REVISITING THE VARIANCE GAMMA MODEL Óscar G. Arnaiz ♣ Universidad de Zaragoza

In the last few years it has become usual to invert the distribution function transform to obtain the probability elements involved in option prices. Heston, Bakshi-Cao-Chen, Bates and Carr-Geman-Madan-Yor among others use this procedure. The method has an inconvenient: the integrand involved has a singularity at the origin, so the Fast Fourier Transform can not be applied. Besides, a deviation from existent literature (based on the arbitrage approach) has arisen (Geman-Madan-Yor). Under this new perspective, prices are typically expressed as Lévy processes and the continuity and normality can be recovered by means of a time change. The time thus considered is random, representing a measure of the economic activity. This paper is concerned about both discoveries. For models where the time is random, we develop a method to obtain expressions computable by means of the Fast Fourier Transform. The method, based on complex variable theory, exploits the numerical restrictions imposed on the model parameters by the “convexity correction”. Firstly, we analyse the Variance Gamma model. An alternative method to derive the density function is proposed, as well as an expression for the European call computable by FFT. This expression also permits pricing options accurately by interpolation of elementary functions. The method can be generalised to a wide class of processes useful in finance. The processes are those in the generalised gamma convolution, obtaining expressions computable by FFT. Some important examples like the CGMY and the CMY models are provided.

♣

[email protected] Iam very grateful to Pedro Miana for his assistance. To Javier Sesma for his assistance, help and support. To José A. Cristóbal for technical assistance and his class notes. And of course, to Vicente Salas.

Although the Black-Scholes model remains as the most extended option pricing model, it has well known limitations broadly investigated 1 . These limitations are in connection to the central role played by the Brownian motion in the model, because the gaussian assumption contradicts the markets reality in several ways. The biases may be explained in terms of the superior moments of the underlying price distributions, clearly apart from the normality postulated by Black-Scholes’73 (BS further on). Such distributional features have been studied using time series of stock and option markets, characterising the last ones the so-called volatility smile. Some important alternatives proposed in the literature are the jump-diffusion models, the stochastic volatility models and models based on Lévy processes (with finite or infinite arrival rate of jumps) like the Variance Gamma model. The Merton model2 introduce stock price discontinuities, because continuity of pure diffusions is actually a drawback. The resultant model leads to non-exact expressions for pricing. Besides, the high number of parameters (five at the minimum) complicates its use 3 . Moreover, as pointed out in McCulloch’78, it does not provide the inherent cohesion of other jump models. Jump-diffusion models capture relatively well the smile shown by the short dated options (DasSundaram’98 or Bates ’96) In the stochastic volatility models, the volatility follows a diffusion (Hull-White’87 or Heston’93a among others) These models are able to capture the smile, but just qualitatively. They always fail in explaining its magnitude. Bates’96 shows that the mean-reverting volatility submodel cannot explain the smile evidence of implicit excess kurtosis except under implausible parameters. Why are stochastic volatility models unable to explain the markets reality? Empirically it has been noted that as sampling period decreases, the kurtosis of the returns increases. That is just opposite to what happens in the models where the volatility is driven by a diffusion. Locally, diffusion models are gaussian, in sharp contrast to reality. So for short dated options, this kind of models is expected to work badly. That is the reason of introducing a jump component in modeling stock price dynamics, as it is noted in Bakshi-Cao-Chen’97.

1

See Rubinstein (’85, ’94), for example. Merton’76 is the first of this kind of models , contemporary to Cox-Ross’ pure-jump models. 3 Standard Maximum Likelihood procedures in estimating jump -diffusion models are no longer valid (Honoré ’98) 2

