Variance Dependent Pricing Kernels in GARCH Models

Transcription

1 U.U.D.M. Projec Repor 01:0 Variance Dependen Pricing Kernels in GARCH Models Amir Hossein Khalilzadeh Examensarbee i maemaik, 30 hp Handledare och examinaor: Maciej Klimek epember 01 Deparmen of Mahemaics Uppsala Universiy

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3 Absrac In his hesis we sudy hree of he sriking conribuions of eve Heson o he conex of opion valuaion. In he coninuous ime sochasic volailiy model of Heson (1993) and discree ime GARCH model of Heson and Nandi (000), we invesigae he whole procedure owards obaining he closed form formula for he value of he European call opion. We hen shape our discussion in he work of Chrisoffersen, Heson and Jacobs (011) on he role of pricing kernel in opion valuaion. They have developed an opion pricing model ha incorporaes he aversion o variance risk in he pricing kernel. Among reasons ha feaure heir work in he conex of opion pricing is a new version of Heson and Nandi's GARCH model which no only overcomes he difficulies in he esimaion of original sochasic volailiy model of Heson, bu also is able o produce a U-shaped pricing kernel observed in empirical sudies.

4 Acknowledgemen Firs and foremos, I would like o express my umos graiude o my advisor Professor Maciej Klimek for inroducing his subjec o me, and for his houghful and deailed commens. I would have been los wihou his coninuous suppor no only for his hesis bu also during he whole period of my sudy a Uppsala Universiy. My sincere hanks also go o Dr. Erik Eksröm for his suppor as he coordinaor of he program, and o Dr. Örjan enflo, Dr. Jörgen Ösensson and Professors a he IT deparmen of Uppsala Universiy for sharing heir professional knowledge wih me. I am also graeful o he sudy counselors in he mahemaics deparmens of Uppsala Universiy, for helping he deparmen o run smoohly and for assising me in many differen ways. I is difficul o oversae my graiude o my parens who gave me srengh o chase my dreams.

5 Conens 1 Inroducion Overview ylized Facs of Financial Reurns Volailiy mile GARCH Models of Volailiy No Arbirage Argumen and he Black-choles Pricing Equaion Characerisic Funcion and Fourier Inversion Theorem Pricing Kernel ochasic Volailiy Models Overview Fully ochasic Volailiy Models Consrucion of he Model Mean Reversion Propery ome Popular Models Heson (1993) Model GARCH Diffusion Model Heson s Opion Pricing Model Overview Theoreical Consideraion No Arbirage Argumen and he Pricing Equaion Marke Price of Volailiy Risk Characerisic Funcion and he Closed-form oluion Limiaions and Furher Developmens... 36

6 4 Heson-Nandi s Opion Pricing Model Overview Heson-Nandi GARCH Diffusion Model The Risk Neural GARCH Process and he Closed form Formula Limiaions and Furher Developmens Chrisoffersen-Heson-Jacobs s Opion Pricing Model Overview Theoreical Consideraion Coninuous Time Risk Neuralizaion and he Pricing Kernel Discree Time Risk Neuralizaion and he Pricing Kernel Conclusion... 5 References... 54

7 Chaper 1 Inroducion 1.1 Overview The lieraure on he opion pricing models has winessed exensive progress since he invenion of Black and choles model (1973). A considerable amoun of his effor has been channeled in providing more realisic assumpions for he underlying price process (see Bakshi, Cao and Chen (1997) for a collecion of models). One of he mos successful price processes is he bivariae diffusion model of Heson (1993). Despie being successful, he Heson s sochasic volailiy model is difficul o esimae because he volailiy is no observable. This problem was successfully addressed by a GARCH framework in Heson and Nandi (000). Ye anoher problem o be ackled in opion valuaion models was due o he pricing kernel which was a funcion of he index reurn only. Recenly, by incorporaing variance risk premium in he pricing kernel, Chrisofesson e al (011) have solved his problem. This projec is aimed o highligh he crucial role of pricing kernel in he conex of opion valuaion. To do so, we invesigae he opion valuaion framework in he V model of Heson (1993), GARCH model of Heson and Nandi (000) and finally he recen work of 1

8 Chrisofesson e al (011). We focus on he derivaion of he closed-form opion price in he Heson (1993) work in chaper 3, whereas, in chaper 4 we mosly discuss on he physical and risk neural processes in he Heson and Nandi's work (he valuaion echniques are almos he same). Finally, in chaper 5 we alk less abou he physical and risk neural processes as well as he valuaion echniques, bu, pu more emphasis on he consrucion of he pricing kernel in Chrisofesson e al (011) which disinguishes he valuaion model among oher models. In he following secions, we briefly explain echniques and conceps needed for he subsequen chapers. 1. ylized Facs of Financial Reurns Properies of he financial daa used for valuaion of opions show he way one has o ake in order o consruc or improve he underlying models. Therefore, knowing hese properies helps us o beer undersand why differen sochasic volailiy models (e.g. Heson or GARCH diffusion) are employed in opion valuaion as we will see in he following chapers. ylized facs refer o a se of common feaures of financial daa. In fac, hey are some saisical properies common across various insrumens and markes which have been revealed by empirical sudies since almos a cenury ago. Here we only lis some of hese properies which are more relevan o his sudy, and refer ineresed reader o Con (001) for an inclusive survey. 1. Absence of serial correlaion: excep a high frequency, here is no linear auocorrelaion in reurns.. Fa-ailness: The uncondiional disribuion of reurns has faer ails han ha expeced from a normal disribuion, meaning ha, using he normal disribuion for he purpose of

9 modeling financial reurns, we will underesimae he number and magniude of crashes and booms. 3. Volailiy clusering: here exiss significan serial correlaion in volailiy of reurns. This means ha a large (posiive or negaive) reurn ends o be followed by anoher large (posiive or negaive) reurn. 4. Leverage effec: he sock reurns are ypically negaively correlaed wih he volailiy. This suggess ha he dep-equiy raio or leverage of he firm increases when he sock price declines. 5. Asymmery: here exiss negaive skewness in he uncondiional disribuion of reurns, meaning ha, exreme negaive reurns are more likely o happen han exreme posiive reurns. Properies wo and five exis, a lower inensiy, even afer correcing for volailiy clusering. 1.3 Volailiy mile Before going hrough opion pricing wih sochasic volailiy in he nex chapers, we essenially need o know why sochasic volailiy models are so imporan in he conex of opion valuaion. Time varying volailiy has been one of he long lasing issues in financial economics. Early commens on his include Mandelbro (1963). Therefore, he assumpion of consan volailiy in he geomeric Brownian moion governing he securiy price process in he Black-choles world was suspeced soon afer he breakhrough of Black and choles (1973). Laer on, in paricular afer he 1987 marke crash, he opions daa observed in he marke found no o be in harmony wih he Black-choles prices. The value of volailiy ha equaes hese wo prices, called 3

