A JUMP-DIFFUSION WITH STOCHASTIC VOLATILITY AND INTEREST RATE

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1 Journal of ahemaics and Saisics 9 (): 43-5, 3 ISSN Science Publicaions doi:.3844/jmssp Published Online 9 () 3 (hp:// A JUP-DIFFUSION WIH SOCHASIC VOLAILIY AND INERES RAE Paiboon Peeraparp and Pairoe Saayaham Insiue of Science, School of ahemaics, Suranaree Uniersiy of echnology, Narkorn Rachasima, hailand Receied 3--6, Reised 3--7; Acceped ABSRAC In his sudy, we presen he applicaion of ime Changed Ley mehod o model a jump-diffusion process wih sochasic olailiy and sochasic ineres rae. We apply he Lewis Fourier ransform mehod as well as he risk neural expecaion pricing mehod o derie a formula for a European opion pricing. hese combining mehods gie quie a shor roue o derie he formula and make i efficien o compue opion prices. We also show he calibraion of our model o he real marke wih global and local opimizaion algorihms. Keywords: ime Changed Ley Process, Calibraion, Sochasic Ineres Rae, Sochasic Volailiy, Jump- Diffusion, Black and Scholes (BS). INRODUCION he success of Black and Scholes (BS) model ress on he ease of compuaion and raceabiliy: i has a closed form soluion and allows for dynamic hedging. Howeer, he BS model fails o explain many aspecs of he real disribuion of an asse reurn. Since hen, here hae been coninued effors o make a beer model o describe a model in he real world saring from eron s jump-diffusion, Heson s sochasic olailiy and Baes sochasic olailiy wih a jump-diffusion. While he search for beer models for dynamics of asse prices coninues, he praciioner s communiy usually focuses more on he uilizabiliy of he models o be jus as imporan as how he models can describe he dynamic of an asse in he real world. he issues are on he compuaion, calibraion and raceabiliy of he model. In his sudy, we will apply he ime Changed Ley mehod o model a jump-diffusion wih sochasic olailiy and sochasic ineres rae which is quie differen from he ypical mehod ha describes he dynamic of he model by separaed componens of he howeer creaes he sochasic olailiy by changing he ime of a pure Ley process o a random sochasic ime. In he compuaion par, he Lewis Fourier ransform mehod was used o calculae he opion prices on he complex plane which works seamlessly wih he ime Changed Ley mehod where he measure changed is defined in he complex domain. he opion formula by he Lewis () mehod comes ou in a single Fourier inegral form which helps o reduce he compuaion ime compared wih he oher mehods which normally generae wo inegrals. We also calibrae he model o he real marke prices using boh global and local search mehods o find he minimum of he discrepancies beween he marke prices and model prices o obain he opimal parameers of he model. he resul is repored in he las secion... he odel... ypical odel he ypical risk neural model for jump-diffusion wih sochasic olailiy and sochasic ineres rae can sochasic facors. he ime Changed Ley mehod, be described by Equaion : Corresponding Auhor: Paiboon Peeraparp, Insiue of Science, School of ahemaics, Suranaree Uniersiy of echnology, Narkorn Rachasima, hailand Science Publicaions 43 JSS

