arXiv:1011.1175v1 [q-fin.PR] 4 Nov 2010 o hsmdln lsdfr omlsfrtepropaga- the for formulas form knowledge, closed our no of model best the this To for 29]. exponen- [28, an model follows Vasicek tial stochastic the that assume and extended price is asset process. the model volatility both Heston the in the processes [25] In arbitrary jump with a price. with asset obtained. extended the are is for model options Heston exotic the some [27] are In for diffusions jump prices [11] and in to applied treated example For contribute methods pricing. transform we Fourier Thereby to on added work dynamics. existing been price the has asset process the jump SV to arbitrary that of an propagator where space gen- Fourier model a the of to propagator model SV space Fourier eral the extend to possible long-term the for for especially provides smiles. structure, SV term while the skews of maturities strong calibration short reproduce at to smiles possible and it param- make dependent Jumps time implied eters. the using calibrate without to surface, possible in volatility it jumps makes of SV and combination and [15] the returns Tankov that and motivate Cont [26] example, Gatheral For models. of class and [19–25]. [14–18]; models diffusion process jump Levy SV models on the diffusion based (f) jump models the (e) (d) [11–13]; in- [7–10]; stochastic models and SV rate the terest (c) stochastic [4–6]; the (b) models [3]; (SV) volatility models in- volatility local models fails the the popular (a) improve clude, others, [2] and Among modify Merton Many model. to Black-Scholes and conducted phenomena. been empirical [1] have important studies Scholes some and reflect Black to of theory eeaie rcn omlsfrsohsi oaiiyju volatility stochastic for formulas pricing Generalized sa plcto,w netgt oe hr we where model a investigate we application, an As it makes that method a present will we article this In latter the on focus will we 26] 19–21, [15, by Inspired pricing option pioneering the that known well is It nlss h eut bandi h rsn omls are formalism agr present in the in volatilities, obtained the s results of jump The the distribution given analysis. beyond a lognormal fluctuations with a th non-Gaussian diffusion yields extend include jump and to incorporate model, to importance part p exponential this In jump the of terms. that as diffusion such and so drift models, themselves, usual adap prices the to with option method conjunction a the present We and diffusion. propagators a and equation drift differential Fokker-Planck describe a only of terms in recast be ahitga ehiusfrtepiigo nniloption financial of pricing the for techniques integral Path .INTRODUCTION I. Q,Uiesti nwre,Uiestisli ,21 A 2610 1, Universiteitsplein Antwerpen, Universiteit TQC, Q,Uiesti nwre,Uiestisli ,21 A 2610 1, Universiteitsplein Antwerpen, Universiteit TQC, ya aoaoyo hsc,HradUiest,Cambridg University, Harvard Physics, of Laboratory Lyman h xoeta aie model Vasicek exponential the .Z in n .Lemmens D. and Liang Z. L. Dtd oebr5 2010) 5, November (Dated: .Tempere J. σ which in lc-coe tcatcdffrnileuto (SDE): equation differential stochastic Black-Scholes evld n nlyacnlso sgvni eto V. section in given is ap- conclusion to the a model finally for Vasicek And exponential ranges valid. the parameter be in give In made also