10 implied volailiy, revealed he answer o he disharmony, he smile. Volailiy smile is simply he relaion beween he implied volailiy and he srike price (or some funcion of he srike price) of he opion. More precisely, he erm volailiy smile is used for FX markes or equiy index opions as he graph urns up a eiher ends, whereas, for opions such as equiy opions he graph is downward sloping and herefore he erm volailiy skew is ofen used. From he perspecive of disribuion of reurns, volailiy smile could be hink of as he fa ail of opionimplied reurn disribuion ha reconciles he empirical disribuions of spo reurns wih he risk neural disribuion underlying opion prices. 1.4 GARCH Models of Volailiy In his secion we lay he foundaion for he Heson and Nandi (000) opion pricing model by describing he well-known discree ime volailiy model, GARCH. Engle (198) invened Auoregressive Condiional Heeroskedasiciy (ARCH) model o rea he serial correlaion in square reurns explained in he previous secions. GARCH model as an exension of his model inroduced by Bollerslev (1986), successfully addressed he fa-ail feaure presen in he disribuion of reurns. We base our inroducion o hese models on Jondeau e al (007). Generally, a volailiy model can be srucured as follow x = µ ( θ ) + ε, ε = σ ( θ ) z, (1.1) where µ ( θ ) = E x F 1, σ ( θ ) = E ( x µ ( θ )) F 1. (1.) 4

11 The dynamic of condiional mean µ ( θ ) in (1.) is usually assumed o be an ARMA(p,q) process, σ ( θ ) is he model for condiional variance, θ is he vecor of parameers and F 1 is he informaion se available a ime. According o (1.1), ε has a ime varying volailiy condiioned on he informaion available a ime 1. Finally, z has a disribuion wih mean zero and variance one, such as sandard normal or suden-. In case of GARCH model z is a srong whie noise process. In a volailiy model, volailiy can be described in eiher of hese wo ways: as an exac funcion of a se of variables (e.g. GARCH models) or as a sochasic funcion (e.g. sochasic volailiy models). In he case of GARCH(p,q) model he dynamic of volailiy is as follow p q = + i i + j j i = 1 j = 1 (1.3) σ ω α ε β σ For his model o be well defined and he condiional variance o be posiive, he parameers mus saisfy ω > 0, α 0, i = 1,..., p, β 0, j = 1,..., q. In addiion o hese consrains, when i j + β j < 1which ensures he covariance saionary of ε, one can obain he p q α i= 1 i j= 1 uncondiional variance as p q σ = ω / 1 αi β j i= 1 j= 1 In he GARCH models, posiive and negaive pas values have a symmeric effec on he condiional variance, which is agains he empirical resuls suggesing ha bad news end o be followed by larger increases in volailiy han equally large posiive reurns. everal 5

12 parameerizaions (e.g. EGARCH, TGARCH and GJR) addressed his asymmery in he response of volailiy o shocks. We do no go furher on he deails of hese models. Insead, we briefly explain wo oher models proposed by Engle and Ng (1993), Non-linear GARCH and Vecor GARCH, as hey are similar o he Heson and Nandi GARCH model of chaper 4. The condiional variance in he NGARCH(p,q) has he following form p q = + i i + i + j j i= 1 j= 1 ( ) (1.4) σ ω α ε γσ β σ whereas he VGARCH has he following condiional variance p q = + i i i + + j j i= 1 j= 1 ( ) (1.5) σ ω α ε σ γ β σ 1.5 No Arbirage Argumen and he Black-choles Pricing Equaion In his secion we sudy some basic conceps in opion pricing needed for he advanced argumens saed in chaper 3 onward. More precisely, his secion reviews he no arbirage argumen of Black and choles (1973) and concludes wih heir well-known pricing equaion. In he on arbirage approach we assume a geomeric Brownian moion for he dynamic of he underlying asse and hen derive he dynamic of he derivaice asse. A risk-free porfolio using he derivaive and he underlying will be composed and in he nex sep in order o avoid he arbirage opporuniy, he insananeous reurn of he porfolio mus be equal o he risk-free rae of ineres. We hen arrive a a parial differenial equaion which has he price of he derivaive asse as is soluion. 6

13 Black and choles assumed ha he sock price follows a geomeric Brownian moion d = µ d + σ db (1.6) Where d is he insananeous price change, µ is he consan expeced reurn, σ is he consan volailiy of he sock reurn process and B is he Brownian moion or Weiner process such ha db ~ N(0, d ). We denoe he price of a European call opion a ime wih exercise price K and ime of mauriy T on he underlying asse by C (, ). Applying Io s lemma o his funcion we obian he following dynamics C C 1 C C dc d µ σ σ = db (1.7) Now he value of he porfolio consising of one uni of he call opion and a shor posiion of C unis in he sock would be C Π = C wih he dynamics C d Π = dc d (1.8) ubsiuing dc from (1.7) and d from (1.6) C 1 C d Π = + σ d 0dB + (1.9) To avoid arbirage opporuniy he reurn of his porfolio mus be he same as he risk-free rae of ineres 7

14 C d Π = rπ d = r C d C 1 = + σ C d (1.10) Finally C C 1 C + + = 0 r σ rc (1.11) olving his equaion along wih is boundary condiion which is he final payoff of he call opion gives us he price of he European call opion a ime. we laer use hese conceps and noaions when we deal wih opion pricing in he presence of sochasic volailiy. 1.6 Characerisic Funcion and Fourier Inversion Theorem Wha made Heson's opion pricing model superior o is preceding works was he closed-form formula he obained via he characerisic funcions. In his secion, we review he basic concep of characerisic funcion and refer o hem in he proceeding chapers. Definiion. Le X be a random variable wih probabiliy disribuion F X. The characerisic funcion of X or FX is he funcion φ defined for real u by iux iux φ ( u ) = E[ e ] = e df ( x ) (1.1) X X or for disribuion F X wih a probabiliy densiy funcion f X iux φ ( u ) = e f ( x ) dx (1.13) X 8 X

