Sample Level 2 Editing

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1 Sample Level 2 Ediing A Laice Model for Opion Pricing Under GARCH-Jump Processes ABSTRACT This sudy posis a need for an innovaive discree-ime laice model This sudy inegraes he GARCH opion pricing ree of Richken and Trevor (1999) and he jump-diffusion opion pricing ree of Amin (1993) o obain a new discree-ime laice model o value opions in circumsances where when he underlying processing of value opions follows a mixure of a GARCH and a disconinuous jump process. In his regard, Richken and Trevor s (1999) Garch opion pricing ree demonsrably proves complemenary o he funcion of Amin s (1993) jump diffusion pricing ree. Moreover, his The assumpion of a GARCH-jump model provides a beer descripion for he underlying process given is consisency wih he empirical resuls,; in which he disconinuiies in he sample pah of financial asses are found even afer allowing for condiional heeroskedasiciy. The laice model can also adap o he Duan, Richken, and Sun s (2006) GARCH-jump model in pricing American- syle opions,. and Furhermore, he Amin (1993), he Richken and Trevor (1999), he Cox, Ross, and Rubinsein (1979), and along wih he Kamrad and Richken s (1991) models, are nesed o inerwoven wih our generalized laice model. Numerical resuls of our model Our numerical resuls are consisen wih he resuls of he Mone Carlo simulaions for pricing European opions under he GARCH-jump process. For Wih respec o American opions, our resuls illusrae daa indicaes ha he early exerciseearly-exercise premium decreases wih he increases of he jump inensiy, or he mean of he jump magniude. In addiion, he resuls for he implied volailiy show ha he generalized model can capure he volailiy smile as well as and he erm srucure of volailiy effecs observed in opions markes. Keywords: GARCH opion model, jump-diffusion opion model, GARCH-jump opion model, laice model

2 A Laice Model for Opion Pricing Under GARCH-Jump Processes I. INTRODUCTION The sochasic process naure of sock prices is a crucial issue of pivoal imporance for opion valuaions. Convenional assumpions have reaed his diffusion process as coninuoussock prices are convenionally assumed o follow a diffusion process wih coninuous pah. For example, The seminal Black-Scholes (1973) opion pricing model assumes ha sock prices follow he Geomeric Brownian Moion (GBM), in which he condiional disribuion of sock (coninuously compounded) reurns is normal. Such conclusions are qualified, hough, by However, empirical evidence, which indicaes ha biases, such as he high level of kurosis and non-zero skewness found in sock reurn disribuions, are embedded in his assumpioncanno be discouned as influencing facors. As a resul, Consequenly, observable marke opions prices observed in he marke, relaive o he Black-Scholes model, generally exhibi end o manifes in cerain paerns, such as he volailiy smile and he erm srucure of volailiy effecs. One of an imporan and A widely adoped remedy oward his issue is holds ha he condiional variance in he underlying process is sochasic or ime varying, where he asse (uncondiional) reurns disribuion displays in skewness as well as in addiion o lepokurosis. Along In his vein, he GARCH (Generalized Auoregressive Condiional Heeroskedasiciy) -ype of processes are discree-ime models able o capure he phenomenon of sochasic changes in volailiy over ime. Under he GARCH framework, Duan (1995) applies he concep of locally risk-neural valuaion relaionship o develop an opion pricingopion-pricing model, hereby and solvesing he opion price by large sample simulaion mehods. The complicaion of opion pricing under he GARCH process is largely aribuable mainly due o is pah-dependen characerisics,. which cause When pricing American-syle opions, he resuling non-recombined effec leads o boh an 1