simulations. proximation we MC with section option comparisons vanilla this as European well to as devoted pricing, is Vasicek IV exponential Section an follow model. to ap- assumed where an is models present volatility we diffusion the with III, jump model section for a In that propagator proximative price. of of asset propagator propagator the the in the jumps to extending model SV for general method the simulations. present MC model. with checked extended are this results also in last leads formulas these which Also pricing price form asset closed model the this to in of jumps propagator with the the to extended extend model we this method of mentioned propagator we is above approximation simulations the the Using (MC) which exam- valid. for Carlo for ranges Monte parameter see Using specify finance to [32–34]). about applied ple information physics vanilla more for from (for model and methods this propagator of for the prices use for option Making approximative formulas derive yet. form we 31] exist closed 30, price [9, methods option integral vanilla path the or tor ( t easm htteastpieprocess price asset the that assume We hsppri raie sflos nscinI we II section In follows. as organized is paper This I EEA RPGTRFORMULAS PROPAGATOR GENERAL II. sbhvn tcatclyoe ie olwn an following time, over stochastically behaving is ) eetwt eut rmsuperstatistical from results with eement dta,cneunl,ngetjmsand jumps neglect consequently, that, nd hce ihMneCrosimulations. Carlo Monte with checked clr efcso tcatcvolatility stochastic on focus we icular, r otybsdo oesta can that models on based mostly are s r omlsfrbt h path-integral the both for formulas t lc-coe oe,admoreover and model, Black-Scholes stecntn neetrt n h volatility the and rate interest constant the is dS oessaetknit con in account into taken are rocesses rcn omlsadpropagator and formulas pricing e z itiuin hsmdli of is model This distribution. ize ( pdffso oesapidto applied models diffusion mp .AbtayS models SV Arbitrary A. t tepn egu and Belgium ntwerpen, = ) tepn Belgium ntwerpen, rS ,M 02138. MA e, ( t ) dt + σ ( t ) S ( t ) dW 1 S ( ( t t ) olw the follows ) , (1) 2 arbitrary : Given the same arbitrary SV process (2), the new prop- agator of this model, denoted by J (xT , σT ,T x0, σ0, 0), dσ(t)= A(t, σ(t))dt + B(t, σ(t))dW2(t). (2) satisfies the new Kolmogoroff forwardP equation| (see for instance [35]) Here and in the rest of the article Wj = Wj (t),t { ≥ 0 (j = 1, 2) are two correlated Wiener processes such ∂ J ∂ j 1 2 } P = r λm σ J that Cov[ dW1(t) dW2(t)]= ρ dt. ∂T ∂x − − − 2 T P T     Eq.(1) is commonly expressed as a function of the lo- 2 greturn x(t) = ln S(t), which leads to a new SDE: 1 ∂ 2 ∂ + 2 σT J + [ A(T, σT ) J ] 2 ∂xT P ∂σT − P 1 2 2   dx(t)= r σ (t) dt + σ(t)dW1(t). (3) 1 ∂ 2 − 2 + 2 B (T, σT ) J   2 ∂σT P To deal with the pricing problem, we need to solve for ∂2  the propagator of the joint dynamics of x(t) and σ(t). +ρ [σT B(T, σT ) J ] ∂xT ∂σT P The propagator, denoted by (xT , σT ,T x , σ , 0), de- ∞ P | 0 0 + scribes the probability that x has the value xT and σ has +λ [ J (xT J) J (xT )] ̟(J)dJ.