COMPARISON OF STOCHASTIC VOLATILITY MODELS: EMPIRICAL STUDY ON KOSPI 200 INDEX OPTIONS. 1. Introduction

Transcription

1 Bull. Korean Mah. Soc , No. 2, pp DOI /BKMS COMPARISON OF STOCHASTIC VOLATILITY MODELS: EMPIRICAL STUDY ON KOSPI 200 INDEX OPTIONS Kyoung-Sook Moon, Jung-Yon Seon, In-Suk Wee, and Choongseok Yoon Absrac. We examine a unified approach of calculaing he closed form soluions of opion price under sochasic volailiy models using sochasic calculus and he Fourier inversion formula. In paricular, we review and derive he opion pricing formulas under Heson and correlaed Sein- Sein models using a sysemaic and comprehensive approach which were derived individually earlier. We compare he empirical performances of he wo sochasic volailiy models and he Black-Scholes model in pricing KOSPI 200 index opions. 1. Inroducion Based on no arbirage argumens, Black and Scholes [3] and Meron [13] derived a parial differenial equaion for he value of European sock opions. The Black-Scholes B-S model assumed ha he asse price follows a geomeric Brownian moion wih a consan volailiy. Because of is simpliciy and analyical racabiliy, he B-S model is widely used among praciioners for pricing opions. However, a number of empirical sudies documened sysemaic abnormaliies and biases abou he asse reurns and resuling opion prices in he B-S framework. Among hem, he mos well-known bias of he B-S model is he so-called volailiy smile or skew. The implied volailiies from he marke prices of opions vary wih respec o srike prices and mauriies, conrary o he assumpion of consan volailiy in he B-S model. Empirical sudies found ha asse reurns usually have a higher kurosis and a heavy ail compared o he normal disribuion which is assumed by he B-S model. Received March 31, Mahemaics Subjec Classificaion. Primary 91B28, 65C20. Key words and phrases. opion pricing, sochasic volailiy model, Heson model, correlaed Sein-Sein model, KOSPI 200 index opion. This work was suppored by he Korea Research Foundaion Gran funded by he Korean Governmen MOEHRD, Basic Research Promoion Fund KRF C This work was suppored by he research fund R of Seoul Developmen Insiue. 209 c 2009 The Korean Mahemaical Sociey

2 210 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON Anoher aspec of previous empirical finding is ha asse reurns and is volailiy are negaively correlaed alhough he effecs on opion pricing have been considered as a less crucial and primary issue in marke and academia. There have been abundan aemps o inroduce new models o improve he resricive B-S framework. Among he lis, here are jump-diffusion models, CEV models, sochasic volailiy models, local volailiy models, and exponenial Lévy models, which are by no means exhausive ye and creae an overwhelming lis of references. One of major direcions is o relax he B-S assumpion of deerminisic volailiy. To capure he volailiy smile effec, i is assumed ha he volailiy is governed by a separae sochasic process which is ermed as a sochasic volailiy SV model. Various SV models have been sudied in he lieraure such as Hull and Whie [8], Sco [18], Wiggins [21], Johnson and Shanno [9], Melino and Turnbull [12], Sein and Sein [20], Heson [7], and Nicolao and Venardos [15]. When dealing wih opion pricing models, i is imporan o have analyical soluions for pricing derivaives in a closed form. Among hese SV models menioned above, Sein-Sein model and Heson model are used more exensively. They admi analyic soluions and hence are compuaionally efficien. Beween wo SV models here are wo disinc advanages which Heson s model can offer. Heson s model allows for nonnegaive volailiy and nonzero correlaion beween he asse price and volailiy, while Sein and Sein s model doesn. Sein and Sein [20] assumed ha he volailiy follows a mean revering Ornsein-Uhlenbeck process and presened explici closed form soluions for asse price disribuions and opion prices. Bu is major weakness was he assumpion of zero correlaion beween asse reurns and is volailiy. Heson [7] developed anoher sochasic volailiy model, which follows a mean-revering square-roo process and asse reurns and is volailiy are correlaed. A closed form soluion for opions was derived using an analyical expression of he characerisic funcion of log reurns of underlying asses. Sco [19] provided a jump diffusion model incorporaing sochasic volailiy and sochasic ineres, each of which follows a mean-revering square roo processes such as Heson s volailiy process. He derived a closed form soluion for opion price using sochasic calculus and he Fourier inversion formula. Schöbel and Zhu [17] applied he analogous echnique as in Sco [19] o exend he Sein-Sein model o he case where asse reurn and is volailiy are correlaed. To be more precise, i was assumed in Schöbel and Zhu [17] ha he volailiy follows a mean-revering Ornsein-Uhlenbeck process and is correlaed wih asse reurns. Closed form soluions for opion prices are obained by employing he Fourier inversion formula. In Bakshi, Cao, and Chen [1], hey conduced a comprehensive empirical sudy on a relaively rich class of opion pricing models which incorporae Heson s SV, sochasic ineres rae, and Meron s random jumps which sill