2 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 ds = (r λ k y)d + dw + (Y )dn, S wih ln Y disribued as Normal( µ, ), J J E[Y ] = exp( µ + ) k, J J y and d = k( θ )d + σ dw, wih dw dw = ρd, dr = α( ω r )d + β r dw r Here W, W and Science Publicaions () r W are he Brownian moions associaed o he underlying asse process, he ariance process and he ineres rae process respeciely. he process S is he underlying asse price process and r is he insananeous risk free rae process. he process N i is a Poisson process wih jump frequency λ and independen from he oher processes. he jump size Y - is disribued as described aboe. he jump is included in he model o make shor erm implied olailiy cure seep as indicaed by he empirical sudies. he process is he ariance process, k is he speed of mean reersion, θ is he mean of long erm ariance and σ is he olailiy of he ariance process. he ariance process is he square roo process known as CIR process. o explain he leerage effec, he negaie correlaion is usually inroduced beween he underlying asse and he ariance as shown aboe. he ineres rae process is also a CIR process bu wih differen parameers and independen from he oher processes... he ime Changed Ley Approach We derie he Fourier opion pricing using he Lewis mehod. here is a ariey of Fourier ransform Pricing mehods bu we choose his mehod as is inegraion domain is on he complex plane. his complex domain will correspond o he domain for he ime changed Ley process. Anoher nice feaure of his mehod is ha i produces a formula in a single inegraion form compared wih he ypical approaches which produce wo inegraions such as he approach in Saayaham and Inarasi (). his single inegral reduces compuaion ime of opion prices in he calibraion process. During our calculaion, we also apply he odular approach, pioneered by Zhu (), which employs he rule of independence of characerisic funcions o wrie he characerisic funcion as produc of each characerisic funcion of an independen sochasic facor. his approach will help us o handle each sochasic facor independenly which resuls in he reducion in he dimensions of he calculaion. 44 Our dynamic of sock price will be an exponenial Ley process which is drien as Equaion : S = S exp( r ds + L ), = exp(x + r ds + L ) s s () Here S is he sock price a ime, S = exp(x ) is he price of sock a ime =, r is a risk free rae process and L is a Ley process wih exp(l ) being a maringale. Le us assume he characerisic funcion of L, φ (z) = E [exp(izl )] is well defined for α<im(z)<β L where α and β are real numbers and z is a complex number. he Fourier ransform of he payoff for a European call opion srike a K wih a payoff funcion a he mauriy, max(s -K) or (S -K) + can be compued as: + H(z) = exp(izx)(exp(x) K) dx, = exp(izx)(exp(x) K)dx, ln K exp(iz + )x exp(izx) = ( K ), iz + iz iz+ K =, Im(z)> z iz Here exp (X ) = S or ln K (3) X = X + r d + L. In Equaion 3, Ĥ(z) is defined in he region where he imaginary par of Fourier ransform ariable z, is greaer han. he corresponding generalized inerse Fourier ransform H(x) for he payoff funcion is defined below: izi + izx H(x) ˆ π e H(z)dz izi = (4) In Equaion 3 and Equaion 4, we exend he ransform ariable z o ake a alue in he complex domain ha is defined in he generalized Fourier ransform sense. Gien, Ĥ(z) is well defined on he plane where he imaginary par of z is greaer han, he inegraion in Equaion 4 is jus he line inegraion on a complex plane paralleled o he real axis wih any Im(z)>. From he Fundamenal heorem of Asse Pricing, he no arbirage condiion is equialen o he exisence of a risk neural measure where a discouned asse price is a maringale. Based on his Fundamenal heorem, we can wrie he alue of European call opion a = as risk neural expecaion of he discouned payoff as: JSS