15 We will be assuming ha X is such ha FX is coninuous and φ ( u X ) du <. For insance, characerisic funcion of 1 X ~ N ( µ, σ ) is φ ( u ) = X exp( µ iu ) σ u. However, he righ hand side of (1.13), wih u replaced by u, is known as he Fourier ransform of f where f is any inegrable funcion. Noe also ha here are differen definiions of Fourier ransform such as ˆ( ) π iξ X = ( ) f ξ e f x dx R ˆ( ) iυ X f υ = e f ( x ) dx R 1 ˆ( ) iω X f ω = e f ( x ) dx π R (1.14) for any inegrable funcion f. Using characerisic funcion i is possible o characerize disribuions ha canno be described by a probabiliy disribuion funcion. Using he inversion heorem, which we bring i here wihou proof, i is possible o recover he disribuion funcion from is characerisic funcion. Inversion heorem. Under he earlier assumpions abou X 1 1 exp( ix ) φx ( ) FX ( x ) = Re d π i. (1.15) 0 Using his heorem he following equivalen expressions can also be drived 9

16 1 1 exp( ix) φx ( ) FX ( x) = d π i 1 exp( ix) 1 = FX (0) φx ( ) d π i 1 ix f X ( x) = e φx ( ) d π (1.16) imilar o he las line in (1.16), for he Fourier ransforms in (1.14) we have ˆ ξ ˆ ξ ξ πiξ X πiξ X f ( ) = e f ( x) dx f ( x) = e f ( ) d R ˆ iυ X 1 iυ X f ( υ) = e f ( x) dx f ( x) e fˆ = ( υ) dυ π R ˆ 1 iω X 1 iω X f ( ω) = e f ( x) dx f ( x) e fˆ ( ω) dω π = π R R R R We are herefore equipped wih flexible formulas which enable us o derive he densiy funcion from he characerisic funcion and vice versa. Using (1.16) for he example of normal random variable X ~ N ( µ, σ ) we can ge he densiy as 1 iux µ iu 0.5σ u f X ( x) = e e du π We will come back o his laer in chaper 3 and 4. In our conex, he Fourier ransform is called characerisic funcion when he funcion used in he inegraion is a densiy. There are echniques (Fas Fourier Transform) in he conex of Fourier analysis ha make he compuaions of he inegrals more efficien. 10

17 1.7 Pricing Kernel Discussing on he pricing kernel ineviably akes us o he uiliy heory. In heory, a uiliy funcion U is an increasing (posiive firs derivaive) and concave (negaive second derivaive) funcion ha akes some observable variable such as consumpion (of sock for insance) and gives he uiliy of his consumpion ha is no observable. Neverheless, individuals have differen uiliy funcions. Three ypes of uiliy funcions concave, convex and linear are assigned o risk averse, risk seeker and risk neural individuals, respecively. These hree funcions however share a common feaure, called non-saiaion propery, ha is, hey increase wih he increase in he wealh, or more wealh is preferred o less wealh. This means ha he firs derivaive of uiliy funcion is always posiive. Wha differeniae he hree forms of uiliy funcion is he risk preference of individuals ha is refleced in he second derivaive of U. The risk averse (lover) invesor has a concave (convex) uiliy funcion which makes he second derivaive negaive (posiive), and he risk neural invesor has a linear uiliy funcion and herefore second derivaive of zero for is uiliy funcion. I is herefore possible o analyze risk aiudes of invesors by looking over heir uiliy funcions. One of he conceps ha link he uiliy funcion of an invesor o wha is happening in he real world is he pricing kernel. Pricing kernel, as we will discuss i below, is he raio beween he risk neural and he hisorical densiies. From asse pricing, we have he price of a securiy a ime zero in an equilibrium model as ( ) = Ε ψ (1.17) P P0 T M T 11

18 where ψ ( T ) is he payoff funcion of securiy a mauriy T, which could be he sock price plus is dividend for a sock, or, he payoff of he call opion a mauriy for an opion. P Ε is he expecaion wih respec o he physical or hisorical measure P condiional on he informaion se available a ime zero, and MT is sochasic discoun facor which we explain i furher below. For he derivaion of equaion (1.17), which is he firs-order condiion for an opimal consumpion and porfolio choice, we refer o Cochrane (001). Cochrane (001) also explains ha MT is sochasic because oherwise (i.e. in case of no uncerainy) we could discoun he payoff wih he gross risk free rae 1 R as P ψ ( ) f = and 0 R f T 1 for a cerain riskier asse as P 0 ɶ ψ ( T ) facor. o, = Ε R ɶ where Rɶ 1 is an asse specific risk-adjused discoun MT is considered as a general discoun facor ha incorporaes all risk adjusmens, and is he same for each asse. ochasic discoun facor could also be shown as M T U ( T ) = β U ( ) 0 (1.18) where β is a fixed discoun facor and U is he firs derivaive of he uiliy funcion. We now ry o obain he form of pricing kernel in case of Black and choles model, and in chaper 5 we will see how his form will be generalize for appropriaely explaining opions daa. Recalling he price of an opion under risk neural measure Q, and following Delefsen e al (010) we can wrie 1

19 ( ) = Ε Q rt P0 e ψ T ( T ) ( ) q ( ) P rt = Ε e ψ T p T (1.19) Hence, from (1.17)-(1.19) we have U ( ) β s = e U ( ) 0 rt ( ) ( ) q s p s (1.0) where r is he risk free rae and q( s) p ( s) is he pricing kernel. ince boh physical and risk neural disribuions in he Black and choles opion pricing model have log-normal disribuion, wih r replaced by µ in geomeric Brownian moion (1.6) for he risk neural disribuion, we have 1 log s ɶ µ { ( ) } ɶ σ ( ) 1 1 ɶ µ = µ σ / + log 0 p( s) = exp, s πσɶ ɶ σ = σ Therefore, one can derive he pricing kernel as µ r s σ ( ) {( µ )( µ σ ) σ } Mɶ ( s) = exp r + r T / ( ) (1.1) 0 which, as we expec from he Black and choles model, has form of he derivaive of he power uiliy funcion. However, he fac ha we canno observe his monoonically increasing and concave uiliy funcion from he marke daa, has given rise o wha is called he pricing kernel puzzle. Pricing kernel puzzle firs arose by Jackwerh (000). To explain his puzzle, we could say ha, heoreically, he marginal uiliy of invesors (pricing kernel) should be decreasing when he 13

