyberbolic model of Eberlein e al. (998). his aricle focuses on he dela hedging performance of he skewness and kurosis adjused Black-Scholes model of Corrado and Su (996) and Brown and Robinson (00). Despie he inensive empirical research on differen opion pricing models, surprisingly lile is known abou he hedging performance beyond he Black- Scholes model. Previous sudies on hedging performance have focused on differen ime-varying volailiy opion pricing models. he hedging performance of sochasic volailiy opion pricing models is invesigaed, e.g., in Bakshi e al. (997, 000), Nandi (998), and Lim and Guo (000) whereas Dumas e al. (998), Engle and Rosenberg (000), Coleman e al. (00), and Lim and Zhi (00) examine opion hedging under deerminisic volailiy models. A bi surprisingly, previous sudies indicae ha alhough ime-varying volailiy opion pricing models clearly ouperform he Black-Scholes model in erms of pricing [see e.g., Bakshi e al. * sami@uwasa.fi Valuable commens by Eva Liljeblom, Jussi Nikkinen, and Bernd Pape are graefully acknowledged. ISSN Global EcoFinance All righs reserved. 6

2 Vähämaa 7 (997)], such models do no necessarily improve he hedging performance. Apparenly, he hedging performance of opion pricing models which relax he normaliy assumpion has no been invesigaed. his aricle aims o fill his void by examining he dela hedging performance of he skewness and kurosis adjused Black-Scholes model. he remainder of he aricle is organized as follows. he skewness and kurosis adjused Black-Scholes model is presened in Secion. In secion 3, he FSE 00 index opion daa used in he empirical analysis are described. Secion presens he mehodology applied in he aricle. Empirical findings are repored in Secion 5. Finally, concluding remarks are offered in Secion 6.. SKEWNESS AND KUROSIS ADJUSED BLACK-SCHOLES MODEL Corrado and Su (996) and Brown and Robinson (00) use a Gram-Charlier series expansion of he sandard normal densiy funcion o derive an expanded Black-Scholes opion pricing formula wih explici adjusmen erms for nonnormal skewness and kurosis. he skewness and kurosis adjused price of a call opion is r c = SN( d) Ke N( d σ ) + µ Q () 3 Q3 + ( µ 3) where Q Q [( σ d ) n( d) σ N ( )] 3 d = Sσ + 3! 3 3 / [( d 3σ ( d σ ) n( d) σ N( )] d = Sσ +! ln d = ( S / K ) + ( r + σ / ) σ and N ( ) denoes he cumulaive sandard normal disribuion funcion, n ( ) is he sandard normal densiy funcion, S is he price of he underlying asse, K is he srike price of he opion, σ is he volailiy of he underlying asse, r is he risk-free ineres rae, is he ime o mauriy of he opion, and µ 3 and µ denoe he sandardized coefficiens of skewness and kurosis, respecively. he firs wo erms of equaion () consiue he Black-Scholes (973) opion pricing formula whereas he addiional erms µ Q and 3 3 ( µ 3) Q measure he effecs of nonnormal skewness and kurosis on he opion price, respecively. he skewness and kurosis adjused Black-Scholes model in equaion () is paricularly convenien from a hedging poin of view since i yields closed-form soluions for he hedge raios. By definiion, he dela is obained by aking he firs parial derivaive of c wih respec o S. hus, he skewness and kurosis adjused dela can be wrien as c δ = = N( d ) + µ 3 q 3 + ( µ 3) q () S where Q3 3 = = σ 3 / N d S 3! σ φd ( d ) + + σ φ n( ) q3