(9) −∞ P − −P the value σT at a later time T given the initial values x0 Z and σ0 respectively at time 0. It satisfies the following If we write the propagator of the arbitrary SV model Kolmogoroff forward equation: as a Fourier integral (here and below, i is the imaginary 2 unit) ∂ ∂ 1 2 1 ∂ 2 P = (r σT ) + 2 σT ∂T ∂x − − 2 P 2 ∂x P (xT , σT ,T x0, σ0, 0) T   T P | 2   +∞ ∂ 1 ∂ 2 dp ip(xT −x0) + [ A(T, σT ) ]+ B (T, σT ) = e F (σT , σ0,r,p,T ), (10) ∂σ − P 2 ∂σ2 P −∞ 2π T T Z ∂2   then the propagator of arbitrary SV jump diffusion mod- +ρ [σT B(T, σT ) ] , (4) ∂xT ∂σT P els can be written as with initial condition J (xT , σT ,T x0, σ0, 0) P +∞ | dp ip(xT −x0) U(p,T ) (xT , σT , 0 x0, σ0, 0) = δ(xT x0) δ(σT σ0). (5) = e F (σT , σ0,r,p,T ) e ,(11) P | − − −∞ 2π Z where B. SV jump diffusion models +∞ − U(p,T )= λT e ipJ 1+ ip eJ 1 ̟(J)dJ. −∞ − − A general SV jump diffusion model is obtained by Z adding an arbitrary jump process into the asset price   (12) process (see for instance [20]). That is, equation (1) be- The proof of this statement is given in the Appendix comes . Note the relation between propagators (10) and (11). The only difference between them is the factor eU(p,T ). J dS(t)= S(t)dt + σ(t)S(t)dW1(t)+ e 1 S(t)dN(t), If this is applied to the propagator of the Heston model − (6) [9], the propagator of the Heston model with jumps is where N = N(t),t 0 is an independent Poisson pro- obtained. This propagator is similar as the one derived in cess with intensity{ parameter≥ } λ> 0, i.e. E[ N(t)] = λt. Ref. [25]. Furthermore the above described method can The random variable J with probability density ̟(J) be combined with the method described in Ref. [9] for describes the magnitude of the jump when it occurs. finding the propagator of a model including both SV and Here the risk-neutral drift = r λmj is no longer stochastic interest rate. In particular extending the result the constant interest rate r, rather− it is adjusted by a of Ref. [9] for the Heston model with stochastic interest compensator term λmj , with mj the expectation value rate to include jumps again only involves multiplying the of eJ 1: propagator with eU(p,T ) as in (11). In the next section, as − +∞ an example of the method of this section the volatility of mj = E eJ 1 = (eJ 1)̟(J)dJ, (7) the asset price will be assumed to follow an exponential − −∞ − Z Vasicek model.   so that the asset price process constitutes a martingale under the risk neutral measure. And the logreturn x(t) III. EXPONENTIAL VASICEK SV MODEL follows a new SDE: WITH PRICE JUMPS 1 dx(t)= r λmj σ2(t) dt + σ(t)dW (t)+ JdN(t). − − 2 1 The Heston model assumes that the squared volatility   (8) follows a CIR process which has a gamma distribution 3 as stationary distribution. This assumption should be where the Lagrangians are given by: compared with market data. Attempts to reconstruct the stationary probability distribution of volatility from the 2 1 2z β(¯a−z) γ z data (among others, see Refs. [28, 29, 33]) y˙ + 2 e + ρ γ + 2 e generally agree that the central part of the stationary [y,z] = , (20) L h 2(1  ρ2) e2z  i volatility distribution is better described by a lognormal −2 distribution. [z ˙ β (¯a z)] β [z] = − − . (21) Due to the different structure in path-behavior be- L 2γ2 − 2 tween different models, Schoutens, Simons and Tistaert find that the resulting exotic prices can vary significantly The first step in the evaluation of (19) is the integration [17]. So an investigation into an