20 aggregae wealh of an economy rises. This means ha behavior of invesor is similar o he risk averse invesor wih a concave and increasing uiliy funcion. Empirical sudies however proved he converse rue. This sudy does no concern abou various parameric and non-parameric mehods ha allows one o derive he pricing kernel, insead, we would like o menion ha a successful opion pricing model should be able o explain he pricing kernel puzzle. We leave he deailed discussion o be coninued in chaper 5, bu, briefly, Chrisoffersen e al (011) generalized he pricing kernel and showed ha he naural logarirhm of such kernel is no decreseaing bu has a U-shape, and hen developed an opion pricing model which is able o accouns for such U-shaped pricing kernel. 14

21 Chaper ochasic Volailiy Models.1 Overview Of he highly conroversial failures of he Black and choles model (1973), which has aged as long as he model iself, is he volailiy smile problem discussed in he previous chaper. I is ou of he scope of his hesis and raher difficul o give a full accoun of aemps done o address his problem. Various auhors enhanced he lieraure by providing a richer srucure for he price process compare o he geomeric Brownian moion in he Black-choles model. Here we describe and group some major effor and hen narrow down he discussion o sochasic volailiy models and GARCH diffusion models which will be he focal poins of his work in he nex chapers. Much research has been done in providing a more realisic descripion for asse price dynamics in he Black-choles model. The simple geomeric Brownian moion has been evolved in wo aspecs. Firsly, in Meron (1976) a Poisson process accompanied he diffusion process of he price dynamics. Meron s work, which cares abou he jumps in he price process, was highly acknowledged and followed by various works such as Cox and Ross (1976), Jones (1984), Ball 15

22 and Torous (1985), Aase (1988) and Kou (00) among ohers. econdly, he consan volailiy assumpion was relaxed hrough he general class of discree and coninuous sochasic volailiy models such as Hull and Whie (1987), co (1987), Wiggins (1987), Heson (1993), Dupire (1994), Derman and Kani (1994), Duan (1995), Heson and Nandi (000), Chrisoffersen, Heson, and Jacobs (006) and Barone-Adesi, Engle and Mancini (008). The sochasic volailiy jump diffusion model as a combinaion of hese wo ypes of models was firs emerged in Baes (1996). The model was suppored by various sudies hereafer, for insance Bakshi, Cao and Chen (1997) and Andersen, Benzoni and Lund (00). Duffie, Pan and ingleon (000) added jump o he dynamics of he volailiy in he Baes s model. This model was also suppored by a number of sudies, for example Eraker, Johannes and Polson (003). Here we skip disscusing jump diffusion models, bu, using he erminology given in Rebonao (004), clarify which sochasic volailiy models we aimed o sudy. In general, here can be wo disinc sources of sochasic behavior for volailiy. The firs sems from he funcional dependence of volailiy on he underlying sochasic price process. The second is due o he fac ha he volailiy is allowed o be shocked by a second Brownian moion, only imperfecly correlaed, if a all, wih he Brownian moion driving he price process. Models displaying sochasic behavior in he volailiy originaing from boh of hese sources are called fully sochasic volailiy models of which he Heson model is a good example. Models for which volailiy is sochasic only due o he firs source can be described as resriced sochasic volailiy models or he wellknown local volailiy models. The erm resriced migh refer o he fac ha in his models volailiy is fully correlaed wih he sock price which is no he case in realiy. The disincion beween hese wo ypes is imporan, because as we will discuss laer, a risk-neural valuaion canno in general be used o obain a unique opion price for fully sochasic volailiy models 16

23 (while i can be done in he resriced sochasic volailiy case). We exclude he resriced models from our discussion from now on. Firsly, we discuss he general consrucion of he fully sochasic volailiy models (hereinafer V models), and briefly inroduce some popular models such as V model of Heson (1993), and hen GARCH diffusion model of Heson and Nandi (000).. Fully ochasic Volailiy Models By he explanaions we provided in chaper 1, we can idenify wo reasons ha drew he aenion of researchers o ackle he specificaion of consan volailiy in he sock price dynamics of he Black-choles model and invening sochasic dynamics for volailiy. Firsly, he random characer of volailiy which had been well documened by saisical analysis of sock price hisory. econdly, he well-known discrepancy beween Black-choles esimaed European opion prices and marke prices, ha is, he smile curve. Here we discuss he general framework of V models bu leave he pricing issues o be addressed in he nex chaper...1 Consrucion of he Model In V models he asse price saisfies he sochasic differenial equaion d = µ d + σ db (.1) 1 where = ( ) is he volailiy process, obviously posiive and no perfecly correlaed wih he Brownian moion. In fac i has he random componen of is own. Therefore, conrary o he diffusion model in he Black-choles model, here here is a bivariae diffusion model. This 17

24 has implicaion in pricing as i requires he analysis o be done for an incomplee marke. We will come back o his in deails in he nex chaper. We complee he bivariae consrucion by (,, ) (,, ) dy = p Y d + q Y db (.) where, = and [ 1,1] is he correlaion coefficien. I is however convenien o wrie = + 1 where W is he Wiener process and, =0. The correlaion, which is found o be negaive in he empirical sudies, is also someimes called leverage effec... Mean Reversion Propery In mos of he models, volailiy ends o ge back o he mean level of is long run disribuion. More precisely, here is a linear pull-back in he drif of he volailiy process. Following closely he noaion in Fouque e al (000), he drif erm in hese models looks like ( ) where >0 (noe ha <0 indicaes ha he process is mean fleeing) is he speed of mean reversion and is he long-run mean level of. To show his, by assuming he volailiy of volailiy o be zero (i.e. he case of a deerminisic process) we have ( ) ( ) dy = a m Y d Y = m + Y m e 0 a I is clear ha as ime ges large, moves owards. We do no go furher on his and refer ineresed readers o Rebonao (004) chaper 13 or Fouque e al (000) chaper for sronger resuls, and also chaper 4 of he laer for saisical echniques o esimae he rae of mean 18

25 reversion from hisorical asse prices. We end his secion by menioning ha he Ornsein- Uhlenbeck models, see below, are he classic way o describe his propery...3 ome Popular Models The fac ha volailiy is no observable has made i challenging o choose he righ model. I is however common o pick a model which produces posiive volailiy, has he mean reversion propery and provides closed-form formulas for European opions. The following hree models are common models for process and able below, aken from Fouque e al (000), shows ha hey have been frequenly used by researchers for he purpose of modeling volailiy in opion pricing. Lognormal (LN): = + Ornsein-Uhlenbeck (OU): = ( ) + Feller or Cox-Ingersoll-Ross (CIR): = ( ) + This process firs invesigaed by Feller (1951) bu was inroduced in finance for modeling ineres rae by Cox, Ingersoll and Ross (1985). 19