3 Vähämaa 8 ( d ) ( φ σ + r + ln( S K )) Q 3 n φ dn q = = N( d ) + 3 / σ σ n( d ) + / S! σ σ ( 3 ) ln( S K ) φ = r σ + ( ) ( 7 ) σ σ + ln( S K ) r ln( S K ) φ = r rσ + σ +. ( d ) he skewness and kurosis adjused dela in equaion () consiss of he Black-Scholes dela, N(d), and wo addiional erms, µ 3q and 3 ( µ 3) q, which measure he effecs of nonnormal skewness and kurosis, respecively. Figure illusraes he impac of skewness and kurosis on he dela of an a-he-money call opion. I can be noed ha he dela is a decreasing funcion of boh skewness and kurosis. Figure. Impac of Skewness and Kurosis on he Dela,0,5,0,05,00 0,95 0,90 0,85-0,80-0, Skewness 5 0,5 3 Kurosis δ(skabs)/ δ (BS) Noes: he figure plos he raio of skewness and kurosis adjused dela (SKABS) o Black-Scholes dela (BS) of an a-hemoney call opion for differen levels of skewness and kurosis. 3. DAA he daa used in his aricle conain selemen prices of he European-syle FSE 00 index opions raded a he London Inernaional Financial Fuures and Opions Exchange (LIFFE). he sample period exends from January, 00 o December 8, 00. he selemen prices for he FSE 00 index opions and he closing prices for he underlying implied index fuures are obained from he LIFFE. he risk-free rae needed for he calculaion of he delas and for he dela hedging experimen is proxied by he hree-monh LIBOR (London Inerbank Offered Rae) rae. wo exclusionary crieria are applied o he complee FSE 00 index opion sample o consruc he sample used in he empirical analysis. Firs, opions wih fewer han 5 or more han 0 rading days o mauriy are eliminaed. his choice avoids any expiraion-relaed unusual price flucuaions and minimizes he liquidiy problems ofen affecing he prices of long-erm opions. Second, opions wih moneyness greaer han.0 or less han 0.90 are eliminaed. Moneyness is defined as he raio of fuures price o srike price for call opions and srike price o fuures price for pu opions. he moneyness crierion is applied because deep ou-of-hemoney and in-he-money opions end o be hinly raded. he final sample conains selemen prices on opions wih 5 o 0 rading days o mauriy and moneyness beween 0.90 and.0. his sample is considered o be a represenaive sample of he mos acively raded index opion conracs. he sample is pariioned ino hree moneyness and wo ime o mauriy caegories. An opion is said o be ou-of-he-money (OM) if he moneyness raio is less han 0.97, a-he-

4 Vähämaa 9 money (AM) if he raio is larger han 0.97 and less han.03, and in-he-money (IM) if he raio is greaer han.03. An opion is said o be shor-erm if i has less han 0 rading days o expiraion and long-erm oherwise.. MEHODOLOGY A cenral issue in he empirical esing of opion pricing models is he esimaion of he unobservable model parameers. Following he sandard approach of simulaneous equaions, he vecor of model parameers, Φ, is esimaed by minimizing he sum of squared deviaions beween he observed marke prices and heoreical opion prices min Φ N [ c i cˆ i ( Φ) ] i= (3) where N is he number of opion price observaions on a given day for a given mauriy class, c and ĉ are he observed and heoreical opion prices, respecively, and Φ = { σ } for he Black-Scholes model and Φ = { σ, µ 3, µ } for he skewness and kurosis adjused model. he dela hedging performance of he Black-Scholes and skewness and kurosis adjused models is invesigaed by consrucing a self-financed dela-hedged porfolio wih one uni shor posiion in an opion, δ unis of he underlying asse, and B unis of a risk-free bond. he value of he porfolio Π a ime is Π = δ S + B c () A he beginning of he hedging horizon B0 = c0 δ0s, and hus, 0 Π 0 = 0. he dela hedging performance of he wo delas is examined in one day and one week hedging horizons using daily rebalancing of he hedge porfolio. A each hedge-revision ime he hedge parameer is recompued and he posiion in he bond is adjused o B r = e ( B + S δ δ ) (5) he dela hedging error ε from hedge-revision ime - o is calculaed as ε = δ r c + e B S (6) and he oal hedging error during he hedging horizon τ is given as he hedging horizon Π, he value of he porfolio a he end of ε τ = = ε = Π (7) wo commonly used error saisics, mean absolue hedging error (MAHE) and roo mean squared hedging error (RMSHE) are used o analyze he dela hedging performance of he models. o avoid any disribuional assumpions abou he error saisics, boosrapping is used o es wheher he hedging errors from he wo models are saisically significanly differen.

