Variance-GGC asset price models and their sensitivity analysis

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1 Variance-GGC asset price models their sensitivity analysis Nicolas Privault Dichuan Yang Abstract his paper reviews the variance-gamma asset price model as well as its symmetric non-symmetric extensions based on generalized gamma convolutions GGC. In particular we compute the basic characteristics decomposition of the variance-ggc model, we consider its sensitivity analysis based on the approach of [8. Key words: variance-gamma model, variance-ggc model, sensitivity analysis. Mathematics Subject Classification: 6E7; 6G51; 6J65; 6G52; 62P5; 91B28. 1 Introduction Lévy processes play an important role in the modeling of risky asset prices with jumps. In addition to the Black-Scholes model based on geometric Brownian motion, pure jump jump-diffusion processes have been used by Cox Ross [5 Merton [13 for the modeling of asset prices. More recently, Brownian motions time-changed by non-decreasing Lévy processes i.e. subordinators have become popular, in particular the Normal Inverse Gaussian NIG model [1, the variancegamma VG model [12 [11, the CGMY/KoBol models [4, [3. he normal inverse Gaussian NIG process [1 can constructed as a Brownian motion time-changed by a Lévy process with the inverse Gaussian distribution, Nicolas Privault Division of Mathematical Sciences, School of Physical Mathematical Sciences, Nanyang echnological University, 21 Nanyang Link, Singapore nprivault@ntu.edu.sg Dichuan Yang Department of Mathematics, City University of Hong Kong, at Chee Avenue, Kowloon ong, Hong Kong, dyang5@student.cityu.edu.hk his research was supported by Singapore MOE ier 2 grant MOE ARC 3/13. 1

2 2 Nicolas Privault Dichuan Yang whose marginal at time t is identical in law to the first hitting time of the positive level t by a drifted Brownian motion. he variance-gamma process [12, [11 is built on the time change of a Brownian motion by a gamma process, has been successful in modeling asset prices with jumps in addressing the issue of slowly decreasing probability tails found in real market data. he CGMY/KoBol models [4, [3 are extensions of the variance-gamma model by a more flexible choice of Lévy measures. However, this extension loses some nice properties of variance-gamma model, for example variance-gamma processes can be decomposed into the difference of two gamma processes, whereas this property does not hold in general in the CGMY/KoBol models. In [6 the variance-gamma model has been extended into a symmetric variance- GGC model, based on generalized gamma convolutions GGCs, see [2 for details a driftless Brownian motion. In this paper we review this model propose an extension to non-symmetric case using a drifted Brownian motion. GGC rom variables can be constructed by limits in distribution of sums of independent gamma rom variables with varying shape parameters. As a result, the variance-ggc model allows for more flexibility than stard variancegamma models, while retaining some of their properties. he skewness kurtosis of variance-ggc processes can be computed in closed form, including the relations between skewness kurtosis of the GGC process of the corresponding variance-ggc process. In addition, variance-ggc processes can be represented as the difference of two GGC processes. On the other h, the sensitivity analysis of stochastic models is an important topic in financial engineering applications. he sensitivity analysis of time-changed Brownian motion processes has been developed the Greek formulas have been obtained by following the approach in [8. In addition, the sentivity analysis of the variance-gamma, stable tempered stable processes has been performed in [9 [1 respectively. As an extension of the variance-gamma process, we study the corresponding sensitivity analysis of the variance-ggc model along the lines of [9. In the remaining of this section we review some facts on generalized gamma convolutions, GGCs including their variance, skewness kurtosis. We also discuss an asset price model based on GGCs its sensitivity analysis. Wiener-gamma integrals Consider a gamma process γ t t R+, i.e. γ t t R+ is a process with independent stationary increments such that γ t at time t > has a gamma distribution with shape