3 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 + V = E [exp( r d)(s K) ], x + E [exp( r d)(e K) ], = izi+ izi = E [ exp( r d) exp( iz(x + r d + L ))H(z)dz], ˆ π izi+ L izi = E [exp( (iz + ) r d))]exp( izx ) φ ( z)h(z)dz ˆ π Science Publicaions (5) he expecaion E is he expecaion under a risk neural measure. he hird line is deried from he second line by replacing he payoff funcion wih he corresponding generalized Fourier ransform in Equaion 4. In Equaion 5, here we suppose he ineres rae process is independen from he oher processes, herefore we can wrie he expecaion ou from he oher erms. Here we will apply he ime Changed Ley mehod o generae a model wih sochasic olailiy and sochasic ineres rae as in Equaion. ha is our model is drien by: N S = S exp( r ds + X + ln y λk ), s k y = s = θ + σ ds wih d k( )d dw (6) where, r, y, λ k y and are defined as in Equaion and X is he ime changed Ley process which will be defined laer. Compared wih Equaion, he Ley par is Equaion 7: N L = X + ln Y λk (7) k y = X + J. he erm N ln Yk λ k y = J is a compensaed compound Poisson process. his ime changed Ley process X is consruced by wo sochasic processes, a subordinaor and an underlying process. he subordinaor, as an increasing process, is a funcion of calendar ime o sochasic ime. he underlying process is normally a pure Ley process. For our case, he subordinaor is now defined o be a process Equaion 8: = ds (8) s where, s is defined as in Equaion. he underlying Ley process is he risk neural Brownian moion wih drif rae r Equaion 9: 45 ds = S (r d + dw ) (9) whose he log reurn can be described by: lns / S = r + W = r + X () As menioned preiously, he ime changed Ley process X can be generaed by subsiuing for in Equaion. So X has he form Equaion : X = W () By assumpion of he independence beween he ime changed Ley process and he jump process, we may wrie he characerisic funcion of Ley process φ ( z) L from Equaion 5 as he produc of he characerisic funcion of he ime changed Ley process and he characerisic funcion of he compensaed compound jump process Equaion : φ ( z) = φ ( z) φ ( z) () L X J We denoe φ (z) and φ (z) J as he characerisic X funcions of he ime changed Ley process and of he compensaed Poison process respeciely. So we now need o calculae each componen of Equaion 5 ha are E [exp( (iz + ) rd)], φ (z) and X φ J (z). he characerisic of he ime changed Ley process is deried as: φ = X iz(w ) (z) E [e ], iz(w ) +ψ (z) ψ (z) X X = E [e ], = [ ψ ] E exp( X(z) ), = E [exp( ψx(z)sds] (3) From he second line o he hird line aboe, we apply he measure change defined as he complex alue Radon- Nikodym deriaie (he deail of his measure can be found in chaper 8 of Zhu () where Equaion 4: d () = exp(izx + ψ X (z)) (4) d JSS

4 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 and ψ (z) X denoes he characerisic exponen of he underlying process X. his measure allows us o wrie he characerisic funcion of he correlaed process (in our case, he underlying process is designed o correlae wih he ariance) as he Laplace ransform under he new measure. ha is Equaion 5: izx E [e ] ψ E [e (z) ] X = (5) Here he characerisic exponen of he Ley process ψ x (z) of he process ( W ) is (iz + z ). o calculae in Equaion 6, we need o find he dynamic of under his new measure by he Girsano heorem (heorem..4 in Zhu ()). he dynamic of is: d = (kθ k )d + σ dw, wih k k iz = ρσ Science Publicaions (6) he ariable W is he Brownian moion associaed o he ariance process under his new measure. Here is he deriaion of he aboe equaion. Based on Girsano heorem, gien a measurable space (Ω,F,), an Io process for he dynamic of : d = k( θ )d + σ dw Denoe as a exponenial maringale under measure defined by: wih E [ ] =. d = () = exp(izx + ψ (z)ds) s X d Wih X = W = sdw sds and ψ X(z) = (iz + z ). hen can be expressed as: d () = exp(iz( sdws sds) s( (iz z )ds), d + + s s sz ds), = exp( iz dw = γ + γ γ exp( dws ds) wih =iz 46 By he assumpion, he underlying process is correlaed wih he ariance process or dw dw = ρ d. hen we hae he following resuls: defines he Radon Nikodym deriaie. ha