3 COMPARISON OF STOCHASTIC VOLATILITY MODELS 211 admi opion pricing formulas in a closed-form. As menioned in heir work, random jumps or sochasic ineres raes added o SV model do no improve overall pricing performances and he SV model alone reduces pricing errors significanly, while he random-jump feaure improves he finess of shor-erm opions and he sochasic ineres rae feaure improves he fi of long-erm opions. More recenly Lin, Srong, and Xu [11], Zhang and Shu [22], Chen and Gau [4], and Dupoye [6] invesigaed empirical performances of Heson s model on FTSE 100 index opions, S&P 500 index opions and currency opions respecively. For pricing KOSPI 200 index opions, Kim and Kim [10] examined four differen classes of sochasic volailiy models including GARCH model, Variance Gamma model, and Heson model. They found ha he Heson model ouperforms he oher models for in-sample pricing, ou-of-sample pricing, and hedging. Anoher poin of view for improving pricing performances is o sress he relevance of nonzero correlaion beween he asse reurn and is volailiy. This phenomena was observed and examined earlier in Rubinsein [16], Baes [2] and Nandi [14]. In paricular, Nandi [14] found subsanial improvemens in pricing ou-of-he-money opions and overall pricing performance allowing nonzero correlaion in he empirical sudy for Heson s model. The objecive of his work is wo-fold. The firs is o sugges a unified way of deriving he closed form soluions for well-known and popular wo versions of SV model using sochasic calculus and he Fourier inversion formula employed individually by Sco [19] and Schöbel and Zhu [17]. We derive he exising formulas for opion prices under Heson and correlaed Sein-Sein models in a unified mehod. For his purpose, we mainain our discussions in he presen work o be self-conained o a grea exen. The second is o examine wo exising SV models and o analyze he empirical performances of he opion prices on KOSPI 200 index beween wo SV models and he B-S model. The paper is organized as follows: In Secion 2 we describe he general seings of Heson and correlaed Sein-Sein SV models. In Secion 3 we derive and presen he opion pricing formulas for he wo SV models in a comprehensive way. In Secions 4.1 and 4.2, we discuss he KOSPI 200 index opions daa and he resuls from parameer esimaion for wo SV models. In Secion 4.3 we invesigae model performances by analyzing ou-of-sample pricing errors for wo SV models and B-S model. Finally in Secion 5 we summarize he resuls The Heson model 2. The sochasic volailiy models Le {S } 0 denoe he price of he underlying asse on a suiable probabiliy space Ω, F, P and saisfy he sochasic differenial equaion 1 ds = µs d + v S dw,

4 212 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON where v is a variance process and µ is a consan expeced rae of reurn. We assume ha v follows Cox-Ingersoll-Ross CIR [5] process: 2 dv = κ θ v d + σ v dz, where θ is a long-run average of v, κ is a rae of mean reversion, σ is called volailiy of volailiy. Here θ, κ, σ are consans and W and Z are wo Brownian moions wih a correlaion coefficien ρ [ 1, 1], i.e., dw dz = ρd. As in Heson [7], we choose he marke price of he volailiy risk proporional o he volailiy. By shifing o a risk-neural measure and applying Girsanov heorem o 1 and 2 we have he following equaions under a risk neural measure Q, 3 where ds = rs d + v S dw, dv = κ θ v d + σ v dz, κ = κ + λ, θ = κθ κ + λ, dw dz = ρd. The variance moves oward he long-run average variance θ, wih he speed deermined by κ. We use he sochasic differenial equaion 3 for he evoluionary model of he underlying asse o describe Heson s model. I is convenien o wrie 4 W = ρz + 1 ρ 2Ẑ, where Ẑ is a sandard Brownian moion independen of Z The correlaed Sein-Sein model Le {S } 0 denoe he price of he underlying asse on a probabiliy space Ω, F, P and saisfy he sochasic differenial equaion 1. To simplify he noaions, we use u = v for a volailiy process and assume ha u follows a mean-revering Ornsein-Uhlenbeck process, 5 du = κ θ u d + σd Z, where κ, θ and σ are consans. As in Schöbel and Zhu [17], we consider he exension of he Sein-Sein model and assume ha W and Z are correlaed Brownian moions wih a correlaion coefficien ρ [ 1, 1]. Again following sandard pracice o ake a risk-neural measure Q, we assume ha he asse price and volailiy are governed as following wo processes; 6 ds = rs d + u S dw, du = κ θ u d + σd Z.