3 exponenial increase in he nodes of he laice model and heir associaed emporal sepshus he number of nodes in he laice model grows exponenially along wih he increase of ime seps when pricing American-syle opions. Realizing Cognizan of his complexiy, Richken and Trevor (1999) developed an efficien laice algorihm o price, in paricular American syleamerican-syle opions, under he discree-ime GARCH process. Furher, Cakici and Tapyan (2000) provide a more efficien mehod by modifying he modified Richken and Trevor s (1999) model, and hey poin ou hus highlighing a faser ha he convergence rae in combinaion wih a more efficien is faser and he compuaional ime is more efficien under heir modified mehod. Alhough Aracive as he GARCH processes may beare aracive, however, hey are no unable o accoun for describe he occasional and large discree any changes embedded in asse price behaviors. An alernaive model, ha can capable of capureing he fa-ail effec and skewness in he reurn disribuion caused by he occasionally large movemens in asse price, is he jump-diffusion process. Meron (1976) proposed a jump-diffusion opion pricing model, where a Poisson jump process is added o he Geomeric Brownian Moion for he underlying asse price. The jump-diffusion model is based on he assumpion ha sock reurns are generaed by a mixure of processes, which includesing small, coninuous, incremenal flucuaions o prices movemens generaed by a Wiener process, and large, infrequen price jumps peraining o nonsysemaic risk,. generaed by a Poisson processa Poisson process generaes he laer. In paricular, The jump process is used o specifically describe he abnormal price variaions vibraions in price due o he arrival of imporan new informaion peraining o abou he financial marke ha urn ou has more han a marginal effec on price. The model can provide an explanaion for empirically observedable disribuions of sock price changes ha exhibi skewness, lepokurosis, and faer ails in boh sides, relaive o he lognormal assumpion in he Black-Scholes model. Besides Meron (1976), Cox and Ross (1976), and Ahn and Thompson (1988) 2

4 also derive opion pricing models by assuming disconinuous jump processes for he price of underlying asses. Their models are similar comparable o Meron s (1976), insofar as hey ha can provide a reasonable explanaion for underesimaion of he observaions ha prices of ou-of-money and close o mauriy opions are underesimaed. They also faciliae comparable evaluaions of and he effeciveness of shor-erm hedging sraegies based on he dynamic porfolio adjusmen, suggesing hey are is overesimaed when associaed wih convenional pricing models.. The Meron s (1976) model is a quasi-closed form model for pricing European opions. In order To price a wider range of opions, Amin (1993) developsed a discree-ime laice model, which alhough derived is rigorously exended from he Cox, Ross, and Rubinsein s (1979) binomial ree, also marks an exension of is analyical capaciy. Amin focuses on wha happens when he underlying asse follows he Meron s jump-diffusion process. He assumes ha he sock price can move up or down by one ick in each discree period, as was posulaed by as he Cox, Ross, and Rubinsein s (1979) model (in each discree period, [where a ick is he minimum possible change in he sock price)]. However, he asse price can also change on accoun for of a rare even (jump). Therefore, he also permis he underlying asse price o change by muliple icks in a single period. This muliple-ick jump is he discree ime counerpar of he coninuous ime jump componen. Based on he risk-neural valuaion argumen and he assumpion ha he jump risk is diversifiable, his model is demonsraed o be able o weakly converges wih o he heoreical opion values under some mild regulariy condiions. Recenly, Chang and Fu (2001) invesigaed he opion pricing on raded asses, in seings where when eiher he underlying asse follows a jump diffusion process, or he volailiy of he underlying asse is assumed o be sochasic. They exended he lieraure by combining he ransformaion echnique of Hilliard and Schwarz (1996) and he discree-ime jump-diffusion model of Amin (1993), yielding o develop a bivariae binomial ree model. Furhermore, Kou (2002) offers a double exponenial 3

5 jump-diffusion model for he purpose of opion pricing. He assumes he asse price follows a Brownian moion plus a compound Poisson process, wih jump sizes regisering as a doubled exponenial increase in disribuionbeing double exponenial disribued. Under he assumpion, he model can explain abou he asymmeric lepokuric feaure and he volailiy smile effec in opions markes. Moreover, i can lead o analyic soluions o many opion-pricing problems, including plain vanilla opions, sandard ineres rae derivaives, and some pah-dependen opions. Commen [PEH1]: CHECK: Unclear. Should i read somehing insead such as, he working assumpion of his model is ha i can explain he asymmeric ec? Among he sochasic volailiy models, he GARCH-ype of model is one of he mos comprehensive and popular widely adoped models used o capure smooh persisen changes in volailiy. However, i is no able o explaining he large discree changes embedded in asse reurns. A racable credible alernaive is o incorporae a jump process ino a GARCH-ype model, and indeed, researchers have explored is possibiliies. Jorion (1988) combined an ARCH model wih a jump componen o empirically examine foreign exchange raes and sock reurns. SimilarlyLikewise, Vlaar and Palm (1993), and and Nieuwland, Vershchoor, and Wolff (1994) adoped a consan jump inensiy-garch model o capure foreign exchange rae dynamics. Commen [PEH2]: CHECK: his is he second ime I believe his erm has appeared. Do large and discree need o be disinguished in his conex, as logically somehing ha is large is less discree han a smaller eniy? Perhaps use of and/or as a linkage poin migh clarify he inended meaning? Furhermore, Lin and Yeh (2000) modify he Jorion s (1988) model o derive a new jump-diffusion-garch model, and provide empirical ess on he Taiwan sock marke o examine wheher disconinuous ime pahs exis or no. In heir empirical sudies, boh Jorion (1988) and and Lin and Yeh (2000), all found ha he combined models could provide a beer explanaion for he behavior of financial asse prices. Recenly, Duan, Richken, and Sun (2004) esed he GARCH-JumpGARCH-jump model using he S&P 500 daa as a research sample. Their research and shows ha he inclusion of jumps significanly improves he fi of hisorical ime series of for he S&P 500 daa. In opion pricing, Duan, Richken, and Sun (2006) propose a new GARCH-jump model which akes he correlaed sysemaic jump ino accoun and solves opion prices by simulaion approaches. However, o he bes of our knowledge, here is currenly no a model available ha can value American syleamerican-syle Commen [PEH3]: CHECK: They were able o solve opion prices by simulaion approaches because hey had already aken he correlaed sysemaic jump ino accoun? If no, his senence would be beer served by breaking down ino separae, clear componens. 4