alternative model which over all y paths. Because the action is quadratic in y, fits market data better is meaningful. this path integration can be done analytically and yields Furthermore the model will serve here both to demon- strate the use of path integral methods in finance and to illustrate the method of section II. LN (yT ,zT y ,z ) P | 0 0 When σ(t) is assumed to be an exponential Vasicek − T L[z]dt 1 = ze 0 process (used for example by Chesney and Scott [36]), D T Z R 2π(1 ρ2) e2zdt this results in the following two SDEs − 0 β(¯a−z) γ 2 y −y + 1 T e2qz dt+ρ T + ezdt − [ T 0 2 0 0 ( γ R 2 ) ] R 2(1−ρ2) RT e2z dt dS = rS dt + σ S dW1, (13) e 0 . (22) 1 × R dσ = σ β [¯a ln σ]+ γ2 dt + γσdW . (14) − 2 2   Note that the probability to arrive in (yT ,zT ) only de- pends on the average value of the volatility along the This model has a lognormal stationary volatility distri- path z(t), in agreement with Ref. [36]. With the help of bution and we will denote it by the LN model, the prop- a Fourier transform, we rewrite the preceding expression agator for this model will be denoted by . In this LN as follows model ln σ(t) is a mean reverting process,P with β the spring constant of the force that attracts the logarithm of asset volatility to its mean reversion levela ¯. Again γ LN (yT ,zT y0,z0) is the volatility of the asset volatility. As far as we know, P +∞ | dp ip(y −y ) − T L[z]dt there is no closed form option pricing formula for this = e T 0 ze 0 −∞ 2π D model. In this section, we will give an approximation Z Z R − 2 2− − − (1 ρ )p ip T e2z dt+ipρ T β(¯a z) + γ ez dt for the propagator of this model. In the next section we e 2 0 0 ( γ 2 ) .(23) will give an approximation for the vanilla option price × R R and determine a parameter range for which the approxi- mation is good. The derivation starts with the following If ζ(t)= z(t) a,¯ then ζ(t) is close to zero because z(t) substitutions: is a mean reverting− process with mean reversion levela ¯, 2 ρ This motivates the approximation eζ 1+ ζ + ζ . This y(t) = x(t) ez(t) rt, (15) ≈ 2 − γ − type of approximation is akin to expanding the path in- z(t) = ln σ(t), (16) tegral around the saddle point up to second order in the fluctuations, as in the Nozieres-Schmitt-Rink formalism where x(t) is defined as before. This leads to two uncor- [37] extended to path-integration by Sa de Melo, Ran- related equations: deria and Engelbrecht [38]. Now we can work out the remaining path integral in (23) 1 β(¯a z) γ dy = e2z ρ − + ez dt − 2 2− − −2 − γ 2 − T L[z]+ (1 ρ )p ip e2z −ipρ β(¯a z) + γ ez dt     0 2 ( γ 2 ) z 2 ze   +e 1 ρ dB1, (17) D R − Z [ζ˙+βζ]2 2 dz = β (¯a z) dt + γdB2, (18) − T − β + A e2ζ +Bβζ eζ − Bγ eζ dt p 0 2γ2 2 2 2 − = ζe   D R where B1 and B2 are two uncorrelated Wiener processes. Z 2 2 γ2M γ2M ω ζ + − ζ + −β ζ2 −ζ2 T ω2 0 ω2 ( T 0 ) Since these equations are uncorrelated, the propagator "    # 2γ2 LN (yT ,zT y0,z0) is given by the following path integral = e P | β−ω−A+Bγ2 γ2M2 + T e 2 2ω2 LN (yT ,zT y0,z0) h i 2 × γ2M γ2M − P | ω ζ + − ζ + e ωT T ω2 0 ω2 T T −      − L[y,z]dt − L[z]dt ω 2 − −2ωT = z ye 0 e 0 , (19) e γ (1 e ) (24), D D × πγ2(1 e−2ωT ) Z Z R  R r − 4 where 4 A = (1 ρ2)p2 ip e2¯a, (25) − − 1 B = ipρ ea¯,  (26) 3.5 γ Bγ2 3 ω = β2 +2γ2 A + Bβ , (27) − 4 s   2.5 Bγ2 M = A + Bβ . (28) − 2 2