3 Variance-GGC asset price models their sensitivity analysis 3 parameter t probability density function e x x t 1 /Γ t, x >. We denote by gtdγ t, 1 the Wiener-gamma stochastic integral of a deterministic function g : R + R + with respect to the stard gamma process γ t t R+, provided g satisfies the condition log1 + gtdt <, 2 which ensures the finiteness of 1, cf. 1.2, page 35 of [7 for details. In particular, there is a one-to-one correspondence between GGC rom variables Wienergamma integrals, Proposition 1.1, page 352 of [7. Generalized gamma convolutions A rom variable Z is a generalized gamma convolution if its Laplace transform admits the representation E[e uz = exp t log 1 + u µds, u s where µds is called the horin measure should satisfy the conditions,1 logs µds < 1, s 1 µds <. Generalized gamma convolutions GGC can be defined as the limits of independent sums of gamma rom variables with various shape parameters, cf. [2 for details. In particular, the density of the Lévy measure of a GGC rom variable is a completely monotone function. From the Laplace transform of Z we find E[Z = t 1 µdt, the first central moments of Z can be computed as

4 4 Nicolas Privault Dichuan Yang E[Z E[Z 2 = t 2 µdt, E[Z E[Z 3 = 2 t 3 µdt, 3 E[Z E[Z 4 = 3Var[Z t 4 µdt. As a consequence we can compute the Skewness[Z = E[Z E[Z3 Var[Z 3/2 = 2 t 3 µdt Var[Z 3/2, Kurtosis[Z = E[Z E[Z4 Var[Z 2 = t 4 µdt Var[Z 2 of Z. We refer the reader to Proposition 1.1 of [7 for the relation between the integr in a Wiener-gamma representation the cumulative distribution function of the associated generalized gamma convolution. Market model sensitivity analysis As an extension of the model of [9 to GGC rom variables we consider an asset price process S defined by the exponent S = S exp θ gsdγ s + τ Θ + Z + cθ,τ, of a variance-ggc process, i.e. gsdγ s is a GGC rom variable represented as a Wiener-gamma integral, Θ is an independent Gaussian rom variable, Z t t R+ is another GGC-Lévy process, θ R, τ, >. In section 3 the sensitivity E[ΦS of an option with payoff Φ with respect S to the initial value S in a variance-ggc model is shown to satisfy where L := S E[ΦS = 1 S E[ΦS L, 2θ gs f 2 sdγ s θ gs f sdγ s + τ η + 2 f sdγ s f sds + ηθ. θ gs f sdγ s + τ η

5 Variance-GGC asset price models their sensitivity analysis 5 for any positive function f : R +,a η >. In heorem 1 we will compute this sensitivity as well as orther Greeks based on the model parameters θ τ. he remaining of this paper is organized as follows. In section 2 we introduce a model for Brownian motion time-changed by a GGC subordinator. he variance, skewness kurtosis of variance-ggc processes are calculated in relation with the corresponding parameters of GGC processes, several example of variance-ggc models are considered. A Girsanov transform of GGC processes is also stated. he sensitivity analysis with respect to S, θ τ is conducted in section 3. 2 Variance-GGC processes Given W t t R+ a stard Brownian motion θ R, σ >, consider the drifted Brownian motion Bt θ,σ := θt + σw t, t R +. Next, consider a generalized gamma convolution GGC Lévy process G t t R+ such that G 1 is a GGC rom variable with horin measure µds on R +. We define the variance-ggc process Yt σ,θ t R+ as the time-changed Brownian motion Y σ,θ t := B θ,σ G t, t R +. he probability density function of Yt σ,θ is given by f σ,θ Y x = 1 t σ x θy 2 exp 2π 2σ 2 h t y dy, x R, y y where h t y is the probability density function of G t, cf. Relation 6 in [11. he Laplace transform of Yt σ,θ [ E exp uy σ,θ t = is e uy f Yt ydy 4 = Ψ Gt θu σ 2 = exp t where Ψ Gt is the Laplace transform of G t. 2 u2 log 1 + θu σ 2 u 2 /2 s µds, his construction extends the symmetric variance-ggc model constructed in Section 4.4, page of [6. In particular, the next proposition extends to variance-ggc processes Relation 8 in [11, [12, which decomposes the variance-