is: d = () d he new Brownian moion W is defined by: Subsiuing we hae: under he measure dw = dw γ dw dw = dw iz ρ d dw in he hird equaion of Equaion 6, d = k( θ )d + izρσ d + σ dw, = θ + σ k k iz (k k )d dw, wih = ρσ hen we can sole for he characerisic of he ime changed Ley process as: φ = exp( C() D() ), X kθ ge σ g d wih C() = (k + d) ln[ ], d k + d e D() =, d σ ge d= (k ) + ψ σ, g = X (k + d) (k d) Here is he deriaion of. X (7) φ From Equaion 3, according o he Feynman-Kac heorem, he characerisic of X : φx (z) = E [exp( ψ(z)sds] will saisfy he following Parial Differenial Equaion (PDE): JSS

5 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 φ (,) φ (,) φ (,) = k ( θ ) + σ ψ (z) φ(x,), wih φ (, = ) = X gien ha d = k ( θ )d + σ dw he equaion for D() is called Riccai equaion which is a nonlinear ordinary differenial equaion. o make i simple we will wrie he equaion for D() as: D() = + P D() RD () where, P = ψ X, = k and R = σ. u ' he soluion for D() will be D() = where u R u will saisfy he following auxiliary differenial equaion: P' u '' + [ + ]u ' + PR = P he general soluion for he aboe equaion is: herefore: α β u() = Ae + Be, where A and B are cons an, P' P' [ + ] + [ + ] 4PR wih α = P P, P' P' and β = P P [ ] [ ] 4PR α β Aα e + Bβe α β Ae + Be R() D() =. Replace P,,R and we hae: + α = and β = k d k d where d= (k ) + ψ σ Wih D() =, we can sole for: X hen C() can be soled from D() and equaion. he compuaion of sochasic ineres rae par is similar o he calculaion of characerisic of ime changed Ley par. We can wrie Equaion 8: E [exp( (iz + ) r d)] = exp(g()r + H()), d α + d e wih G() = ( ), β d ge d αω ge H() = ( d) ln( ), α + β g α + d d = α + (iz + ) β and g= α d (8) he las par is he characerisic of he compensaed compound Poison process which is compued below Equaion 9: N J k Y φ (z) = E [exp(iz( ln Y λk )], N exp(e [(iz ln Y k )] iz k Y), = λ = exp( izλ (exp(u J + J) ) + ( λ(exp(izu J z j) )).3. Calibraion (9) Calibraion is he process o obain a model s parameers ha mach o he marke prices of he opions. his model s parameers generally will be differen from parameers esimaed by he saisical mehods. he saisical parameers reflec he pas dynamic of he underlying asse and no guaranee ha he models buil ou of his parameers are arbirage free. Conrary o he saisical mehod, he parameers from he calibraion are arried wih he principles of no arbirage a leas for he raded opions ha included in he calibraion process. he differences of hese wo mehods reflec in he inesors risk preferences, hedging coss and iews of he paricipans in he marke which is no be able o be capured by a saisical mehod. Gien we can obsere he prices of he opion from he marke, he equaion for he alue of he opion based on he risk-neural aluaion is Equaion : d k + d j e (k + d) D() =,wih g d = σ ge (k d) p + V ( θ,,k) = E [exp( r d)(s K) ] () Science Publicaions 47 JSS

6 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 Here θ p is he se of he model parameers, and K are he ime o mauriy and srike of he obsered opion respeciely. he dynamic of S is described as a parameric model under he risk neural measure. If we can obain he prices of opions a any ime for all K, we could deermine he parameers of he dynamic of S by he aboe equaion. Bu his is impossible in he real marke where we hae limied prices of opion in any mauriy, one possible way is o minimize he discrepancies beween he aailable marke prices and model prices generaed from a parameric model. herefore he problem of calibraion has ransformed o an opimizaion problem for he leas square of he discrepancies. he scenario is ha he marke prices consiss of he price of European call opions spanning a se of