5 COMPARISON OF STOCHASTIC VOLATILITY MODELS 213 Noe ha we coninue o use he same parameers as in 5 o avoid displaying oo many insignifican parameers. Here W and Z are correlaed Brownian moions wih a correlaion coefficien ρ [ 1, 1], i.e., dw d Z = ρd. 3. The opion pricing formulas For boh SV models considered in he presen work, he price of European call opion can be expressed as a condiional expecaion of a discouned payoff under he risk-neural measure Q, 7 E Q [e rτ maxst K, 0 F ], where τ T is he ime o expiry, r is he consan riskless ineres rae, and K is a given srike price. Here F is defined by he smalles σ-algebra generaed by {W s, Z s : s } for he Heson model and {W s, Z s : s } for he correlaed Sein-Sein model respecively. In he presen secion, we invesigae a sysemaic and unified mehod o provide closed-form soluions for European call opion prices under wo SV models. Firs, we consider he Heson s model under risk-neural measure Q and wrie 8 C Heson S, v, T = E Q [e rt maxs T K, 0 F ] = E Q [e rt S T 1 {ST >K} F ] KE Q [e rt K1 {ST >K} F ] C 1 S, v, T KC 2 S, v, T. Again under he correlaed Sein-Sein model governed by 6, we obain following analogous expression for European call opion price, 9 C S S S, u, T = E Q [e rt maxs T K, 0 F ] = E Q [e rt S T 1 {ST >K} F ] KE Q [e rt K1 {ST >K} F ] C 1 S, u, T K C 2 S, u, T. I is worhwhile o noe ha C 1 and C 1 sand for he values of asse-or-nohing call opion for each SV model. Similarly, C 2 and C 2 denoe he values of cash-or-nohing call opion for each model Asse-or-nohing opions Le us denoe he log of he asse price x = ln S. For Heson s model, by change of he measure, dq 1 dq = e rt S T,

6 214 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON we have C 1 S, v, T = E Q [e rt S T 1 {xt >ln K} F ] = E Q 1 [e r 1 {xt >ln K} F ]E Q [S T e rt F ] = E Q1 [1 {xt >ln K} F ]S F 1 S. The characerisic funcion f 1 of x T under Q 1 condiioned on F can be defined by 10 f 1 φ = E Q 1 [e iφx T F ] = EQ [e iφx T S T e rt F ] E Q [S T e rt F ] = EQ [e iφx T S T e rt F ] S e r = e x rt E Q [e 1+iφx T F ]. Consequenly, by applying he Fourier inversion formula, we obain 11 F 1 = exp iφ ln K Re f 1 φ dφ π iφ 0 for Heson s model. Therefore, he asse-or-nohing call opion can be compued by 12 C 1 S, v, T = S F 1. By repeaing he same argumen for he correlaed Sein-Sein model, we have 13 C1 S, u, T = S F1, where F 1 and f 1 are defined analogously as in 10 and Cash-or-nohing opions The value of a cash-or-nohing call opion for Heson s model can be expressed as he following, 14 C 2 S, v, T = E Q [e rt 1 {xt >ln K} F ] = e rt Qx T > ln K F e rt F 2. Now we calculae he characerisic funcion f 2 of x T under Q condiioned on F by 15 f 2 φ = E Q [e iφx T F ]. Once we compue he characerisic funcion f 2, F 2 can be calculaed using he Fourier inversion formula: 16 F 2 = exp iφ ln K Re f 2 φ dφ. π iφ 0

7 COMPARISON OF STOCHASTIC VOLATILITY MODELS 215 Again for correlaed Sein-Sein model, we wrie along he similar lines, 17 C2 S, u, T = e rt F2, where f 2 denoes he characerisic funcion of x T under Q condiioned on F, and F 2 denoes he expression in 16 wih f 2 replacing f The opion pricing formulas for he Heson model We now assume Heson s model, under which S and v follow he equaions in 3 under Q. To derive he formulas for f 1 and f 2 in 10 and 15, i is convenien o obain he Fourier ransform of x T under Q condiioned on F in a general seing. The proof of he lemma is given in Appendix. Lemma 3.1. Assume ha he value of he underlying asse price and is variance saisfy 3 under Q. Then, for z = 1 + iφ or iφ wih φ real and τ = T, we have 18 fz = E Q [e zx T F ] = exp z{x + rτ ρ σ v + κ θ τ} where he funcions Aτ and Bτ saisfy 19 and Aτ = s 2 γ + κ + s 2 e τγ γ κ + 2s 1 e τγ 1 s 2 e τγ 1σ 2 + γ κ + e τγ γ + κ, 20 Bτ = 2θ κ wih σ 2 exp Aτv + Bτ, [ ] 2γe τγ+κ /2 ln s 2 e τγ 1σ 2 + γ κ + e τγ γ + κ γ = κ 2 2s 1 σ 2, s 1 = z ρ σ κ z2 1 ρ 2, s 2 = z ρ σ. Once we compue he Fourier ransform 18, we drive he value of European call opion under he Heson SV model. Theorem 3.2 Heson Model. Assume ha he underlying asse price and is variance saisfy 3 under Q. Then European call opion pricing formula for he Heson model is given by C Heson S, v, T = S F 1 e rτ KF 2, where K is he srike price and r is he riskless ineres rae, and τ = T is he ime o mauriy. Here f 1 φ = f1 + iφ exp x rτ, f 2 φ = fiφ, F i = π and f is defined in exp iφ ln K Re f i φ dφ, iφ