6 6 Nicolas Privault Dichuan Yang gamma process into the difference of two gamma processes. Here, we are writing Y t as the difference of two independent GGC processes, i.e. Y t becomes an Extended Generalized Gamma Convolution EGGC in the sense of Chapter 7 of [2, cf. also 3 of [14. Proposition 1. he time-changed process Y t can be decomposed as Y t = U t W t, where U t W t are two independent GGC processes with horin measures µ A µ B which are the image measures of µdt on R + respectively, by the mappings s Bs := θ σ θ 2 σ σ 2 + 2s, s R +, s As = θ σ θ 2 σ σ 2 + 2s, s R +. Proof. From 4, the Laplace tranform of Y t can be decomposed as [ E exp uyt σ,θ = exp t log 1 u 1 + u µds Bs As = exp t log 1 + u µds t log 1 u µds As Bs = exp t log 1 + u µ A ds t log 1 u µ B ds s s = E[e uu t E[e uw t. uy σ,θ t he Laplace tranform of Y t can also be decomposed as [ E exp = exp t = exp t log 1 + u s log 1 + u µ A ds t s µ B ds t log 1 + u s log 1 u µ B ds s, 5 µ A ds where µ B is the image measure of µ B by s s, in particular, Y t is an extended GGC EGGC with horin measure µ A + µ B in the sense of Chapter 7 of [2. In the next proposition we compute the variance, skewness kurtosis of variance-ggc processes. Proposition 2. We have i Var[Y 1 = θ 2 Var[G 1 + σ 2 E[G 1,

7 Variance-GGC asset price models their sensitivity analysis 7 ii Skewness[Y 1 = E[G 1 E[G σ/θ 2 Var[G 1 2Var[G 1 + σ/θ 2 E[G 1 3/2 = θ 3 2 Skewness[G 1 Var[G 1 3/2 Var[Y 1 3/2 θσ2 Var[G 1, 6 Var[Y 1 3/2 iii Kurtosis[Y 1 = 3 + 3θ 4 E[G 1 E[G 1 4 3Var[G 1 2 8θ 2 Var[G 1 + σ 2 E[G θ 2 σ 2 E[G 1 E[G 1 3 /4 + σ 4 Var[G 1 θ 2 Var[G 1 + σ 2 E[G = 3 + θ 4 Kurtosis[G 1 3Var[G Var[Y σ 2 θ 2 Skewness[G 1Var[G 1 3/2 4Var[Y σ 4 Var[G 1 Var[Y 1 2. Proof. Using the horin measure µ A + µ B of Y t 3 we have Var[Y 1 = = = t 2 µ A dt + t 2 µ B dt 1 A 2 t µdt + θ 2 +tσ 2 µdt t 2 = θ 2 Var[G 1 + σ 2 E[G 1, E[Y 1 E[Y 1 3 = 2 t 3 µ A dt + 2 E[Y 1 E[Y 1 4 = 6 = t 2 µ dt + = B 2 t µdt t 3 µ B dt θ 3 + θσ 2 θ 2 /σ 2 + 2t µdt = θ 3 2 E[G 1 E[G θσ 2 Var[G 1, t 3 t 4 µ A dt + 6 t 4 µ B dt t 2 µ + dt 2 θ 4 + θσ 2 4θ 2 /σ 2 + 8t/2 2 + σ 4 4θ 2 /σ 2 + 8t 4 /2 t 4 µdt

8 8 Nicolas Privault Dichuan Yang = θ 2 +tσ 2 t 2 2 µdt 3θ 4 + 6σ 2 θ 2 t + 4σ 4 t 2 t 4 µdt + 3 = 3 8 θ 4 E[G 1 E[G 1 4 3Var[G 1 2 θ 2 +tσ 2 t 2 2 µdt θ 2 σ 2 E[G 1 E[G σ 4 Var[G 1 + 3θ 2 Var[G 1 + σ 2 E[G 1 2, this yields 6 7. Girsanov theorem Consider the probability measure Q λ defined by the Radon-Nikodym density dq λ dp := eλy E[e λy = 1 λa e λy = e λy +a log1 λ, λ < 1, 8 cf. e.g. Lemma 2.1 of [9, where Y is a gamma rom variable with shape scale parameters a,1 under P. hen, under Q λ, the rom variable Y t has a gamma distribution with parameter a,1/1 λ, i.e. the distribution of Y t /1 λ under P. In the next proposition we extend this Girsanov transformation to GGC rom variables. Proposition 3. Consider the probability measure P f defined by its Radon-Nikodym derivative dp f dp = e f sdγ s E[e f sdγ s = e f sdγ s + log1 f sds, where f : R +,1 satisfies log 1 + f t dt <. 9 1 f t Assume that g : R + R + satisfies 2, log1 + ugs f sds >, u >. hen, under P f, the law of gsdγ s is the GGC distribution of the Wiener-gamma integral gs 1 f s dγ s