2

For a detailed analysis of the term structure of the volatility smile, see Das-Sundaram’98. Another drawback concerning to both kinds of models is pointed out in Bates’96. The stochastic volatility/jump-diffusion model analysed exhibits parameter instability. This is a consequence of its infinite variation. As Madan’99 points out, the lack of robustness of diffusions is an inherent property of infinite variation processes 4 , and he strongly advocates against the use of this kind of processes as models of price dynamics. Finally we have the VG model and in general pure-jump models with finite or infinite arrival rates (the VG belongs to the first class). The corresponding logprice processes are infinitely divisible with independent increments, so they may be described by their Lévy-Khintchine densities. The hyperbolic model (Eberlein-Keller-Prause’98) is another example, but it has several disadvantages if compared to the VG model. The Variance Gamma model is a three-parameter one that eliminates the volatility smile in the strike direction. Its pricing accuracy lies generally between 1 and 3 per cent, considering strikes up to 20 to 30% out-the-money and maturities as short as two days sometimes (see Madan’99). The density function and the call price may be expressed as closed forms in terms of special functions, and the parameter estimation may be performed by Maximum Likelihood. The formulas derived for the VG model have a disadvantage: the functions involved, expressed as power series, are computationally expensive. To price options using the closed form may be slower than computing prices by numerical procedures employing Fourier-inversion5 . Some drawbacks of these numerical methods are commented on below. The objective of this paper is to overcome these limitations, providing an alternative method to obtain simple formulas. By means of complex variable arguments, we will firstly provide an alternative derivation of the VG logprice density function. Then, we will obtain an alternative integral form for the European call. It will be an integral expression of elementary functions that may be directly performed by FFT, making unnecessary to introduce external factors like in Carr-Madan’98. This new formulation has other advantages. It makes possible to value options by interpolation. This possibility for option pricing is accurate, being low its computational cost because elementary functions are involved. Furthermore, an alternative closed form simpler than the one proposed in Madan-Carr-Chang’98 can be obtained. The VG model represents a particular case of a wide class of processes. The method is generalised and some important examples are provided. The rest of the paper is structured as follows. In the next section we introduce the VG model. In Section 2 the density function and the call price are obtained employing an alternative method based on complex variable theory. The VG prices are interpolated in Section 3. Section 4 generalises the method, providing expressions for some useful pure-jump models. Section 5 presents the main conclusions.

4

ε >0

A process of finite variation requires that the integral

∫

0

xk ( x )dx be finite. In such processes, the sum of their moves

in absolute magnitude is finite. 5

See Carr-Madan’98.

3

1.- THE VG MODEL The VG model generalises the Brownian Motion as logprice process and it is explained in its general form in Madan-Carr-Chang ’98. That paper generalises the two previous formulations: Madan-Seneta ’90 considers symmetric returns and Madan-Milne’91 is intended to option pricing, by skewing the former in a general equilibrium setting. In the VG model, the statistical and the risk-neutral price dynamics are given in terms of the VG process. The price dynamics will be precisely described below, after defining and characterising the VG process. The VG process is a pure jump model, and their three parameters σ ,θ , v take into account the variance, skewness and kurtosis of the price process. Specifically, it is obtained as a Brownian motion (BM) with drift evaluated at a random time γ (t ) 6 : X t = θγ ( t ) + σ Wγ (t ) [0] being Wt a standard BM and γ (t ) a gamma process evaluated at t. The BM requires no further explanation. The gamma process is a infinitely divisible one, obtained by adding independent increments which follow a gamma random variable. The density function of a random gamma variable of parameters (a, p) is given by: a p p−1 − ax f γ ( t) ( x ) = x e [1] Γ( p) So γ (t ) is a non-decreasing process distributed as a gamma random variable of parameters a = 1 / v , p = t / v , and may be approximated as a compound Poisson process. It may also be described by means of its characteristic function, univocally obtained by the inverse Fourier transform of the density function given above:   1 φ γ ( t ) (u ) =    1 − iuv / µ 

t /ν

[2]

As infinitely divisible process, it may also be characterised by means of its Lévy measure: µ  2 e − v x µ dx, x > 0 vx k γ ( x) dx =  0, x ≤ 0 The integral of this function is infinite, so the gamma process has infinite activity. As k(0) is also infinite, the measure is concentrated near the origin. The coefficient of the diffusion term − u 2 / 2 in the Lévy-Khintchine representation of its characteristic function is zero, and then the process is a purejump one. By evaluating a BM with drift at a gamma random time, we obtain the VG process. Its density function is not as simple as in the gamma process, but its characteristic function and Lévy density are. 6

This process is employed by Heston’93b for modeling price dynamics.

4

The characteristic function may be evaluated integrating the Brownian characteristic function with respect the variance gamma measure f γ ( t) ( x ) dx , f γ ( t) ( x ) given above. It yields to the simple expression :  1 φ VG (u ) =  2 2  1 − iθvu + (σ ν / 2)u

  

t /ν

[3]