8 216 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON 3.4. The opion pricing formulas for he correlaed Sein-Sein model By employing he analogous echniques, we compue he characerisic funcions for he correlaed Sein-Sein model. The proof of he lemma is given in Appendix. Lemma 3.3. Assume ha he underlying asse price and is volailiy saisfy 6 under Q. Then for z = 1 + iφ or iφ wih φ real and τ = T, we have 21 fz = E Q [e zx T F ] = exp z{x + rτ 1 2 ρ{ σ 1 u 2 + στ}} 1 exp 2 Gτu2 + Hτu + Iτ, where he funcions Gτ, Hτ and Iτsaisfy Gτ = 1 σ sinhγ 1 τ + γ 2 coshγ 1 τ 22 2 κ γ 1, coshγ 1 τ + γ 2 sinhγ 1 τ and 23 Hτ = 1 σ 2 γ 1 κ θγ 1 γ 2 γ 3 + γ 3 sinhγ 1 τ + γ 2 coshγ 1 τ κ θγ 1, coshγ 1 τ + γ 2 sinhγ 1 τ 24 Iτ = 1 2 lncoshγ 1τ + γ 2 sinhγ 1 τ κτ + κ2 θ 2 γ1 2 γ3 2 sinhγ 1 τ 2 σ 2 γ1 3 coshγ 1 τ + γ 2 sinhγ 1 τ γ 1τ + κ θγ 1 γ 2 γ 3 γ 3 σ 2 γ1 3 wih coshγ 1 τ 1 coshγ 1 τ + γ 2 sinhγ 1 τ γ 1 = κ 2 + 2s 1 σ 2, γ 2 = 1 γ 1 κ 2s 3 σ 2, γ 3 = κ 2 θ s 2 σ 2, s 1 = 1 2 z2 1 ρ z1 2 κ ρ σ 1, s 2 = z κ θ ρ σ 1, s 3 = 1 2 z ρ σ 1. Using Lemma 3.3, we can similarly obain he value of European call opion under he correlaed Sein-Sein model as follows: Theorem 3.4 Correlaed Sein-Sein Model. Assume ha he underlying asse price and is volailiy saisfy 6 under Q. Then European call opion pricing formula for he correlaed Sein-Sein model is given by C S S S, u, T = S F1 e rτ K F 2,,

9 COMPARISON OF STOCHASTIC VOLATILITY MODELS 217 where K is he srike price and r is he riskless ineres rae, and τ = T is he ime o mauriy. Here f 1 φ = f1 + iφ exp x rτ, f 2 φ = fiφ, F i = π and f is defined in exp iφ ln K Re f i φ dφ, iφ 4. Empirical performances on KOSPI200 index opions Based on he closed form soluions presened in Theorem 3.2 for he Heson model and in Theorem 3.4 for he correlaed Sein-Sein model we examine and compare he relaive empirical performances of wo SV models and B-S model in pricing KOSPI 200 opions. To perform he empirical es, i is essenial o esimae he unobservable srucure parameers and he spo volailiy using an efficien mehodology. We use he mos popular approach of minimizing he weighed sum of squared pricing errors beween he models prices and marke prices for esimaion of parameers. We assess and analyze wo disinc ypes of ou-of-sample pricing errors for KOSPI 200 opions under wo SV models and B-S model Daa descripion The KOSPI200 Korea Sock Price Index 200 opion was developed a he Korea Sock Exchange KSE in June 1994 by selecing 200 socks from a broad range of indusry groups of socks lised on he KSE. Our empirical sudy is based on European syle KOSPI 200 opions over he period from Jan. 2, 2004 hrough Sep. 21, For each day in he sample, he las repored bidask price quoes and closing prices are recorded which are used for differen purposes. To enhance he efficiency of our analysis, we eliminae opions prices from our daa o avoid unnecessary bias and noises. We exclude opions wih mauriies less han 6 days or more han 90 days, and opions wih prices lower han 0.2. Moreover opions wih he moneyness S/K is less han 0.8 or larger han 1.2, i.e., very deep ou-of-money or deep in-he-money are discarded. The prices which are no saisfying he arbirage resricion are excluded, i.e., C Marke S D e rt K where C Marke is he observed marke prices and D is a presen value of a sum of he fuure dividends during remaining ime T. Following he convenion of KSE, he 91-day cerificae of deposi CD 91-day rae is used as he risk-free ineres rae. Through a series of daa filering, a oal of 17,843 call opions over he period 879 days are recorded. We caegorize he opion daa according o heir