26 Auhor Correlaion ( ) Process Hull-Whie =0 LN co =0 OU ein-ein =0 OU Ball-Roma =0 CIR Heson 0 CIR I is imporan o noe ha boh ein and ein (1991) and Heson (1993) used he OU model bu Heson applied Io s lemma o he OU process and obained he square roo process of Cox- Ingersoll-Ross (1985). Following he purpose of his sudy we discuss he Heson model and GARCH diffusion model in more deails. This also would be a warming up for our discussion on pricing opions hrough hese models which is he opic of he nex wo chapers. Oher models inroduced for modeling he dynamic of volailiy are Consan Elasiciy of Variance Model, ABR volailiy model, Chen model, 3/ model and chönbucher s sochasic implied volailiy model Heson (1993) Model Of he conribuions of Heson s V model compare o previous works is ha i allows for shocks o reurn and volailiy o be negaively correlaed. This is very imporan because i makes he reurn disribuion skewed which is, as we discussed in chaper 1, one of he sylized facs of equiy index reurns. econdly, i provided he opion valuaion under sochasic volailiy wih he firs closed-form soluion for European opions and consequenly caugh he aenion of boh academia and indusry since hen. Assuming σ = Y in (.1), by applying Io s lemma o he square of he process 0

27 = + We ge direcly he following DE for Y namely =( ) +, which can be wrien also as he CIR process = ( ) +. We can herefore wrie he Heson s bivariae diffusion process as follows d = µ d + Y db 1 ( ) dy = κ θ Y d + σ Y db (.3) Where κ is he mean reversion rae, θ is he long run mean and, =. We will come back o his model in chaper 3 where we show how he Heson s opion pricing formula can be derived..3 GARCH Diffusion Model The focus of he discussion in his secion is on he Heson and Nandi (000) GARCH diffusion model. Engle and Musafa (199) and Duan (1995), among ohers, also successfully used GARCH model o price opions bu none of hese works could provide a closed-form soluion for he price of opion. We, herefore, inroduce only he GARCH diffusion model of Heson and Nandi (hereinafer HN) as i is also he main pillar of his sudy. We will come o he pricing problem in chaper 4 and revisi he model in chaper 5 again in order o see how he new pricing kernel is employed in his model. 1

28 The following GARCH(1,1) process governs he dynamics of he spo asse price in he HN s diffusion model over ime seps of lengh ln = ln + r + λ y + y z (.4) ( ) y = ω + β y + α z γ y (.5) where consan is he coninuously compounded ineres rae for any ime inerval of lengh and is a sandard normal disurbance. is he condiional variance of he log reurn beween and, and is known from he informaion se a ime. The srucure of he price process in (.4) has wo imporan implicaions: Firsly, he expeced spo reurn exceeds r by an amoun proporional o he condiional variance [ ] = where =. econdly, he reurn premium per uni of risk is proporional o he level of volailiy = +. I is also imporan o look a he following wo characerisics of he variance process (.5): Firsly, recalling from chaper 1 i is clear ha he equaion for he variance process is slighly differen from he sandard GARCH(1,1) process of Bollerslev (1986) bu similar o he NGARCH and VGARCH models of Engle and Ng (1993).

29 econdly, he parameers and deermine he kurosis and skewness of he disribuion of logreurns, respecively. The parameer makes i possible for shocks o have asymmeric influence. Tha is, a large negaive shock hrough raises he variance more han a large posiive shock. In general, given posiive, posiive value for makes he correlaion beween he variance process and he spo reurn negaive [, ]=. Following Foser and Nelson (1994), Heson and Nandi (000) showed ha heir proposed mean and variance equaions (.4) and (.5) have a coninuous ime diffusion limi as ends o zero =( + ) + = ( ) + where = is he variance per uni of ime and is exacly he square-roo process of Feller (1951), CIR (1985) and Heson (1993). We end his secion by saying ha he mean and variance equaions (.4) and (.5) are he physical processes by which we consruc he risk-neural process of he HN s opion pricing model in chaper 4. 3

30 Chaper 3 Heson s Opion Pricing Model 3.1 Overview ylized facs of financial reurns explained in chaper 1 are an argumen agains he assumpion of consan variance of asse reurns. This problem had been well-documened by researchers, e.g. Mandelbro (1963), before he breakhrough of Black and choles (1973). oon afer ha, much evidence emerged showing ha volailiy smile assers ha marke prices of opions can be obained by employing a reurn disribuion wih faer ails han he normal disribuion. Therefore, he smile problem, in addiion o he exising facs before Black and choles, highlighe he problem of non-normaliy and consan variance assumpion again and cenered he aenion of researchers. In chaper we breifly reviewed effors in he conex of opion pricing o model he smile and inrodcued he mos successful of hem, ha is, sochasic volailiy models. In his chaper we ake one sep furher and clearly discuss ha, having such models in hand, how he sandard Black-choles model was exended o accoun for sochasic volailiy. We do so by focusing on he brillian work of Heson (1993) which exended he Black-choles model o he Heson opion pricing model by providing he firs closed-form 4

31 soluion compare o is preceding works. The (semi) closed-form soluion was due o using characerisic funcion, which is a special case of he Fourier ransform. 3. Theoreical Consideraion In chaper 1 we showed ha how he fundamenal parial differenial equaion in he classical Black-choles model can be derived given he geomeric Brownian moion. We again follow he no-arbirage sraegy and derive he Heson opion pricing formula bu we noe ha here we need o use he bivariae Io s lemma because here are wo sources of randomness in he Heson model. We expec a parial differenial equaion for he pricing funcion which has wo space dimensions of price and volailiy. We also expec he pricing funcion depend on he value of he volailiy which is no observable. This has however implicaion in he hedging argumen as we will see in he following subsecion No Arbirage Argumen and he Pricing Equaion The ask here is o consruc a hedged porfolio of asses ha can be priced by he no-arbirage principle. Unlike he Black-choles case in chaper 1, wo derivaives are required o obain a risk-neural porfolio. This is because boh of he random sources in our diffusion model have o be balanced. Therefore, we consider shor posiion of he call opion C hedged by purchasing α unis of he underlying asse and β unis of he second opion D wrien on he same underlying. C and D however have differen specificaions. Recalling from previous chaper Heson s bivariae diffusion is d = µ d + v db (3.1) 1 5