We see that also the integral over the final value ζT can Propagator be done, yielding the marginal probability distribution: 1.5

LN (xT x0, ζ0) 1 P +∞ | dp − − ζ0 − ip[xT x0 rT ]+B(e 1) = e 0.5 −∞ 2π Z 2 γ2M βζ2−ω ζ + 0 0 ω2 β−ω−A+Bγ2 γ2M2   + + T 0 e 2γ2 2 2ω2 −1.5 −1 −0.5 0 0.5 1 1.5 × Ξ h i x − x − T 0 e γ2[2ω+(β−ω+Bγ2)(1−e 2ωT )] , (29) − × 1+ 1−e 2ωT [β ω + Bγ2] 2ω − q FIG. 1: Propagator P(xT |x0,ζ0) as a function of xT −x0. The where full curves are our analytical results, while the symbols repre- γ2M sent Monte Carlo simulations. T = 0.25y, ρ = 0 (crosses). T Ξ = ω 2Bγ2N + ω(N )2 (β + Bγ2)N 2 = 1y, ρ = -0.5 (circles). T = 5y, ρ = 0.5 (triangles). For the − ω2 −   other parameters the following values are used for the three 2 4 4 − B γ Bγ M figures: β = 5, a¯ = −1.6,γ = 0.5,r = 0.015. +(1 e 2ωT ) − 2 − ω  (β + Bγ2)γ4M 2 + , (30) 2ω3  2 2 γ M γ M −ωT we find that, for different xT values, the absolute values N = (ζ + )e . (31) − ω2 − 0 ω2 are all in the order of 10 7 or even smaller. In section The goodness of this approximative propagator needs IV B we come back to the discussion concerning the good- the support from MC simulations because of the lack of ness of our approximation. a closed form solution. According to the discussion of Section II, an extension Figure 1 shows the propagators as a function of xT x0, S of this model to the one with price jumps is straightfor- i.e., ln T . The full curves come from expression− (29), S0 while the marked ones are MC simulation results, with ward: the new marginal probability distribution would time to maturity ranging from three months to five years, be: and correlation coefficients 0, 0.5 and 0.5 respectively. − Here and in the rest of the article we will set σ0 equal to the long time average of the volatility: LNJ (xT 0, ζ ) P | 0 γ2 +∞ dp − − ζ0 − σ0 = lim E [σ(t)] = exp a¯ + , (32) ip[xT x0 rT ]+B(e 1) t→∞ 4β = e   −∞ 2π Z 2 γ2M which seems a reasonable choice. For these MC simula- βζ2−ω ζ + 0 0 ω2 β−ω−A+Bγ2 γ2M2   + + T tions 5,000,000 sample paths are used. e 2γ2 2 2ω2 h i It is seen that our analytical results fit the MC simu- × Ξ 2 − 2 − −2ωT lations quite well. Actually, using the parameters of Fig. e γ [2ω+(β ω+Bγ )(1 e )] − 1, and putting expression (29) for those three cases into × 1+ 1−e 2ωT [β ω + Bγ2] the left hand side of the Kolmogorov backward equation: 2ω − +∞ −ipJ − J − qλT −∞ [e 1+ip(e 1)]̟(J)dJ ∂ 1 ∂ 1 ∂2 e , (34) P + r e2(ζ0+¯a) P + e2(ζ0+¯a) P × R − ∂T − 2 ∂x 2 ∂x2   0 0 2 2 ∂ 1 2 ∂ ζ0+¯a ∂ βζ0 P + γ P2 + ρe γ P =0, (33) − ∂ζ0 2 ∂ζ0 ∂x0ζ0 where the same notations as in Eq.(29) are used. 5