10 218 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON moneyness S/K and mauriy and describe heir characerisics in Table 1. On he basis of moneyness opion daa consis of 32.9% OTM, 32.4% ATM, and 34.7% ITM opions. The sudy shows ha he average call opion price ranges from for shor-erm and deep OTM opions o for shor-erm and deep ITM call opions. In urns ou ha on average we reain abou 26 call opion prices observed on each day. Table 1. KOSPI200 Call Opion Daa Days-o-Mauriy Moneyness Overall OTM S/K < {227} {1398} {1335} {2960} 0.94 S/K < {687} {1218} {1009} {2914} ATM 0.97 S/K < {931} {1163} {938} {3032} 1.00 S/K < {876} {1091} {774} {2741} ITM 1.03 S/K < {735} {972} {627} {2334} S/K {1089} {1673} {1100} {3862} Overall {4545} {7515} {5783} {17843} This able repors average opion price, sandard error in parenheses, and he number of opions in braces, for each moneyness-mauriy caegory. The sample period exends from January 2, 2004 o Sepember 21, Daily informaion from he las ransacion prices of each opion conrac is used o obain he summary saisics. Moneyness is defined as S/K, where S denoes he spo price and K denoes he srike price. OTM, ATM, and ITM denoe ou-of-he money, a-hemoney, and in-he-money opions, respecively Parameer esimaion To compue opion prices for Heson s model, one needs o esimae he vecor of spo variances v 0 and srucural parameers, Φ = κ, θ, σ, ρ, v 0.

11 COMPARISON OF STOCHASTIC VOLATILITY MODELS 219 Table 2. Parameer Esimaion Model B-S σ imp Parameers S-S κ θ σ ρ u Heson κ θ σ ρ v This able repors he daily average and is sandard error in parenheses of he esimaed parameers. The Black-Scholes model in which a single implied volailiy is esimaed across all srikes and mauriies on a given day and he SV models in which parameers are esimaed by minimizing he weighed sum of squared pricing errors beween model and marke opion prices for each day. For he counerpar of Sein-Sein model, he vecor of he spo volailiy u 0, and srucural parameers, Φ = κ, θ, σ, ρ, u0, are inpu parameers o be esimaed from opion daa. Following he sandard pracice by mos researchers and praciioners, we use he leas-square-minimizaion mehod o obain he parameers implied by opion daa. We describe he esimaion procedure as follows. Firs, we collec N opion values on he same underlying asse in he same day. These opions have differen imes o mauriy and srike prices. be he mid price of bid-ask spread of he i-h opion on day, and Ci, Model be eiher he Heson s or he correlaed Sein-Sein model prices of he i-h opion on day. The parameer se Φ and Φ are hen deermined by N 25 min ω i C Mid i, Ci, Model 2 for = 1, 2,..., T d, i=1 Le C Mid i, where T d is he oal number of days in he sample. Here we ake he observed mid price Ci, Mid and he weigh funcion ω i as following C Mid i, = Cbid i, + Ci, ask, ω i = 2 C ask i, 1 Ci, bid where Ci, bid and Ci, ask are he bid and ask prices of he i-h opion a he day. If he bid-ask spread of a paricular opion is grea, hen we have a wider range of marke prices around he mid-price which necessarily imply small liquidiy as a resul. I is quie reasonable o ake less weighs for such opions in esimaing he implied parameers.