9 Variance-GGC asset price models their sensitivity analysis 9 under P. Proof. For all u >, we have E Pf [exp u gsdγ s [ = E exp u gsdγ s + f sdγ s + log1 f sds [ = E exp f s ugsdγ s exp log1 f sds = exp log1 + ugs f sds exp log1 f sds = exp log 1 + ugs ds 1 f s [ gs = E exp u 1 f s dγ s. Note that 8 is recovered by taking gs = 1 [,a s f s = λ1 [,a s for λ,1, i.e. G = gsdγ s is a gamma rom variable with shape parameter a we have E Pf [e ug = 1 + u a = E[ exp u 1 λ 1 λ G, u >, λ < 1. Next we consider several examples particular cases. Gamma case In case the horin measure µ is given by µdt = γδ c dt, where δ c is the Dirac measure at c > we find the variance-gamma model of [12. Here, G t, t >, has the gamma probability density φ t x = c γt xγt 1 e cx, x R +, Γ γt with mean variance γt/c γt/c 2, G t becomes a gamma rom variable with parameters γt,c. In this case, the decomposition in Proposition 1 reads Ψ Yt u = 1 σ 2 u 2 tγ = 1 σu tγ 1 + σu tγ, 2c 2c 2c

10 1 Nicolas Privault Dichuan Yang we have µ A dt = µ B dt = γδ 2c/σ dt, thus U t t R+, W t t R+ become independent gamma processes with parameter γt, 2c/σ. he mean variance of U 1 are E[U 1 = t 1 µ A dt = σγ 2c Var[U 1 = E[U 1 E[U 1 2 = t 2 µ A dt = γσ 2 2c. Symmetric case When θ = we recover the symmetric variance-ggc process Y t := B σ G t, t R +, defined in Section 4.4, page of [6, i.e. the time-changed Brownian motion is a symmetric variance-ggc process. Here, Y t is a centered Gaussian rom variable with variance σ 2 G t given G t, where Bt σ is a stard Brownian motion with variance σ 2. he Laplace transform of Y t in Proposition 1 shows that Y t decomposes into two independent processes with same GGC increments since µ A µ B are the same image measures of µdt on R +, by s 2s/σ. Variance-stable processes Let G t t R+ be a Lévy stable process with index parameter α,1 moment generating function hs = e sα. In this section we consider a non-symmetric extension of the symmetric variance stable process considered in Section 4.5, pages of [6. he horin measure of the stable distribution is given by µdt = ϕtdt = α π sinαπtα 1 dt, cf. page 35 of [2. By Proposition 1, Y t can be decomposed as Y t = U t W t, where U t W t are processes with independent stable increments horin measures

11 Variance-GGC asset price models their sensitivity analysis 11 µ A dt = ϕ A tdt = α 1 π sinαπσ 2 t + θ 2 σt θ/σ2 θ 2 α 1 2σ 2 dt, µ B dt = ϕ B tdt = α 1 π sinαπσ 2 t θ 2 σt θ/σ2 θ 2 α 1 2σ 2 dt. In the symmetric case θ = we find σ µ A dt = ϕ A tdt = µ B dt = ϕ B tdt = σ 2 2 t 2 tϕ dt = α sinαπ 2 2 α 1 π σ 2α t 2α 1 dt, i.e. 2U t /σ 2W t /σ are stable processes of index 2α. Note that the skewness kurtosis of G t Y t are undefined. Figure 1 presents a simulation of the variance-stable process Fig. 1 Sample paths of variance-stable process with α =.99. Variance product of stable processes Here we take G 1 = Z 1/α X α where Z is a Γ γ,1 rom variable X α is a stable rom variable with index α < 1. he MGF of G 1 is hs = 1 + s α γ, cf. page 38 of [2, i.e. G 1 is a GGC with horin measure µdt = ϕtdt = 1 π γαt α 1 sinαπ 1 +t 2α + 2t α cosαπ dt,