5 Vähämaa 0 5. EMPIRICAL RESULS able presens he summary saisics of he model parameer esimaes. he esimaed parameers of he skewness and kurosis adjused model indicae ha he reurn disribuion of he FSE 00 index is fa-ailed and negaively skewed wih mean skewness and kurosis esimaes of 0. and 3.8, respecively. Ineresingly, he implied volailiy esimaes under he skewness and kurosis adjused model end o be higher han he Black-Scholes implied volailiies. able. Summary Saisics of he Esimaed Model Parameers BS SKABS σ σ µ 3 µ Mean Median Minimum Maximum Sd. Dev Noes: he able repors he summary saisics of he model parameer esimaes. he model parameers are esimaed by minimizing he sum of squared deviaions beween he observed marke prices and heoreical opion prices. he parameers σ, µ 3, and µ represen volailiy, skewness, and kurosis, respecively. BS and SKABS denoe he Black-Scholes and skewness and kurosis adjused opion pricing models, respecively. he raio of skewness and kurosis adjused dela o Black-Scholes dela for differen level of moneyness is presened in Figure. he raio seems o vary wih moneyness, being below one for OM opions and above one for IM opions, and hus, indicaes ha he skewness and kurosis adjused dela is smaller han he Black- Scholes dela for OM calls and IM pus bu larger for IM calls and OM pus. For AM opions he difference in delas is almos negligible. Similar dela srucure for sochasic volailiy models is documened in Bakshi e al. (000). Figure. he Raio of Skewness and Kurosis Adjused Dela o Black-Scholes Dela,0,0 δ(skabs)/δ(bs),00 0,98 0,96 0,9 0,9 0,90 0,9 0,93 0,95 0,97 0,99,0,03,05,07,09 Moneyness Noes: he figure plos he average raio of skewness and kurosis adjused dela (SKABS) o Black-Scholes dela (BS) for differen levels of moneyness. he delas are calculaed using he model parameer esimaes summarized in able I.

6 Vähämaa he resuls of he dela hedging experimen for he one day hedging horizon are repored in able. he repored numbers are mean absolue hedging errors (MAHE), roo mean squared hedging errors (RMSHE), and he mean difference beween he error saisics of he wo models. he resuls in able indicae ha he skewness and kurosis adjused model performs worse han he Black-Scholes model in erms of dela hedging. he hedging errors produced by he skewness and kurosis adjused model are consisenly larger han he Black-Scholes hedging errors, regardless of he moneyness and mauriy of he opions. he difference in hedging performance appears o be mos disinc for long-erm AM opions. All differences repored in able are saisically significan a he % level. able. Hedging Errors for One Day Hedging Horizon ime o MAHE RMSHE Moneyness Mauriy BS SKABS Difference BS SKABS Difference Full Sample All * * Shor * * Long * * OM All * * Shor * * Long * * AM All * * Shor * * Long * * IM All * * Shor * * Long * * Noes: he repored numbers for each mauriy-moneyness caegory are (i) he mean absolue hedging error (MAHE), (ii) he roo mean squared hedging error (RMSHE), and (iii) he mean difference beween he errors of he wo models. BS and SKABS denoe he Black-Scholes dela and skewness and kurosis adjused dela, respecively. * significan a he 0.0 level and significan a he 0.05 level. able 3. Hedging Errors for One Week Hedging Horizon ime o MAHE RMSHE Moneyness Mauriy BS SKABS Difference BS SKABS Difference Full Sample All * * Shor * * Long * * OM All * * Shor " Long * * AM All * * Shor * * Long * * IM All * * Shor * Long * * Noes: he repored numbers for each mauriy-moneyness caegory are (i) he mean absolue hedging error (MAHE), (ii) he roo mean squared hedging error (RMSHE), and (iii) he mean difference beween he errors of he wo models. BS and SKABS denoe he Black-Scholes dela and skewness and kurosis adjused dela, respecively. * significan a he 0.0 level and significan a he 0.05 level.