32 ( ) σ dv = κ θ v d + v db (3.) where 1 d B, B d = ρ. To keep he noaion simple we rewrie i as d = µ d + σ db (3.3) 1 dv = µ d + σ db (3.4) V V Now, we need o modify he argumen presened in chaper 1. Assuming C (, v, ) and D(, v, ) as he prices of wo call opions, he dynamic of C (and similarly D ) can be obained using he bivariae Io s lemma dc C C C 1 C 1 C C = + µ + µ V + σ + σ V + ρσ σv v v v C σ 1 + db + C σv v db d (3.5) Therefore, he value of he resuling porfolio Π = C α βd has he following evoluion d Π = dc αd βdd C C C 1 C 1 C C = V V V C 1 C + σ db + σv db V µ µ V σ σ V ρσ σv 1 { d σ db } α µ + D D D 1 D 1 D D β + µ + µ + σ + σ + ρσ σ v v D 1 D βσ db βσv db v V V V v d d (3.6) This can be rearranged as 6

33 C d Π = + C + C 1 + C 1 + C + C µ µ V σ σ V ρσ σv αµ v v v D D D 1 D 1 D D β + µ + µ V + σ + σ V + ρσ σv d v v v C D C + σ βσ ασ db + σ 1 V v D βσv v d B d (3.7) In order o neuralize he risk due o 1 db and mus be zero. Therefore, we have wo hedge raios db he expressions in he wo las brakes of (3.7) C v β = D v (3.8) C C D v α = D v (3.9) Now, in order o avoid arbirage apporuniy, reurn of he hedged porfolio mus be equal o he reurn on a risk-free invesmen. Tha is d Π = rπ d ( α β ) = r C D d (3.10) Therefore, if we compare he d erms of (3.7) and (3.10) and also subsiue he values of α and β from (3.8) and (3.9), we ge 7

34 C C C 1 C 1 C C C + µ V + r rc + σ + σ V + ρσ σv v v v = v (3.11) D D D 1 D 1 D D D + µ V + r rd + σ + σ V + ρσ σv v v v v As i can be seen in (3.11) here is a symmery in he he above equaion and he wo sides differ only in he ype of he opion. This means ha here is no cross dependencies on eiher sides of he equaion. In fac, we need o have equaion (3.11) valid for call opion of any mauriy and srike price and herefore each side of he equaliy mus be independen from he ype of opion one considers. We can herefore require each side o be equal o some funcion, say λ (, v, ), which does no depend on he mauriy or srike price of opion C C C 1 C v rc + r + µ V + σ 1 C C C + σ + ρσ σ = λ(,, ) V V v v v v (3.1) Replacing he parameers from (3.1) and (3.), he fundamenal parial differenial equaion is C C C rc + r + [ κ ( θ v ) λ(, v, )] v 1 C 1 C C + v + v + v = 0 σ ρσ v v (3.13) x Now by changing he variable x = ln( ) and considering he call opion, C ( e, ), we ge hen C C C = = x x and C = x C. Therefore, (3.13) becomes 8

35 C C C rc + r + [ κ( θ v ) λ( x, v, )] x v 1 C 1 C C + v + v + v = 0 σ ρσ x v x v (3.14) In equaion (3.14) one can see ha boh r and he erm [ κ ( θ v ) λ( x, v, )] are playing he same role for sochasic sock price and sochasic volailiy, respecively. In he Black-choles case, r is he rae a which he sock price mus grow in order ha he opion price grows a he riskless rae. imilarly, [ κ ( θ v ) λ( x, v, )], which is he risk neural drif of he volailiy process, has he same role for volailiy. The ask from now on is o solve he above equaion. The firs hings herefore we need are he boundary condiions for he European call opion wih srike price K and mauriy T C (, v, r, K, T, ) = max( K,0) T T C (0, v, r, K, T, ) = 0 C (, v, r, K, T, ) = 1 (3.15) The firs wo condiions are obvious bu he hird one means ha he changes in he value of he call opion is equivalen o changes in he value of he underlying given he opion is deeply in he money. Recall ha we lef he funcion λ( x, v, ) oally general. We suffice o say ha i can be inerpreed as volailiy risk premium bu devoe he nex secion o ge more insigh on his funcion. 9

36 3.. Marke Price of Volailiy Risk The erm marke price of volailiy risk refers o λ(, v, ) which appeared in he pricing equaion (3.14). Here we explain why we call i marke price of volailiy risk. uppose ha, in he presence of sochasic volailiy, we consruc a porfolio of he opion C which is hedged wih he underlying asse only, Π = C α. Wha does his mean? imply, his means ha of he wo sources of randomness ha mus be hedged only he random source due o he asse price is hedged. Therefore, we expec ha he unhedged risk due o he sochasic volailiy aracs some reward from he marke. Using (3.3)-(3.5) we ge d Π = dc αd C C C 1 C 1 C C = v v v C C + σ db + σ db αd µ µ V σ σ V ρσ σv 1 V v C 1 C 1 C C = C σ σ V ρσ σv v v dv C + + v α d d d (3.16) C ince we are dela hedging ( α = ) he las erm is zero. Therefore C C 1 C d Π rπ d = + r rc + σ + 1 σ C C C V + ρσ σv d + dv v v v (3.17) Using (3.4) and (3.1) we ge 30

37 C C C d Π rπ d = λ (, v, ) d µ d + dv v v v V C [ λ (, v, ) d µ d dv ] = V + v C = λ (, v, ) d µ d + µ d + σ db v V V V C λ(, v, ) = σv d + db v σv (3.18) The erms in he brake in equaion (3.18) shows ha for one uni of risk due o he random source of volailiy we ge λ σv uni of reward for an invesmen in he marke raher han a bank accoun and has why λ is called marke price of risk (noe ha we could have λ raher han λ σ V because we could define λ λ σv = ). Ineresed readers may wan o consul Rebonao (004) page for a discussion on his funcion from a differen perspecive. We now come back o he pricing problem by discussing how he closed-form soluion can be derived Characerisic Funcion and he Closed-form oluion We now have almos everyhing required o price he European call opion. In his secion we closely follow he procedure in Jondeau, Poon and Rockinger (007). We already know ha he price of he European call opion in is general form is r τ θ = 0 T T T C (, v, K, T,, ) e max( K,0) p(, v ) d (3.19) where τ = T and θ is he vecor of parameer, and p( T, v ) is he condiional probabliy densiy funcion of he price process, condiioned on he earlier informaion abou he price and volailiy. Using he change in variable x = ln( ) we ge 31