12 12 Nicolas Privault Dichuan Yang Y t decomposes as Y t = U t W t, where U t W t are processes of independent product of stable increment horin measures µ A dt = ϕ A tdt = 1 π γασt + θ/σ 2 /2 θ 2 /2σ 2 α 1 sinαπσ 2 t + θ 1 + σt + θ/σ 2 /2 θ 2 /2σ 2 2α + 2σt + θ/σ 2 /2 θ 2 /2σ 2 α cosαπ dt, µ B dt = ϕ B tdt = 1 π γασt θ/σ 2 /2 θ 2 /2σ 2 α 1 sinαπσ 2 t θ 1 + σt θ/σ 2 /2 θ 2 /2σ 2 2α + 2σt θ/σ 2 /2 θ 2 /2σ 2 α cosαπ dt. In the symmetric case µ A dt = ϕ A tdt = µ B dt = ϕ B tdt σ = σ 2 2 t 2 γασ 2α t 2α 1 sinαπ tϕ dt = 2 π2 α α 1 σ 4α t 4α + σ 2α t 2α cosαπ dt. he skewness kurtosis of G t Y t are undefined. Figure 2 presents the corresponding simulation Fig. 2 Sample paths of variance-product of stable process with α =.99 γ =.2.

13 Variance-GGC asset price models their sensitivity analysis 13 3 Sensitivity analysis In this section we extend approach of [8 to the sensitivity analysis of variance-ggc models. Consider B t t R+ a stard one-dimensional stard Brownian motion independent of the Lévy process Y t t [, generated by Y := gsdγ s. Let Θ be a stard Gaussian rom variable independent of Y t t [,. For each t [,, we denote by F t the filtration generated by Θ σy s : s [,t. Let Z t t R+ be a real-valued stochastic process in R independent of Y t t R+ B t t R+. Finally we denote by let Cb nr +;R denote the class of n-time continuously differentiable functions with bounded derivatives, whereas C c R + ;R denotes the space of continuous functions with compact support. Given θ R τ R + we consider the asset price S written as S = S exp θy + τ Θ + Z + cθ,τ, where the function gs : R + R + verifies 2. Remark 1. When θ = the above model reduces to the stard Black-Scholes model, in case θ we find the variance-ggc model by taking Z t t [, to be a GGC process. For example, we can take the Wiener-gamma integral gsdγ s to be a stable rom variable set Z to be another stable rom variable, then the exponent of S t is a variance-stable process. his example will be developed in the next section. he next theorem deals with the sensitivity analysis of the variance-ggc model with respect to S, θ τ, is the main result in this section. Define the classes of functions C L R + ;R := { f CR + ;R : f x C1 + x for some C > }, { DR + ;R := f : R + R : f = n k=1 c k f k 1 Ak, n 1, c k R, f k C L R + ;R, A k intervals of R + }. heorem 1. Let Φ DR + ;R. Assume that the law of Z is absolutely continuous with respect to the Lebesgue measure, with

14 14 Nicolas Privault Dichuan Yang log 1 + gs f k s 1 λ f s k+1 ds <, k = 1,2,3. 1 hen i Delta - sensitivity with respect to S. We have where L = S E[ΦS = 1 S E[ΦS L, 2θ gs f 2 sdγ s θ gs f sdγ s + τ η + 2 f sdγ s f sds + ηθ. θ gs f sdγ s + τ η ii sensitivity with respect to θ. We have [ θ E[ΦS = E ΦS L gsdγ s 1 gs f sdγ s H c + S θ θ,τ E[ΦS, S where H = θ gs f sdγ s + τ η. iii heta - sensitivity with respect to τ. We have τ E[ΦS = E [ΦS L Θ ηh c + S τ θ,τ E[ΦS. S iv Gamma - second derivative with respect to S. We have 2 S 2 where = 1 S 2 E E[ΦS 1 S K = 2θ [ ΦS L 2 1 I H 2K 2 H H 3 + N H M K H 2 S E[ΦS, gs f 2 sdγ s, M = f sdγ s f sds + ηθ, I = 6θ gs f s 3 dγ s, N = 2 f sdγ s f sds + ηθ.