12 220 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON For he Heson and he correlaed Sein-Sein models, Table 2 repors he averages and he sandard errors in parenheses of he parameers, which are esimaed daily. The implied parameers of KOSPI200 index call opion in he sample are shown o be he average values of {κ, θ, σ} = {6.9282, , } for he Heson model and { κ, θ, σ} = {2.5214, , } for he correlaed Sein-Sein model. The average of esimaed correlaion coefficiens are ρ = and ρ = We conform he widely acceped assumpion ha asse reurns and volailiy are negaively correlaed, as documened by many previous resuls such as Rubinsein [16], Nandi [14], and Baes [2] Empirical ess Using esimaion mehology described earlier, we perform ou-of-sample pricing ess for wo versions of SV model and he B-S model. For wo SV models, he implied srucural parameers and spo variance and spo volailiy are compued by minimizing he weighed sum of squared errors beween model prices and mid-prices from he previous day while for B-S model, an average single implied volailiy is compued from he previous day. For measuremen of pricing errors, we demonsrae values of he mean percenage pricing errormpe and mean absolue pricing errormae defined as 1 MPE 2 MAE 1 N 1 N N C Marke i Ci Model Ci Marke i=1 N i=1 C Marke i Ci Model where N is he oal number of opions in a paricular moneyness-mauriy caegory, Ci Marke and Ci Model represen he marke price and he heoreical model price, respecively. These measuremens were popular, e.g., in Bakshi, Cao, and Chen [1]. Table 3 and Table 4 repor he MPE s and MAE s of he Black-Scholes model, he correlaed Sein-Sein model, he Heson model, for each moneynessmauriy caegories. We summarize he resuls and draw some conclusions. From boh pricing error measuremens, wo SV models improve pricing performances significanly over B-S model which is no surprising as was found by Bakshi, Cao, and Chen [1]. In comparison of wo SV models, i is worhwhile o noice ha Heson model performs slighly beer han correlaed Sein-Sein model in he mos caegories, alhough wo SV models have he idenical number of srucural parameers. Pricing improvemens of wo SV models over B-S model are paricularly noiceable for OTM opions in pricing error measuremens. As an example, for long-erm opion wih moneyness less han 0.94, MPE for B-S, Sein-Sein and Heson models are -2.76%, -1.33%, -1.05% and

13 COMPARISON OF STOCHASTIC VOLATILITY MODELS 221 Table 3. The Ou-of-sample Mean Percenage Pricing Error MPE Days-o-Mauriy Moneyness Overall OTM S/K < 0.94 B-S S-S Heson S/K < 0.97 B-S S-S Heson ATM 0.97 S/K < 1.00 B-S S-S Heson S/K < 1.03 B-S S-S Heson ITM 1.03 S/K < 1.06 B-S S-S Heson S/K 1.06 B-S S-S Heson Overall B-S S-S Heson The Black-Scholes model price is compued by using he average implied volailiy from he previous day. The SV model prices are compued by using he implied parameers esimaed from minimizing he weighed sum of squared errors beween he mid price and he model price from he previous day. The repored percenage pricing error is he sample average of he marke price minus he model price, divided by he marke price. The numbers in he parenheses are he sandard errors. MAE for B-S, Sein-Sein and Heson models are , , and respecively. By examining MPE, we find ha regardless of pricing models, OTM opions are overpriced in each mauriy class, whereas he magniude of errors in wo SV models is much smaller han in B-S model. On he basis of

14 222 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON Table 4. The Ou-of-sample Mean Absolue Pricing ErrorMAE Days-o-Mauriy Moneyness Overall OTM S/K < 0.94 B-S S-S Heson S/K < 0.97 B-S S-S Heson ATM 0.97 S/K < 1.00 B-S S-S Heson S/K < 1.03 B-S S-S Heson ITM 1.03 S/K < 1.06 B-S S-S Heson S/K 1.06 B-S S-S Heson Overall B-S S-S Heson The Black-Scholes model price is compued by using he average implied volailiy from he previous day. The SV model prices are compued by using he implied parameers esimaed from minimizing he weighed sum of squared errors beween he mid price and he model price from he previous day. The repored absolue pricing error is he absolue value of he marke price minus he model price wihin each moneyness-mauriy caegory. The numbers in he parenheses are he sandard errors. MPE, he pricing improvemens by wo SV models over B-S model decrease as moneyness increases, i.e., from OTM o ATM and from ATM o ITM in each mauriy group. Conrary o his, he magniude of absolue pricing errors