6 opions. Such a model would need o be capable of capuring he dependency of hose opions ha depend on an underlying asse, wih respec o boh a under a GARCH process wih and a jump componen included is sill lacking. The purpose of his paper is o develop a discree-ime opion pricing model which allows he underlying sock prices o follow a mixure of GARCH process and jump process. To his end we combine he laice algorihms of GARCH model of Richken and Trevor s (1999) GARCH model and he discree-ime ree of jump-diffusion model of Amin (1993). o derive aan inegraed GARCH-jump opion pricing model is hus aained., The new model which provides us an efficien discree-ime laice framework o price, in paricular, American syleamerican-syle opions. I is also demonsraed ha he model can provide more greaer degrees of freedom o explain he skew feaure of he sock reurns disribuion and capure he volailiy smile and erm srucure of volailiy effecs on he opions marke. A he same ime, he new GARCH-jump model, which conainings several nesed models as degeneraed cases, provides us wih an efficien ool o conduc empirical ess on opions pricing. The remainingder of his paper is organized as follows: iin Secion II, we consruc a generalized laice model under he GARCH-jump process and discuss is adapion and degeneraion o he nesed models. In Secion III, we derive opion pricing procedures under from he GARCH-jump laice model. In Secion IV, we proceed produce numerical ansalyses for he GARCH-jump laice model in pricing opions, and hen consider is convergence behavior owards he nesed models. Secion V is he conclusion of his paper. II. THE LATTICE MODEL WITH GARCH AND JUMPS 1. General Framework Suppose ha he price of he sock a ime, under he risk-neural measure, 5

7 wih he ime incremen, follows he following generalized GARCH-jump process: S = m Δ + h Δ X, (1) S +Δ ln where h m = m( h, Δ ) = rf λ( Δ)[ K 1], 2 N ( Δ) ( j) = Z + J j= 1 X, Z ~ N (0,1), ( ) J = J h, Δ ~ N ( μ ( h, Δ), σ ( h, Δ )), ( j) ( j) 2 J J ( j) (, ) ( exp( )) K = K h Δ = E hδ J, and N ( Δ ) is a Poisson process wih he jump inensiy of λ ( Δ) risk-neural measure. Δ under he In he above specificaions, r f is he risk-free ineres rae, and m and h denoe he drif rae and he variance of he sock price process a ime. X is a compound Poisson normal process, which is a mixure of a sandard normal process Z and a Poisson jump process J, wih he jump inensiy of λ( ), and he jump magniude following a normal disribuion, whose mean and sandard deviaion, μ J ( J (h, ) and σ J ( J (h, ), are assumed o be generally dependen on h and. K denoes he average rae of jump plus 1. The variance process of he sock price reurns is assumed o follow he generalized process: where ( ν,, ) h h = f h Δ, (2) ν +Δ +Δ +Δ (ln S+Δ ln S mδ) / hδ E ( X) =, Var ( X ) 6