IV. EUROPEAN VANILLA OPTION PRICING corresponding U(p,T )’s:

−ipν− 1 δ2p2 UM (p,T ) = λT e 2 1 A. General Pricing Formulas − h ν+ 1 δ2 +ip e 2 1 , (41) If we denote the general marginal propagator by − p+ p−i +∞ UK(p,T ) = λT + 1 1+ ipη+ 1 ipη− − dp ip(xT −x0−rT ) U(p,T ) (xT x0, σ0)= e F (p,T ) e ,  − P | −∞ 2π p+ p− Z +ip( + 1) . (42) (35) 1 η+ 1+ η− − then the option pricing formula of a vanilla call option −  with expiration date T and strike price K is given by theC Using expression (38) and results (41), (42) in formu- discounted expectation value of the payoff: las (36), (37) allows to find the price of the vanilla call option for the exponential Vasicek +∞ with price jumps model. −rT xT = e (e K)+ (xT x0, σ0)dxT C −∞ − P | Z K +∞ ip ln −rT B. Monte Carlo simulations (0) dp e S0 (p) = G + i  G , (36) 2 −∞ 2π p Z To test our analytical pricing formula for the LN model, we focus on the parameters that most strongly where influence the approximation. To satisfy the assumption

U(p+i,T ) that quadratic fluctuations around the mean reversion (p) = S0F (p + i,T ) e levela ¯ captures the behavior of the volatility well, the G − Ke rT F (p,T ) eU(p,T ), (37) mean reversion speed β and the volatility γ of asset − volatility are crucial. The substitution τ = γ2t transforms expression (18) and (x)+ = max (x, 0). Here we have followed the derivation outlined in Ref. into [39]. In particular for the LN model F (p,T ) equals: β dz(τ)= [¯a z(τ)] dτ + dB (τ), (43) γ2 − 2 2 γ2M βζ2−ω ζ + 0 0 ω2 β−ω−A+Bγ2 γ2M2   + + T β F (p,T ) = e 2γ2 2 2ω2 showing that it is actually the parameter c = γ2 which h i ζ0 − Ξ determines whether the approximation will be good. For B(e 1)+ 2 2 −2ωT e γ [2ω+(β−ω+Bγ )(1−e )] bigger c values the approximation z (t) a¯ will be better. .(38) ≈ − −2ωT As the correlation parameter ρ controls the skewness × 1+ 1 e [β ω + Bγ2] 2ω − of spot returns, we will also consider the typical negative q and positive skewed cases by taking values 0.5, 0 and At this stage one needs to specify the PDF for the 0.5 for this parameter. On the other hand, the− constant jump sizes. Merton [13] and Kou [12] proposed a normal interest rate r and the mean reversion levela ¯ do not distributed jump size, denoted by ̟M (J), and a asym- influence the accuracy of the result a lot, and we just metric double exponential distributed one, denoted by assume them to be constant values: r = 0.015 anda ¯ = ̟K (J), respectively: 1.6 ln 0.2. These two parameters seem to be quite −reasonable≈ for the present European options. 1 − (J−ν)2 To get an idea of what is a reasonable range for c, ̟M (J) = e 2δ2 , (39) √2πδ2 and since calibration values for the LN model are not − 1 available, we took calibration values from the literature 1 η J ̟K (J) = p+ e + Θ (J) [4, 40] for the Heston model and fitted our model to the η + volatility distribution of the Heston model with those 1 1 J + p− e η− Θ ( J) . (40) parameters. For [4] we obtained c 7 and for [40] c 18. η− − Therefore in Table I we used values≈ for β and γ≈such that c ranges from 4.08 up to 25. We calculated prices For the Merton model ν is the mean jump size and δ is for S0 = 100 and K = 90, 100 and 110. the standard deviation of the jump size. For Kou’s model The comparison of our analytical solution with the MC 0 < η+ < 1, η− > 0 are means of positive and negative solution for a European call option in the LN model as jumps respectively. p+ and p− represent the probabilities shown in Table I suggests that for the above mentioned of positive and negative jumps, p+ > 0, p− > 0, p++ parameter values the relative errors are less than 3% and p− = 1 and Θ is the Heaviside function. most of the time even less than 1%, which is acceptable According to expression (12), it is easy to derive their when we take the typical bid-ask spread for European 6

30 30 30

20 20 20

10 10 10

Call Option Price 0 Call Option Price 0 Call Option Price 0 80 100 120 80 100 120 80 100 120 Strike Strike Strike 0 0 0

−0.5 −1 −1 −1

−2 −2 −1.5 80 100 120 80 100 120 80 100 120 Relative Deviation (%) Relative Deviation (%) Relative Deivation (%) Strike Strike Strike