15 Variance-GGC asset price models their sensitivity analysis 15 Next we state two lemmas which are needed for the proof of heorem 1. Lemma 1. Assume that E[e 2γZ < for some γ > 1. Let f : R,a be a positive function λ,ε for ε < 1/a such that 1 holds. Fix η > suppose that one of the following conditions holds: i he density function of Y = [ gsdγ s decays exponentially, or ii E e 2γ1+θδY < for all δ >. Let also S λ f gs = S exp θ 1 λ f s dγ s + τ Θ + ηλ + Z + cθ,τ, H λ f = λ logsλ f gs f s = θ 1 λ f s 2 dγ s + τ η, H = H, K λ f = λ Hλ f gs f 2 s = 2θ 1 λ f s 3 dγ s, K = K. hen we have the L 2 Ω-limits K λ f λ H λ f lim λ Sλ f H λ f = S H lim = K 2 H 2. Proof. For any λ,ε, we have [ sup E S λ f H λ f 2γ C 1 E [e 2γτ Θ E [ e 2γZ λ,ε sup E[ θ λ,ε gs f s 1 λ f s 2 dγ s + τ 2γ η exp 2γ gs 1 λ f s dγ s C 1 E[e 2γτΘ E[e 2γZ [ a 2γ sup λ,ε 1 λa 2 E gsdγ s + τ 2γ 2γθ η exp gsdγ s 1 λa C 1 E[e 2γτΘ E[e 2γZ [ a 2γ 1 εa 2 E gsdγ s + τ 2γ 2γθ η exp gsdγ s, 1 εa where C 1 is a positive constant. Under condition i or ii above we have [ [ E Y 2γ 2γθ exp 1 εa Y E exp 2γ 1 + θ Y <, 1 εa

16 16 Nicolas Privault Dichuan Yang similarly we have E [e 2γθ 1 εa Y <. Finally, we have E[e 2γZ < by assumption, it is clear that E[e 2γτΘ <. hen S λ f H λ f is L 2γ Ω-integrable, hence S λ f H λ f 2 is uniformly-integrable since γ > 1. herefore, we have proved that S λ f H λ f converges to S H in L 2 Ω as λ. Next, for any λ,ε we have [ sup E[ K λ f /H λ f 2θ 2 2γ sup E λ,ε λ,ε τ gs f 2 2γ s η 1 λ f s 3 dγ s a 2 2γ [ 2γ 1 λa 3 E gsdγ s sup 2θ λ,ε τ 2γ η a 2 2γ [ 2γ 2θ 1 εa 3 E gsdγ s τ 2γ η, [ since E gsdγ s 2γ is finite under Condition i or ii above. herefore K λ f /H λ f 2 2 is uniformly-integrable since γ > 1, this shows that K λ f converges to K /H 2 as λ in L 2 Ω. /H λ f 2 Lemma 2. Assume that E[e 2γZ < for some γ > 1 that 1 holds. Suppose in addition that one of the following conditions holds: 1. he density function of [ gsdγ s decays exponentially; e 2. E 2γ1+θδY < for all δ >, where Y = gsdγ s; then for Φ C 1 b R +,R it holds that i E [ [ Φ S S H = E f sdγ s f sds + ηθ ΦS, ii E[Φ [ S S = E[ΦS L, [ iii E Φ S S gsdγ s = E ΦS L gsdγ s 1 iv E[Φ S S B = E [ΦS L Θ ηh, v If in addition Φ C 2 b R +,R 1 is satisfied then we have Proof. We have E[Φ S S 2 + E[Φ S S [ = E ΦS L 2 1 H H I H 2K 2 H 3 + N H M K H 2 E[ΦS 2 2E[ΦS ΦS 2 + 2E[ΦS 2 gs f sdγ s,.