8 is he sandardized innovaion of he sock price reurn process and E ( X ) = λ( Δ) Δμ ( h, Δ), J Var h h 2 2 ( X ) = 1 + λ( Δ ) Δ ( μj(, Δ ) + σj(, Δ )), are he meanis he mean and variance of he compound Poisson normal process, under he risk-neural measure. If he NGARCH process is considered, he variance process becomes: X,; 2 ( ν,, ) β ( β 1) β (, ) ( (, )) f h Δ = Δ + hδ + h Δ hδ v c h Δ. (3) +Δ Δ The variance srucure imposed in Equaion (3) is a more general nonlinear asymmeric GARCH (NGARCH) model han hose used by Engle and Ng (1993) and Duan (1995). The nonnegaivenon-negaive parameer c ( h, Δ ) signifies a negaive correlaion beween he innovaions of he sock price reurn and is condiional volailiy under he risk-neural measure. To follow he NGARCH process, he parameers have some resricions wih a ypical GARCH process, which include β > 0 0, β 0 1 β (, ) 0 2 h Δ o ensure he posiive condiional volailiy., and Based on he Amin (1993) and Richken and Trevor s (1999) seing, given y = ln( S ), he logarihmic sock price process of y +Δ, wih an incremen of ime Δ, can be approximaed in a laice space as y = y + jγ ; j = 0, ± 1, ± 2,..., (4) +Δ n where γ n is he size of changes in he sock price reurn o be defined laer, and j denoes he index of he number of possible icks changed in he sock price reurn for he GARCH-jump process. In he laice model, we assume ha he sock price changes can be driven by a local componen and a jump componen, where he local means he variaion of he sock price follows he assumpion of a diffusion process, and he jump means ha he sock price can change o an arbirary level, eiher wihin or beyond he local change 7

9 levels. According o he assumpion in Amin (1993), he jump risk is diversifiable and is no priced in he marke, and he local and jump changes are muually exclusive 1. In oour model makes allowance for he simulaneous occurrence of boh kinds of price flucuaion, we furher permi boh of he wo ypes of price changes can occur Commen [PEH4]: CHECK: why? Because i is no priced in he marke? Is i he frequency of risk ha is diverse or he kinds of risks ha are hemselves diverse? simulaneously. 1 Amin (1993) permis he wo price changes are o be muually exclusive for he exposiional convenience. In he coninuous ime limi, i is irrelevan wheher hey are muually exclusive or no. 8

10 Table 7. Early ExerciseEarly-exercise Premium for Duan, Richken, and Sun s (2006) Model This able shows he early exerciseearly-exercise premium for Duan, Richken, and Sun s (2006) model. Parameers for he example are he same as in Table 5, and he number of variances for each node is M=50. Assume furher ha he risk-free ineres rae r f = 0.05, he iniial sock price S 0 = 500, he srike price X = 500, and he ime incremen Δ isare se o one day. We calculae pu opionpu-opion prices for boh European (Panel A) and American (Panel B) syles. As we can be seen in Panel C of he able, he early exerciseearly-exercise premiums are generally small for he parameers used. Panel A: European Pu OpionPu-opion Value Srike Time o Mauriy Price Panel B: American Pu OpionPu-opion Value Srike Time o Mauriy Price Panel C: Early exerciseearly-exercise raio =(American Pu- European Pu)/American Pu Srike Time o Mauriy Price % 0.02% 0.03% 0.04% 0.05% 0.06% 0.08% % 0.02% 0.03% 0.04% 0.04% 0.06% 0.09% % 0.01% 0.02% 0.03% 0.04% 0.06% 0.09% % 0.01% 0.01% 0.02% 0.03% 0.06% 0.09% % 0.01% 0.01% 0.02% 0.03% 0.06% 0.09% 9

11 Table 8. Sensiiviy Analysis for Early ExerciseEarly-exercise Premium This able shows he sensiiviy analysis for opion early exerciseearly-exercise premium wih he GARCH-jump model, wih respec o jump inensiy and magniude parameers. According o Amin (1993), significan jumps obsacle impede he early exercise of American opions. This reduces he early exerciseearly-exercise premium and produces insignifican differences beween American opions and European opions. As shown in his able, we can find he proporion of early exerciseearly-exercise premium increases wih he decreases in he jump inensiy. Similarly, he proporion of early exerciseearly-exercise premium increases wih he decreases in he jump magniude parameer. Parameers used are he same as hose in Table 3, a399nd M=50. All numbers are in percenageile value of he early exerciseearly-exercise value relaive o he American opion value. Panel A: Time o Mauriy T=10 days Mean of Jump inensiy λ Jump magniude μ J Panel B: Time o Mauriy T=50 days Mean of Jump inensiy λ Jump magniude μ J