15 COMPARISON OF STOCHASTIC VOLATILITY MODELS 223 repored in MAE ypically increase as he moneyness increase in each mauriy class across all models, since he opions become more valuable as he moneyness increases. We observe again ha in MAE measuremen, he Heson model ouperforms he oher models in mos of he moneyness-mauriy groups, and he magniude of improvemen is more subsanial for OTM opions han for ITM opions. 5. Conclusions In his work, we review and derive he closed form soluions for he values of European call opions under he Heson s and he correlaed Sein-Sein sochasic volailiy models using a unified approach. The analyic soluions are derived using sochasic calculus and Fourier inversion formula. Pricing performances of wo versions of SV models and B-S model are empirically analyzed using he KOSPI 200 index opion daa from Jan. 2, 2004 hrough Sep. 21, The cross-secional implied parameers are esimaed from minimizing he weighed sum of squared errors beween he marke and he model prices for each model. Through he measuremens of pricing errors, we conclude ha wo SV models ouperform significanly over B-S model and he Heson model performs slighly beer han he correlaed Sein-Sein model. 6. Appendix In his secion, we prove he formulas of he characerisic funcions 18 and 21 for he Heson model in Lemma 3.1 and he correlaed Sein-Sein model in Lemma 3.3 respecively. Proof of he Lemma 3.1 We adop he approach employed in Sco [19] and provide more deailed discussions along he lines. Le us denoe he log of he sock price x = ln S. Then from he sochasic differenial equaion 3, x saisfies dx = r 1 2 v d + v dw. For z = 1 + iφ, we obain 26 fz = f1 + iφ = E Q [ e 1+iφx T F ] = e 1+iφx E Q [ e 1+iφx T x F ] [ = e 1+iφx+rτ E Q e 1+iφ T 1 2 vsds+ T ] vsdw s F. Using he relaion 4, we can wrie T T vs dw s = vs ρdz s + 1 ρ 2 dẑs,

16 224 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON where Z s and Ẑs are independen Brownian moions. We inroduce anoher filraion F, he smalles σ-algebra generaed by {W s, s, Z u, u T }. Then he expecaion in 26 can be wrien [ E Q E [exp Q 1 T ] ] T iφ v s ds exp 1 + iφ vs dw s F F = E [exp Q 1 T iφ v s ds exp 1 + iφρ Since we obain T vs dz s E [exp Q 1 + iφ ] ] T 1 ρ 2 vs dẑs F F. T 1 ρ 2 v s dẑs N 0, T 1 ρ 2 v s ds E [exp Q 1 + iφ ] T 1 ρ 2 vs dẑs F = exp Also from 3, we have iφ2 1 ρ 2 T T v s ds 1 + iφρ vs dz s = 1 + iφ ρ T T dv s k θ v s ds σ = 1 + iφ ρ σ v T v k θ τ iφ k ρ σ As a consequence, we have. T 27 f1 + iφ = e 1+iφx +rτ 1+iφ ρ σ v +κ θ τ E Q [e s 2v T e s 1 where 28 s 1 = 1 + iφ, v s ds. T v sds F ], k ρ σ iφ2 1 ρ 2, s 2 = 1 + iφ ρ σ. Le us define yv,, T = E Q [e s2vt T e s1 vsds F ], wih s 1 and s 2 are defined in 28. Then he Feynman-Kac sochasic represenaion heorem provides us ha y is he soluion of he following PDE 29 y + k θ v y v σ2 v 2 y v 2 + s 1vy = 0

17 COMPARISON OF STOCHASTIC VOLATILITY MODELS 225 wih a final condiion yv, T, T = e s2v. By analogy wih Black-Scholes formula, we guess a soluion of he form: yv,, T = e AT v+bt. By subsiuing he proposed value in 29, we obain he following ordinary differenial equaions ODEs for Aτ and Bτ: A τ = k Aτ 1 2 σ2 A 2 τ s 1, A0 = s 2, B τ = k θ Aτ, B0 = 0. The soluions of he above Riccai equaion have well known formula from he lieraure. Hence f1 + iφ is given by 30 f1 + iφ = e 1+iφx+rτ 1+iφ ρ σ v+κ θ τ e Aτv+Bτ, where τ = T and Aτ and Bτ are defined in 19 and 20 respecively for z = 1 + iφ. For z = iφ, one can apply similar echniques o ge he expression in 18 for fiφ. This concludes he lemma. The similar calculaions are performed for he correlaed Sein-Sein model o ge Lemma 3.3. Proof of he Lemma 3.3 We follow he proof given in Shöbel and Zhu [17]. Le us denoe he log of he sock price x = ln S. Then from he sochasic differenial equaion 6, x saisfies dx = r 1 2 u2 d + u dw. For z = 1 + iφ, we obain fz = f1 + iφ = E Q [ e 1+iφx T F ] [ = e 1+iφx+rτ E Q e {1+iφ T 1 2 u2 sds+1+iφ T usdws} ] F. From Io s formula and he equaion 6, we have { T u s d Z s = 1 ut 2 u 2 σ 2 T 2 κ 2 σ T u s θ u s ds We perform similar calculaions as in he proof of he Lemma 3.1 and o obain f1 [ ] + iφ = e 1+iφx +rτ iφ ρ στ+u2 σ 1 } E Q e s3u2 T T s 1u 2 s +s2usds F }.