FIG. 2: The upper figures show European call option prices in the LN model (left), the LN model with Merton’s jump (middle) and the LN model with Kou’s jump (right). The red curves are our analytical results and the black crosses are the Monte Carlo simulations. The corresponding lower figures give the relative deviations of our analytical results from the MC simulations in the unit of percent. Parameter values S0 = 100, r = 0.015, T = 1, β = 5,a ¯ = −1.6, γ = 0.5, ρ = −0.5, λ = 10, ν = −0.01, δ = 0.03, p+ = 0.3, p− = 0.7, η+ = 0.02, η− = 0.04 are used here. options into account. Here each MC simulation runs plied to the Heston model, leads to similar results as 20,000,000 times. those obtained in Ref. [25], which gives us confidence For the basic LN model we can conclude that we found in the present treatment. The stationary volatility dis- an approximation valid up to 3% for parameter values tribution of the Heston model, however, does not corre- c> 7 (We only checked values of c< 25, but for bigger c spond to the observed lognormal distribution [28, 29, 33] the approximation will only become better), 0.5 <ρ< in the market. The exponential Vasicek model does have − 0.5, T < 1 and 0.9 < K/S0 < 1.1. the lognormal distribution as its stationary distribution. Finally we consider the vanilla call option pricing in Therefore we used this model for the volatility to illus- LN model combined with Merton’s and Kou’s jumps, re- trate the method presented in section II. For this model spectively. Since the jump process is independent from no closed form pricing formulas for the propagator or the approximation we made, we do not investigate the vanilla option prices exist. We first derive approximative goodness of our approximation as thoroughly as in the formulas for the propagator and vanilla option prices for basic LN model (assuming that, if it is good there it will this model without jumps, using path integral methods. be good here). Figure 2 illustrates our analytical results This result was checked with a Monte Carlo simulation, (curves) and the MC simulations (crosses), as well as the proving a parameter range for which the approximation relative errors in the unit of percent. Each MC simula- is valid. We specified a parameter range for which our tion runs 300,000,000 times. These results suggest that pricing formulas are accurate to within 3%. They become β the approximation error is typically less than 2%. And more accurate in the limit γ2 >> 1 where β is the mean due to the fact that whenever the degree of moneyness reversion rate and γ is the volatility of the volatility. Fi- (the ratio of the strike price K to the initial asset price nally we extended this result to the case where the asset S0) is relatively high, the average bid-ask spread tends price evolution contains jumps. to be relatively high for call options [41], our analytical results can serve as an easy way to get a quick estimate that is normally accurate enough for many practical ap- plications. Appendix: Derivation of equations (11), (12).

V. CONCLUSION The proof starts by assuming that a solution for J (xT , σT ,T x , σ , 0) of the form (11) exists. Below P | 0 0 We presented a method which makes it possible to ex- we show that this assumption indeed leads to a so- lution, which in turn justifies the assumption. Since tend the propagator for a general SV model to the propa- ∞ + dp ip(xT −x0) ∂F (σT ,σ0,r,p,T ) gator of that SV model extended with an arbitrary jump −∞ 2π e ∂T equals the right hand process in the asset price evolution. This procedure, ap- side of Eq.(4) and the derivative operators ∂ and ∂ R ∂xT ∂σT 7 have no effect on eU(p,T ), it follows that: the right hand side of Eq.(9) is expressed as

∂ 1 2 +∞ r σT J dp ip(xT −x0) ∂F (σT , σ0,r,p,T ) U(p,T ) ∂xT − − 2 P e e     −∞ 2π ∂T 2 1 ∂ ∂ Z ∞ 2 + dp + 2 σT J + [ A(T, σT ) J ] ip(xT −x0) U(p,T ) 2 ∂xT P ∂σT − P + e F (σT , σ0,r,p,T ) e −∞ 2π 2   Z 1 ∂ 2 +∞ + 2 B (T, σT ) J −ipJ J 2 ∂σT P λ e 1+ ip(e 1) ̟(J)dJ. (A.4) × −∞ − − ∂2  Z +ρ [σT B(T, σT ) J ]   ∂xT ∂σT P This, of course should equal the left hand side of Eq.(9), +∞ dp ip x −x ∂F (σT , σ0,r,p,T ) U p,T which is given by = e ( T 0) e ( (A.1)). −∞ 2π ∂T Z +∞ dp − ∂F (σT , σ0,r,p,T ) j ∂ ip(xT x0) U(p,T ) Adding the term λm J , which is given by e e ∂xT −∞ 2π ∂T P Z ∞ +∞ U(p,T ) + dp − ∂e dp − ip(xT x0) ip(xT x0) U(p,T ) + e F (σT , σ ,r,p,T ) (A.5). λ ipe F (σT , σ0,r,p,T ) e 0 −∞ 2π −∞ 2π ∂T Z Z +∞ (eJ 1)̟(J)dJ, (A.2) Expression (A.4) equals (A.5) when × −∞ − Z ∞ +∞ + ∂U(p,T ) −ipJ J as well as the term λ −∞ [ J (xT J) J (xT )] ̟(J)dJ, = λ e 1+ ip e 1 ̟(J)dJ, P − −P ∂T −∞ − − which is given by Z R   (A.6) +∞ from which the result (12) for U(p,T ) follows. dp ip(xT −x0) U(p,T ) λ e F (σT , σ0,r,p,T ) e −∞ 2π Z +∞ e−ipJ 1 ̟(J)dJ, (A.3) × −∞ − Z 