38 rτ xt C (, v, K, T,, θ ) = e max( e K,0) p( x x, v ) dx T T rτ xt = e e p( x x, v ) dx log K rτ e K p( x x, v ) dx log K T T T T (3.0) Heson hen uses he maringale relaion x rτ xt = e = e e p( x x, v ) dx T T and hen (3.0) urns ou o be e p( x x, v ) C (, v, K, T,, ) dx xt T θ = log K yt e p( y T x, v ) dy T T yt = rτ e K p( x x, v ) dx log K T T (3.1) The firs inegrand in (3.1) is posiive and defines a probabiliy measure ha we denoe by q C (, v, K, T,, θ ) = q ( x x, v ) dx T T log K rτ e K p( x T x, v ) dx T (3.) log K ( log K x, v ) rτ ( T log, ) = P X > 1 T e KP X > K x v now recall from he Fourier inversion heorem of chaper 1 we have 3

39 1 1 exp( iux ) φ( u ) F ( x ) = Re du π iu exp( iux ) φ( u ) P[ X > x ] = 1 F ( x ) = + Re du π iu 0 (3.3) where φ is he characerisic funcion of some densiy. Therefore, we can wrie P1 and P in (3.) as 1 1 exp( iu log K ) φ j ; x,, ( ) Re v u Pj = + du j 1,. π = 0 iu (3.4) The characerisic funcion φ j for j = 1and, is associaed wih he ransiion probabiliies q and p, respecively. The las sep is o idenify he characerisic funcion of hese wo ransiion probabiliies o be able o derive a closed-form soluion for he opion price. To do so, we use he approach of Feynman-Kac which is he classical ool for he case of Black-choles model. Recall ha from previous secion we arrived a equaion (3.14) by changing variable x = ln( ). This parial differenial equaion which is rewrien again below is enough for p C C C rc + r + [ κ( θ v ) λ( x, v, )] x v 1 C 1 C C + v + v + v = 0 σ ρσ x v x v (3.5) bu we do no have ye he parial differenial equaion needed for q. o, o derive i, we pu he soluion (3.) ino (3.5) and hen regroup erms in P1 and P. We hen have 33

40 P P P r + u v + v + v + ( ) ( ) j 1 j j j σρ x x x v P P P a b v + v + = 0 j = 1,. j 1 j j j σ v v (3.6) where, following he assumpion by Heson (1993), he volailiy risk premium is replaced by a linear funcion of variance, ha is λ(, v, ) = λv, and also a = κθ, b = κ + λ σρ, b = κ + λ, u = 0.5, u = Having (3.6) we can use he Feynman-Kac echnique (see he appendix of Heson (1993) for more clarificaion). Here he goal is o obain he characerisic funcion φ j ha saisfies (3.6). Given ha he coefficiens in (3.6) are linear, he characerisic funcion has he form φ j ; x, v, ( u ) = exp[ A j ( τ, u ) + B j ( τ, u ) v + C j ( τ, u ) x ] (3.7) such ha φ u ( iux ) ;,, ( ) exp, and herefore, A (0, u ) = B (0, u ) = 0, C (0, u ) = iu. j x v T T ubsiuing (3.7) and is parial derivaives ino (3.6) we ge j j j 1 1 r + u v C + v C + v B C + a b v B + σ v B A j B j C j = + v + x j = 1,. ( ) ρσ ( ) j j j j j j j j (3.8) which mus hold for for all v and x. Therefore, we ge hree ODEs by seing ( x v ) ( x v ) ( x v ), = (0, 0),, = (1,0),, = (0,1) For he case of C we have C = 0 and by using is boundary condiion w ge C ( τ, u ) = iu. j j o, wha we have o solve is a sysem of wo ODEs, called Ricai equaions 34 j

41 B A j j 1 1 = u + ρσ iub + σ B b B + iu = ab + riu j j j j j (3.9) By solving his sysem, finally, we have he closed-form solion of Heson opion pricing for he European call opion as follow C (, v, K, T,, θ) = P Ke P { rτ 1 exp( iu log K ) φ ( u ) 1 1 j ; x,, Re v Pj = + du j 1, π = 0 iu φ ( u ) = exp[ A ( τ, u ) + B ( τ, u ) v + iux ] j ; x, v, j j d jτ κθ 1 g je A j ( τ, u ) = rui τ + ( b j ρσiu + d j ) τ log, σ 1 g j d jτ b j ρσiu + d j 1 e B j ( τ, u ) =, d jτ σ 1 g je b j ρσ iu + d j g j =, b j ρσiu d j d j = ( ρσ iu b j ) σ ( u jiu u ), u 1 = 0.5, u = 0.5, b1 = κ + λ σρ, b = κ + λ. (3.30) This is he firs closed-form soluion for an opion valuaion model based on coninuous sochasic volailiy model which is very popular in he academia and indusry. An imporan feaure of his model is ha i allows for he correlaion beween he asse spo price and volailiy. 35

42 3.3 Limiaions and furher developmens Despie is populariy, he Heson model has some limiaions. This is mainly due o he unobservabiliy of volailiy. According o Heson and Nandi (000), his makes i impossible o exacly filer a volailiy variable from discree observaions of spo asse prices. Therefore, i is no possible o compue ou-of-sample opions valuaion errors from he hisory of asse reurns. Moreover, Mikhailov and Nogel (003) showed ha he inegrals in he soluion do no always have a convenien convergence behavior. Besides, he model migh need o be exended (such as ime dependen parameers) o beer is performance across large ime inervals of mauriies. I also fails o creae shor erm skew of a magniude observed in he marke and is unable o fi inverse yield curves. In he nex chaper, we discuss he GARCH opion valuaion model of Heson and Nandi (000) which provides a closed-form soluion for European opions and address he main problems due o Heson model as well. Briefly, he variance of spo asse follows a GARCH process and is correlaed wih he asse reurns. The single lag version of his process converges o he Heson sochasic volailiy model as he ime inervals shrink. 36

43 Chaper 4 Heson-Nandi s Opion Pricing Model 4.1 Overview In he previous chaper we described he sochasic volailiy model of Heson (1993) which exended he Black-choles formula o he Heson opion pricing model. In his chaper we sudy he opion pricing framework based on a GARCH diffusion model presened by Heson and Nandi (000). imilar o he former model, his model accouns for he sochasic naure of volailiy as well as he correlaion beween he volailiy and he spo reurns. The model adresses he criicism of he V model of Heson (see secion 3 in he previous chaper and also Heson and Nandi (000) for more deails). Moreover, compared o he exsiing GARCH opion models a ha ime (e.g. Engle and Musafa (199), Amin and Ng (1993) and Daun (1995)) which are simulaion based and compuaionally inensive, his model provides he opion value wih a closed-form soluion. The resuling opion valuaion model differs from he Black-choles and Heson's formula in he sense ha he opion values depend upon he curren and lagged asse prices. Moreover, his model includes he Heson s V model as is coninuous ime limi. 37