18 226 K.-S. MOON, J.-Y. SEON, I.-S. WEE, AND C. YOON where Le us define s 1 = iφ2 1 ρ iφ1 2 κ ρ σ 1, s 2 = 1 + iφ κ θ ρ σ 1, s 3 = iφ ρ σ 1. yu,, T = E Q[e s 3u 2 T e T s 1u 2 s +s 2u s ds F ]. Then he Feynman-Kac sochasic represenaion heorem provides us ha y is he soluion of he following PDE y + κ θ u y u σ2 2 y u 2 s 1u 2 + s 2 uy = 0 wih a final condiion yu, T, T = e s 3u 2. Similarly y can be wrien by yu,, T = e 1 2 GT u2 +HT u+it and he soluion of ODEs for Gτ, Hτ, Iτ are given in 22, 23, and 24 in Lemma 3.3 respecively See Schöbel and Zhu [17] for deails. For z = iφ, we use similar echniques o ge he expression in 21 for fiφ, which concludes he lemma. References [1] G. S. Bakshi, C. Cao, and Z. W. Chen, Empirical performance of alernaive opion pricing models, Journal of Finance , no. 5, [2] D. Baes, Pos-87 crash fears in he S&P 500 fuures opion marke, Journal of Economerics , [3] F. Black and M. Scholes, 1973, The pricing of opions and corporae liabiliies, Journal of Poliical Economy , [4] M. Chen and Y. Gau, Pricing currency opions under sochasic volailiy, 2004 NTU Inernaional Conference on Finance, [5] J. C. Cox, J. E. Ingersoll, and S. A. Ross, A heory of he erm srucure of ineres raes, Economerica , no. 2, [6] B. Dupoye, Informaion conen of cross-secional opion prices: A comparison of alernaive currency opion pricing models on he Japanese yen, Journal of Fuures Markes , [7] S. L. Heson, A closed-form soluion for opions wih sochasic volailiy wih applicaions o bonds and currency opions, The Review of Financial Sudies , [8] J. Hull and A. Whie, The pricing of opions on asses wih sochasic volailiies, Journal of Finance , [9] J. Johnson and D. Shanno, Opion pricing when he variance is changing, Journal of Financia1 and Quaniaive Analysis , [10] I. J. Kim and S. Kim, Empirical comparison of alernaive sochasic volailiy opion pricing models: Evidence from Korean KOSPI 200 index opions marke, Pacific-Basin Finance Journal , [11] Y.-N. Lin, N. Srong, and X. Xu, Pricing FTSE 100 index opion under sochasic volailiy, Journal of Fuures Markes ,

19 COMPARISON OF STOCHASTIC VOLATILITY MODELS 227 [12] A. Melino and S. Turnbull, Pricing foreign currency opions wih sochasic volailiy, Journal of Economerics , [13] R. C. Meron, Theory of raional opion pricing, Bell Journal of Economic and Managemen Science , [14] S. Nandi, How imporan is he correlaion beween reurns and volailiy in a sochasic volailiy model? Empirical evidence from pricing and hedging in he S&P 500 index opions marke, Journal of Banking and Finance , [15] E. Nicolao and E. Venardos, Opion pricing in sochasic volailiy models of he Ornsein-Uhlenbeck ype, Mahemaical Finance , [16] M. Rubinsein, Implied binomial rees, Journal of Finance , [17] R. Schöbel and J. Zhu, Sochasic volailiy wih a Ornsein-Uhlenbeck process: An exension, European Finance Review , [18] L. Sco, Opion pricing when he variance changes randomly: Theory, esimaion, and an applicaion, The Journal of Financial and Quaniaive Analysis , [19], Pricing sock opions in a jump-diffusion model wih sochasic volailiy and ineres raes: Applicaions of Fourier inversion mehods, Mah. Finance , [20] E. Sein and J. Sein, Sock price disribuions wih sochasic volailiy: An analyic approach, Review of Financial Sudies , [21] J. B. Wiggins, Opion values under sochasic volailiy: Theory and empirical esimaes, Journal of Financial Economics , [22] J. E. Zhang and J. Shu, Pricing S&P 500 index opions wih Heson s model, Proceedings, 2003 IEEE Inernaional Conference on Compuaional Inelligence for Financial Engineering, Kyoung-Sook Moon Deparmen of Mahemaics & Informaion Kyungwon Universiy Gyeonggi-do , Korea address: ksmoon@kyungwon.ac.kr Jung-Yon Seon Deparmen of Mahemaics Korea Universiy Seoul , Korea address: sunjy24@korea.ac.kr In-Suk Wee Deparmen of Mahemaics Korea Universiy Seoul , Korea address: iswee@korea.ac.kr Choongseok Yoon Deparmen of Mahemaics Korea Universiy Seoul , Korea address: csyoon@korea.ac.kr