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TABLE I: Comparison of our approximative analytic pricing result and the MC simulation value for the LN model.

Parameter values Relative error K ρ γ β MC value(a) Approx.(b) (b - a)/a (%) 90 -0.5 1.2 7 15.3947 15.2533 -0.9185 8 15.2979 15.1731 -0.8166 10 15.1630 15.0588 -0.6869 0.8 5 15.1079 14.9995 -0.7180 6 15.0307 14.9337 -0.6475 7 14.9776 14.8855 -0.6151 0 0.7 2 15.2486 15.1982 -0.3307 3 15.0259 15.0024 -0.1564 4 14.9190 14.8992 -0.1328 0.5 1 15.2061 15.1576 -0.3187 2 14.9030 14.8882 -0.0996 3 14.7951 14.7865 -0.0577 0.5 0.3 1 14.6051 14.5035 -0.6953 1.5 14.5524 14.4815 -0.4872 2 14.5354 14.4775 -0.3986 0.2 0.5 14.6015 14.5398 -0.4192 0.75 14.5609 14.5098 -0.3519 1 14.5332 14.4981 -0.2418 100 -0.5 1.2 7 9.4541 9.2862 -1.7762 8 9.3720 9.2197 -1.6253 10 9.2599 9.1274 -1.4310 0.8 5 9.1537 9.0346 -1.3006 6 9.0950 8.9869 -1.1886 7 9.0557 8.9534 -1.1291 0 0.7 2 9.5394 9.4975 -0.4395 3 9.2906 9.2704 -0.2174 4 9.1682 9.1513 -0.1840 0.5 1 9.4915 9.4493 -0.4480 2 9.1445 9.1328 -0.1285 3 9.0235 9.0155 -0.0887 0.5 0.3 1 9.0168 8.9081 -1.2051 1.5 8.9302 8.8522 -0.8737 2 8.8886 8.8246 -0.7195 0.2 0.5 8.9655 8.9023 -0.7048 0.75 8.9039 8.8509 -0.5955 1 8.8628 8.8247 -0.4295 110 -0.5 1.2 7 5.2749 5.1365 -2.6237 8 5.2209 5.0916 -2.4758 10 5.1507 5.0335 -2.2756 0.8 5 5.0170 4.9219 -1.8947 6 4.9877 4.8986 -1.7862 7 4.9709 4.8849 -1.7307 0 0.7 2 5.6480 5.5989 -0.8694 3 5.3942 5.3705 -0.4394 4 5.2684 5.2503 -0.3429 0.5 1 5.5967 5.5510 -0.8173 2 5.2475 5.2343 -0.2519 3 5.1253 5.1160 -0.1821 0.5 0.3 1 5.3151 5.2095 -1.9860 1.5 5.2048 5.1289 -1.4589 2 5.1427 5.0812 -1.1966 0.2 0.5 5.2170 5.1576 -1.1380 0.75 5.1449 5.0954 -0.9620 1 5.0960 5.0595 -0.7163

Other parameter values S0 = 100, r = 0.015,a ¯ = −1.6 and T = 1 are used here.