17 Variance-GGC asset price models their sensitivity analysis E[ΦS E[Φ rs + 1 rs 2 S S 2 dr <, since Φ C 1 b R +;R. As for i we have [ [ E ΦS λ f dpλ f = E dp ΦS, 11 F where we define the probability measure P λ f via its Radon-Nikodym derivative dp λ f dp F = eλ f sdγ s e ληθ E[e λ f sdγ s E[e ληθ = eλ f sdγ s + log1 λ f sds+ληθ λ 2 η 2 /2, where f : R,a λ,ε. In this way the GGC rom variable gsdγ s the Gaussian rom variable Θ under P λ f are transformed to gs 1 λ f s dγ s Θ + ηλ under P. [ First we prove that λ E For every ε λ,λ we have ΦS λ f ΦS ε f ΦS 1 = Φ S rε f S rε f ε exists equals the left h side of i. H rε f dr, by the Cauchy-Schwarz inequality we get [ 1 E ε ΦSε f ΦS Φ S S H 1 1 E[ Φ S rε f S rε f 1 + H rε f E[Φ S rε f 2 E[Φ S rε f Φ S 2 Φ S S H dr E[S rε f H rε f S H 2 dr E[S H 2 dr. 12 From the boundedness continuity of Φ S ε f with respect to ε in L 2 Ω, we have E[Φ S ε f 2 < lim E[Φ S ε f ε Φ S 2 =. By Lemma 1 we get that S λ f H λ f converges in L 2 Ω. [ Finally, we take the limit on both sides of 12 as ε. Next we prove that λ E dpλ f dp ΦS exists F

18 18 Nicolas Privault Dichuan Yang equals the right h side of i. For every ε λ,λ the Cauchy-Schwarz inequality yields 1 dp ε f E[ ε dp dp F dp ΦS f sdγ s f sds + ηθ ΦS F E[ΦS 2 E 1 dp ε f ε dp dp F dp f sdγ s f sds + ηθ 2. F It is then straightforward to check that E[ ΦS 2 < exp 1 λ converges to λ f sdγ s + log1 λ f sds + ληθ λ 2 η 2 /2 f sdγ s f sds + ηθ 1 in L 2 Ω as λ tends to since λ 1 e λ f sdγ s 1 converges to f sdγ s in L 2 Ω as λ. We conclude by taking the limit on both sides as λ. For ii we start with the identity [ ΦS λ f First we prove that E H λ f [ λ E ΦS λ f For every ε [ λ,λ we have 1 ΦS ε f ε H ε f ΦS H H λ f = E [ dpλ f dp ΦS. F H exists equals the left h side of ii. 1 Φ S rε f S rε f H rε f 2 ΦS rε f K rε f = H rε f dr, 2 by the Cauchy-Schwarz inequality we get 1 ΦS E[ ε f ε H ε f ΦS Φ S S H 2 ΦS K H H Φ E[ S rε f S rε f H rε f 2 ΦS rε f K rε f H rε f Φ S S H 2 ΦS K 2 H 2 dr

19 Variance-GGC asset price models their sensitivity analysis 19 1 E[Φ S rε f 2 E[S rε f S 2 dr 1 + E[Φ S rε f Φ S 2 E[S 2 dr 1 + E[ΦS rε f 2 E[K rε f /H rε f 2 K /H 2 2 dr 1 + E[ΦS rε f ΦS 2 E[K /H 2 2 dr. We have shown E[ΦS 2 < in the proof of i. hen E[ΦS ε f 2 2E[ΦS ε f S 2 + 2E[ΦS 2 1 2ε 2 E[Φ S rε f S rε f H rε f 2 dr + 2E[ΦS 2 2ε 2 sup Φ x 2 sup E[S ε f H ε f 2 + 2E[ΦS 2 <, x R ε λ where the Cauchy-Schwarz inequality the Fubini theorem have been used for the second inequality. he convergence of S ε f H ε f as ε in L 2 Ω has been proved in Lemma 1. Note that E[ΦS ε f 2 < also implies E[ΦS ε f ΦS 2 as ε. By Lemma 1, we get K ε f /H ε f 2 converges to K /H 2 as ε in L 2 Ω. aking the limit on both sides of 13 as ε. [ Next, we prove that λ E dpλ f dp ΦS exists is equal to the right h F H side of ii. For all p > we have E[H λ f 2p = gs f s θ 1 λ f s 2 dγ s + τ 2p η f 1 ydy < τ η 2p, where f 1 is the density function of gs f s dγ 1 λ f s 2 s. herefore, the moment is uniformly bounded. We conclude as in the second part of proof of i. he proof of iii iv is similar to that of ii. As for iii we have E [ ΦS λ f H λ f [ gs 1 λ f s dγ dpλ f s = E dp ΦS gsdγ s. F H For the first part, the existence of the derivative can be obtained as