7 Vähämaa able 3 presens he hedging resuls for he one week hedging horizon. he resuls seem very similar o he resuls for he one day hedging horizon. Again, he error saisics clearly indicae ha dela hedging under he Black-Scholes models is more effecive han under he skewness and kurosis adjused model. Regardless of he moneyness and mauriy of he opions, boh error saisics are lower for he Black-Scholes dela. Mos of he differences repored in able III are highly saisically significan, wih he only excepions being shor-erm OM and IM opions. 6. CONCLUSIONS his aricle has invesigaed he dela hedging performance of he skewness and kurosis adjused Black- Scholes model of Corrado and Su (996) and Brown and Robinson (00). he empirical ess in he FSE 00 index opion marke show ha he more sophisicaed skewness and kurosis adjused model performs worse han he simplisic Black-Scholes model in erms of dela hedging. he hedging errors produced by he skewness and kurosis adjused model are consisenly larger han he Black-Scholes hedging errors, regardless of he moneyness and mauriy of he opions and he lengh of he hedging horizon. A he firs sigh, hese resuls may seem raher surprising. he resuls are, however, consisen wih he empirical ess on he hedging performance of ime-varying volailiy models [see e.g., Bakshi e al. (997), Dumas e al. (998), Nandi (998), Lim and Guo (000)], and hus, provide furher suppor for he view ha a good opion pricing model is no necessarily a good model for hedging. A poenial explanaion migh be ha, alhough he Black-Scholes dela is likely o be biased, he esimaion error in he dela is relaively small due o he simpliciy of he model. REFERENCES Bakshi, G., C. Cao and Z. Chen (997) Empirical Performance of Alernaive Opion Pricing Models, Journal of Finance, 5, Bakshi, G., C. Cao and Z. Chen (000) Pricing and Hedging Long-erm Opions, Journal of Economerics, 9, Black, F. and M. Scholes (973) he Pricing of Opions and Corporae Liabiliies, Journal of Poliical Economy, 8, Brown, C. and D. Robinson (00) Skewness and Kurosis Implied by Opion Prices: A Correcion, Journal of Financial Research, 5, Coleman,. F., Y. Kim, Y. Li and A. Verma (00) Dynamic Hedging wih a Deerminisic Local Volailiy Funcion Model, Journal of Risk,, Corrado, C. and. Su (996) Skewness and Kurosis in S&P 500 Index Reurns Implied by Opion Prices, Journal of Financial Research, 9, Derman, E. and I. Kani (99) Riding on a Smile, Risk, 7, Duan, J.-C. (995) he GARCH Opion Pricing Model, Mahemaical Finance, 5, 3-3. Dumas, B., J. Fleming and R.E. Whaley (998) Implied Volailiy Funcions: Empirical ess, Journal of Finance, 53, Dupire, B. (99) Pricing wih a Smile, Risk, 7, 8-0. Eberlein, E., U. Keller and K. Prause (998) New Insighs ino Smile, Mispricing and Value a Risk: he Hyperbolic Model, Journal of Business, 7, Engle, R.F. and C. Musafa (99) Implied ARCH Models from Opion Prices, Journal of Economerics, 5, Engle, R.F. and J. Rosenberg (000) esing he Volailiy erm Srucure Using Opion Hedging Crieria, Journal of Derivaives, 8, 0-8. Heson, S. (993a) A Closed Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions, Review of Financial Sudies, 6, Heson, S. (993b) Invisible Parameers in Opion Prices, Journal of Finance, 8, Heson, S. and S. Nandi (000) A Closed-form GARCH Opion Valuaion Model, Review of Financial Sudies, 3, Hull, J. and A. Whie (987) he Pricing of Opions on Asses wih Sochasic Volailiies, Journal of Finance,, Jarrow, R. and A. Rudd (98) Approximae Opion Valuaion for Arbirary Sochasic Processes, Journal of Financial Economics, 0, Lim, K. and X. Guo (000) Pricing American Opions wih Sochasic Volailiy: Evidence from S&P 500 Fuures Opions, Journal of Fuures Markes, 0, Lim, K. and D. Zhi (00) Pricing Opions Using Implied rees: Evidence from FSE-00 Opions, Journal of Fuures Markes,, Melick, W. and C. homas (997) Recovering an Asse s Implied PDF from Opion Prices: An Applicaion o Crude Oil During he Gulf Crisis, Journal of Financial and Quaniaive Analysis, 3, 9-5. Nandi, S. (998) How Imporan is he Correlaion beween Reurns and Volailiy in Sochasic Volailiy Model? Empirical Evidence from Pricing and Hedging in he S&P 500 Index Opions Marke, Journal of Banking and Finance,, Rubinsein, M. (99) Implied Binomial rees, Journal of Finance, 69: 3,