expiraion daes,, N and for each i, he marke quoes for srike K i,,k i. he leas square mehod o find he minimum of differen of prices and he parameers of he model can be described by Equaion : in F( θ ) p N = arg min w [C ( θ,,k ) C(,K )] θ i = j = Science Publicaions ij p i ij i ij () he funcion F(θ) is he objecie funcion wih parameer θ p. he funcion C (θ p, i,k ij ) and C( i,k ij ) are he alue of he call opion generaed by parameers θ p and he obsered price a mauriy i and srike K ij respeciely. he ariable w ij is he weigh associaed o he confidence of he obsered price which aries wih he ega of opion. he deail of weighing can be found in chaper 4. of Poklewski-Koziell (). his leas square problem will hae a leas a soluion gien he domains of parameers are compac. o find he aboe minimum problem, we employ he simulaed annealing mehod which is one of he mos efficien mehod o find a global opimum. In his par, we will calibrae he models o he DAX index opion prices on July 5, as shown in able. Based on Equaion 7-9, our model will hae parameers, ha are, k, θ, σ, ρ, λ, u J, J, r,α, ω and β. We run he calibraion algorihm in ALAB opimizaion oolbox which proides boh a simulaed annealing algorihm and gradien based opimizaion algorihms. We compare our model wih he Jump- Diffusion wih Sochasic Volailiy model (JDSV)..4. Calibraing of he odel he simulaed annealing mehod is he global opimizaion mehod ha replicaes he way he meal is 48 heaed o a suiable emperaure and cooled down slowly o ge he opimum srucure. he suiable emperaure is called he iniial emperaure and he way he emperaure is reduced is called he annealing schedule. he algorihm is summarized below. Sep : Iniializaion Se he iniial soluion x and he iniial emperaure which is high enough for he accepance probabiliy of.8 o.95. Sep : Perurbaion Generae he new soluion x i according o he designed probabiliy disribuion and deermine he difference of objecie funcion f = f(x i )-f(x i- ). Sep 3: Accepance Deerminaion he new poin is esed o accep if i falls on he following crierior: If f < or r < exp(-k f/) where, r is a uniformly disribued random number beween [,]. he erm exp(-k f/) is referred o as he accepance probabiliy where K is normally equals. Repea sep and 3 unil he equilibrium is reached (no much improemen for he objecie funcion in his emperaure). Sep 4: Annealing he emperaure is slowly reduced wih a specific schedule o zero. Repea sep o sep 4 unil he objecie funcion reach a specified goal. he simulaed annealing mehod differs from he gradien based mehods in ha i can aoid rapping in he local minimum by allowing he candidae poin o be acceped een he objecie funcion is worse han he exising objecie funcion as shown in sep. he acceping probabiliy depends on he emperaure which is high when he emperaure is high and low in when he emperaure is a low leel..5. Resuls he simulaed annealing ool, in he ALAB s opimizaion oolbox, proides a lo of opions ha caer o naures of he problem including he emperaure seing, annealing schedule and sopping crierior. We JSS

7 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 hae run he opimizaion he numbers of imes and se he iniial emperaure a 4. For he random search, he Bolzmann generaing funcion seems o be he bes choice o obain he minimum poin o our problem. We compare our model (JDSVSI) wih he Jump Diffusion wih Sochasic Volailiy (JDSV) wih has 8 parameers, ha are, k, θ, σ, ρ, λ, u J and J. he daa poins ha are he in-sample on his calibraion are all he daa in able excep for he columns of monh and 8 monhs ha we use as he ou-of-sample poins. Due o he naure of simulaed annealing mehod, he search for he candidae poins being random can locae only he