44 4. Heson-Nandi GARCH diffusion model In he Heson-Nandi (hencforh HN) model he one period rae of reurn on is assumed o be condiionally log-normally disribued under physical probabiliy measure P R = r + λv + ε, ε = v z, ε F 1 ~ N (0, v ) (4.1) where R ln ( / ) = 1 is he log-reurn beween 1 and, wih v as he condiional variance. F 1 is he informaion se of all informaion up o and including ime, λ is a consan, r is he coninuously compounded riskless ineres rae for a period of lengh one, and z has he sandard normal disribuion. The mean equaion in (4.1) shows ha he average spo reurn depends on he level of risk. Under he normaliy, he condiional expeced rae of reurn equals r + λv, herefore λ can be inerpreed as he uni risk premium. I is also eviden ha he reurn premium per uni of risk is proporional o he volailiy, as in he Cox, Ingersoll and Ross (1985) model. Heson and Nandi also assume ha, under he measure P, ε follows a paricular GARCH(1,1) process ( ) v = ω + βv + α ε v γ v (4.) where ω > 0, α 0, β 0 and γ are consan, and α and γ conrol he kurosis and skewness of he disribuion of he log-reurns, respecively. Following Foser and Nelson (1994), who derived he coninuous ime version of he variance process in he ypical GARCH(1,1) model, one can show ha he variance process in (4.) 38

45 converges weakly o he coninuous ime variance process of Heson (1993) (see Appendix B of Heson and Nandi (000)). The variance process can be expressed in erms of he spo reurns by subsiuing he expression for ε ino he variance equaion (4.) α v = ω + βv + R r λ + γ v (4.3) ( ( ) ) v 1 imilar o he coninuous case, he price process and volailiy process are negaively correlaed ( ) Cov v,ln + 1 = αγv for posiive α and γ. 4.3 The risk neural GARCH process and he closed form formula In order o value an opion we firs need o sae he physical process in erms of risk-neural measure Q. Heson and Nandi propose ha for he processes in (4.1)-(4.), he risk neural * process akes he same GARCH form as before wih λ = 1 / and = + + 1/. This * γ γ λ means ha under measure Q 1 * * R = r v + ε, ε F 1 ~ N(0, v ) ( ) * * * v = ω + βv + α ε v γ v, γ = γ + λ + 1/ (4.4) One can see from (4.4) ha 1 1 F is log-normally disribued under risk neural measure Q. 1 ince RF 1 ~ N( r v, v ), herefore, 39

46 1 1 r v 1 ( 1) + v Q Q R r Ε F = Ε e = e = e F (4.5) 1 We also need o show ha he condiional variances under wo measures are equal, his means ha he variance process in (4.4) mus be equal o P ( 1 ) = ( 1 ) Q Var R F Var R F (4.6) This is desirable because i enables us o observe and esimae he condiional variance under P. To show (4.6) holds, we firs compare he wo physical and risk neural mean equaions and ge ε = ε ( λ + ) v. By subsiuing such ε ino he variance equaion in (4.) we ge * 1 ε v = ω + βv + α γ v v ε ( λ + ) v = ω + βv + α γ v 1 * v 1 = ω + βv ε + α γ + λ + * ( ) v 1 v 1 ε = ω + βv + α γ v 1 * 1 * 1 v 1 (4.7) which proves (4.6). Finally, he value of he call opion is he discouned expeced value of he payoff calculaed using he risk-neural probabiliies. To develop he pricing formula we firs solve for he generaing funcion of he GARCH process. Le f ( u) denoes he condiional generaing funcion of he sock price 40

47 P u f ( u) = f ( u;, T) = Ε T (4.8) which is equivalenly he momen generaing funcion of he ln, i.e. u T P uln T ( ) e f u = Ε For heir GARCH process, Heson and Nandi showed ha he momen generaing funcion akes he log-linear form (see Appendix A in Heson and Nandi (000)) ( ) f ( u) = u exp A( u;, T) + B( u;, T) v + (4.9) 1 where 1 ( α ) A( u;, T) = A( u; + 1, T) + ur + B( u; + 1, T) ω ln 1 B( u; + 1, T) (4.10) and 1 ( u γ ) 1 B( u;, T) = u( λ + γ ) γ + β B( u; + 1, T) + (4.11) 1 αb( u; + 1, T) can be calculaed recursively from he erminal condiion A( u; T, T ) = B( u; T, T ) = 0. ince he generaing funcion of he spo price, f ( u ), is he momen generaing funcion of he logarihm of he spo price, he characerisic funcion of he logarihm of he spo price is simply f ( iu ). Therefore, o use his characerisic funcion we have o replace u by iu in (4.9)-(4.11). The opion pricing formula can be obained by recovering he risk neural probabiliies from he characerisic funcion of he log spo price. We have he closed-form solion of HN opion pricing formula for he European call opion as follow 41

48 [ ] C(, v, K, T ) = e Ε max( K,0) = P Ke P r ( T ) Q r( T ) + 1 T 1 r ( T ) iu * 1 e K f ( iu + 1) P1 = + Re du π iu 0 iu 1 1 K φ( u) * P = + Re f ( iu) du π iu 0 (4.1) where f * ( iu) is he characerisic funcion wih λ and γ in (4.10) and (4.11) replaced by * λ = 1 /, and * γ λ γ = + + 1/, respecively. P1 and P are he risk-neural probabiliies calculaed by inversion of he characerisic funcion f * ( iu) of he logarihm of he spo asse price. The valuaion formula can also be expressed as follow, which makes he compuaion faser r ( T ) Q 1 r ( T ) C(, ) = e Ε [ max( T K,0) ] = ( Ke ) + P1, r ( T ) iu * iu * e K f ( iu + 1) K f ( iu) P1, = Re K Re du π 0 iu iu (4.13) I is imporan o noe ha, conrary o he Black-choles and Heson's V formula, he valuaion formula (4.13) is a funcion of he curren asse price and he condiional variance v which is iself a funcion of he observed pah of he asse price (see (4.3)). Therefore, in conras o he coninuous case (see (3.30)) here is no need for he volailiy o be calculaed by oher mehods. 4.4 Limiaions and furher developmens In chaper we discussed he limiaion of he Heson s V model which addressed by he Heson-Nandi model in he curren chaper. Ye anoher problem o be ackled in opion valuaion models, including he laer model, is due o he pricing kernel which was a funcion of 4