20 2 Nicolas Privault Dichuan Yang [ gs 2 E 1 λ f s dγ f s 2 s gsdγ s E[ λ gs 1 λ f s dγ s a 2 E[ λ gsdγ s 1 λa Similarly,. gs f s 1 λ f s 2 dγ s converges to gs f sdγ s in L 2 Ω as λ. he second part is almost the same as i by uniform boundedness of H λ f. For iv we have [ [ ΦS λ f dpλ f Θ + ηλe H λ f = ΘE dp ΦS. F H For the first part, the existence of the derivative follows from the fact that Θ has a Gaussian distribution. he second part is proved similarly. Finally, for v, define Ψx = Φ xx, by the result of ii we have E[Φ S S 2 = E[Ψ S Φ S S = E[ΨS L E[Φ S S Hence, we obtain the desired equation by differentiating [ at λ =. E ΦS λ f Lλ f H λ f Now we can prove heorem 1. = E [ dpλ f dp F ΦS L H Proof. he proof of heorem 1 uses the same argument as in the proof of Corollary 3.6 of [9. he only difference is that S is a variance-gamma process in the proof of Corollary 3.6 of [9, while S is a variance-ggc process in this proof. When Φ C 2 b R +,R, all four formulas in heorem 1 are direct consequences of ii v in lemma 2, we now extend this result to the class DR + ;R. In general, in order to obtain an extension to Φ in a class R 1 of functions based on an approximating sequence Φ n n N in a class R 2 R 1, it suffices to show that for each compact set K R we have sup E[Φ n S E[ΦS as n, 14 S K

21 Variance-GGC asset price models their sensitivity analysis 21 lim n E[Φ n S 1 E[ΦS L S K S S =. 15 he extension is then based on the above steps, first from Cb 2R +,R to C c R +,R, then to C b R +,R to the class of finite linear combinations of indicator functions on an interval of R. Finally the result is extended to the class of functions Φ of the form Φ = Ψ 1 A where Ψ C L R +,R A is an interval of R +. his shows that are satisfied, the details of each step are the same as in the proof of Corollary 3.6 of [9. References 1. O.E. Barndorff-Nielsen. Processes of normal inverse Gaussian type. Finance Stochastics, 21:41 68, L. Bondesson. Generalized gamma convolutions related classes of distributions densities, volume 76 of Lecture Notes in Statistics. Springer-Verlag, New York, S.I. Boyarchenko S.Z. Levendorskiĭ. Non-Gaussian Merton-Black-Scholes theory, volume 9 of Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co., Inc., River Edge, NJ, P.P. Carr, H. Geman, D.B. Madan, M. Yor. he fine structure of asset returns: an empirical investigation. J. Business, 752, J.C. Cox S.A. Ross. he valuation of options for alternative stochastic processes. Journal of Financial Economics, 3: , H. Geman, D.B. Madan, M. Yor. Asset prices are Brownian motion: only in business time. In Quantitative analysis in financial markets, pages World Sci. Publ., River Edge, NJ, L.F. James, B. Roynette, M. Yor. Generalized gamma convolutions, Dirichlet means, horin measures, with explicit examples. Probab. Surv., 5: , R. Kawai A. Kohatsu-Higa. Computation of Greeks multidimensional density estimation for asset price models with time-changed Brownian motion. Applied Mathematical Finance, 174:31 321, R. Kawai A. akeuchi. Greeks formulas for an asset price model with gamma processes. Math. Finance, 214: , R. Kawai A. akeuchi. Computation of Greeks for asset price dynamics driven by stable tempered stable processes. Quantitative Finance, 138: , D.B. Madan, P.P. Carr, E.C. Chang. he variance gamma process option pricing. European Finance Review, 2:79 15, D.B. Madan E. Seneta. he variance gamma model for share market returns. J. Business, 63: , R. Merton. Option pricing when underlying stock returns are discontinuous. J. of Financial Economics, 3, N. Privault D.C. Yang. Infinite divisibility of interpolated gamma powers. J. Math. Anal. Appl., 452: , 213.