neighborhood of he minimum. We need he gradien based mehod or call here local search o fine une he minimum poin. he minimum of objecie funcions of JDSV and JDSVSI are jus approximaely. and. imes higher compared o he BS s one percen error in implied olailiy as shown in able. hese errors are consisen wih he bid/offer spread in he real marke which is jus abou a lile higher han percen. he errors are due o he facors ha we do no recognize and do no include in he models, he daa error and he landscape of he domain of he problem. he parameers of he calibraion of boh models are shown in able 3 and 4. able. Implied Volailiy Surface of DAX Index Opions on Jul. 5, a Spo of Expiry Jul. Aug. Sep. Dec. ar. 3 Jun. 3 Dec. 3 Jun. 4 ime weeks monh 3 monhs 6 monhs 9 monhs monhs 8 monhs 4 monhs Dae 9 Jul. 6 Aug. Sep. Dec. ar. 3 Jun. 3 9 Dec. 3 8 Jun. 4 enor () Ineres rae (r) Srike able. inimum Objecie Funcion Comparison Obj. funcion Percen oer BS BS wih % implied error Jump wih SV Jump wih SV and SI able 3. JDSV Parameers k θ σ ρ λ u J J Annealing Local search able 4. JDSVSI parameers k θ σ ρ λ u J J r α ω β Annealing Local search able 5. Errors of opion implied olailiy for JDSV Srike weeks monh 3 monhs 6 monhs 9 monhs monhs 8 monhs 4 monhs Σ(error) %Σ(error) 36.6E-5 9.6E-6 9.6E-5.5E-7 4.E-6.9E-5.E-5.9E-5.7E E-4.6E-5.3E-5.5E-7 7.9E-6.96E E-6 4.E-8 5.7E E-4.5E-6.3E-5.5E-6.37E-5 8.E-7.89E-6.76E-5.66E E-5.E E-5.69E-6 5.9E-6 9.E-8 8.4E-6 5.9E-6.5E E-4 6.7E-5 4.E-5 6.5E-6.37E-5 3.6E-7.94E-5.5E-7.6E E E-5.E+.37E-5 9.E-6.E E-5.44E-5 3.8E Σ(error) E-3 %Σ(error) Science Publicaions 49 JSS

8 Paiboon Peeraparp and Pairoe Saayaham /Journal of ahemaics and Saisics 9 (): 43-5, 3 able 6. Errors of opion implied olailiy for JDSVSI Srike weeks monh 3 monhs 6 monhs 9 monhs monhs 8 monhs 4 monhs Σ(error) %Σ(error) 36.53E-4 8.4E E-5 6.4E-7 4.9E-7.5E-7.89E-6.E-8 3.E E E-5 6.5E-6.6E-7.6E-5 4.4E-6 3.3E-5 4.9E-7 7.5E E-5.89E-6 7.9E-6.6E-5 8.4E-6.5E-6.E-5.5E-5.E E-4 3.E-5 5.8E-5.6E-5 7.9E-6.69E-6.3E-5 4.4E-6.6E E-5.5E E-5.44E-5.94E-5 4.9E-7.9E-5 3.6E-7.9E E-4.99E-4.E-5.44E-5.E-5 6.4E E-5.6E-5 6.7E Σ(error) E-3 %Σ(error) he able 5 and 6 show he square discrepancies of boh models implied olailiy wih he corresponding BS implied olailiy. he second las line presens he sum of he square error and he las line is he error adjused by he oal of square error for all columns. As expeced, he JDSV is beer for he shor enor bu he JDSVSI is beer in he longer enor in accordance wih he finding from Abudy and Izhakian (). he oal errors of he ou-of -sample daa poins ha shows on he las line of he column of monh and l8 monhs of boh ables are wihin he aerage.5 excep for he error for monh of JDSVSI model.. CONCLUSION he combining mehods ha we implemen gie a shor roue and sraighforward opion pricing formula compared wih he exising ones which normally are deried by riskless porfolio parial differenial equaions or high dimensional risk neural expecaion mehod. From he calibraion resuls, our model is beer han JDSV for he longer enors bu no for he shor one. 3. REFERENCES Abudy,. and Y. Izhakian,. Pricing sock opions wih sochasic ineres rae. Science Elecronic Publishing, Inc. Lewis, A.L.,. A simple opion formula for general jump-diffusion and oher exponenial ley processes. Science Elecronic Publishing, Inc. Poklewski-Koziell, W.,. Sochasic olailiy models: Calibraion, pricing and hedging. WIS Library. Saayaham, P. and A. Inarasi,. An approximae formula of European opion for fracional sochasic olailiy jump-diffusion model. J. ah. Sai., 7: DOI:.3844/jmssp Zhu, J.,. Applicaions of Fourier ransform o Smile odeling. nd Edn., Springer, Berlin Heidelberg, ISBN-: , pp: 33. Science Publicaions 5 JSS