The notion of equilibrium is implicit in the model, as it is explained in Geman-Madan-Yor ’98 or Madan’99 and it may be summarised as follows: the VG model is obtained by evaluating the BM at a random time, and in Monroe’78 it is demonstrated that it implies to be a semimartingale 7 . And a price process is a semimartingale if and only if No-Free-Lunch-with-Vanishing- Risk is possible (DelbaenSchachermayer’94). This condition is more restrictive than the No-Arbitrage condition. Moreover, as no particular form has been assigned to the risk premium (the price process can be derived from a Lucas-type economy) the equilibrium is ge neral and not only partial. This simple expression allows us to express the VG process in an alternative form. It consists of expressing it as the difference of two gamma processes. Factoring the characteristic function, it may be expressed as: 2

µp t

µ n 2t

    vn 1   1  . φ VG (u ) =   1 − i v p u   1 + i v n u  µn µp     The parameters µ p , µ n , v p , v n may be related to the original ones (see Madan-Carr-Chang ’98) vp

As the gamma process is non-decreasing, the price process can be viewed as the difference of two gamma processes. They take account of the up and down moves of logprices. The VG process is consequently of finite variation8 , which may also be shown by integrating xk(x) in a neighbourhood of zero. The VG Lévy-Khintchine density is simply given by:

 2 − µv n x  µn e n dx , x < 0 θ x v x eσ 2 −  n k VG ( x ) dx =  = e µp vx  µ 2 − vp x  p e dx , x > 0  vp x 

2 θ2 + v σ2

σ

x

dx

The VG has been described and characterised. Now we introduce the statistical and the risk-neutral price dynamics. The statistical price dynamics is given by 7 8

If the price process is a semimartingale, there exists a probability measure under which price processes are martingales. As it has already been commented on, it leads to a better tolerance of parametric heterogeneity.

5

S t = S 0 exp (mt + ωt + X t (σ ,θ , v ) ) [4] where X t is a VG process, m is the statistical mean return and ω is the convexity correction calculated by evaluating the characteristic function at –i 9 , in this case resulting: 1 ω = log( 1 − θv − σ 2 v / 2) [5] v It introduces a restrictio n on the feasible parameters of the model, because ω must be real. This has not been sufficiently pointed out before and it will be used along the paper. The density function can be obtained by integrating the gaussian density with respect the random time density function. It leads to an expression containing the McDonald function (or modified Bessel function of the second kind):

f X t (x ) =

v t /ν

t

  2v 2e x2  2  2π σΓ(t / v )  2σ / v + θ 2  θx / σ

2

−

1 4

 2σ 2 / v + θ 2   x 1 2  −  σ v 2 

Kt

[6]

This returns density permits to test how the model fits stock market prices. The risk-neutral price dynamics (the relevant one in futures and options markets) is obtained in a completely similar manner. It just requires to substitute the statistical parameters by their risk-neutral counterparts. As it is well known, the risk-neutral counterpart to the statistical drift m is the interest rate r, because discounted stock prices under risk-neutral valuation have to be martingales. The European call price is then obtained by calculating the expectation e −rT E RN ( ST − K ) + . This expectation is taken under the risk- neutral probability, so employing risk-neutral parameters. This involves integrating the BS formula with respect the gamma density . This is the method employed in Madan-Carr-Chang ’98. There are other possibilities. One of them is to compute the integral by using the returns density function obtained above. Another is the method proposed in this paper, using the VG characteristic function and complex variable theory. Other possibilities are commented below. The European call price obtained in Madan-Carr-Chang’98 is expressed in closed- form10 in terms of special functions given by:

(

CT = S 0ψ d

1− c1 v

, (α + s )

v 1− c1

)

(

, γ − Ke −rTψ d

1− c2 v

,α

v 1− c2

)

,γ ,

[7]

where s, α , γ , c1 , c 2 , d are functions of σ ,θ , v . The function ψ may be expressed in terms of special functions and is given by:

9

E RN ( S T ) = e rT S 0 ⇒ E RN ( e X T ) = e −ω T ⇒ φ ( −i ) = e −ω T

10

In the referred paper is explained what is understood as a “closed form”.

6

ψ (a , b, γ ) =

− sign (a )

c

+ sign ( a)

c

c

γ + 12

e sign( a )c (1 + u) γ 2π Γ(γ )γ

K γ + 1 ( c) Φ (γ ,1 − γ ,1 + γ ; 1+2u ,− sign ( a) c(1 + u ) ) − 2

γ + 12

e sign( a )c (1 + u) γ K γ − 1 ( c) Φ(1 + γ ,1 − γ ,2 + γ ; 1+2u , −sign ( a) c(1 + u ) ) + 2 2π Γ(γ )(1 + γ )

γ + 12

[8]

e sign( a) c (1 + u ) γ Kγ − 1 ( c) Φ (γ ,1 − γ ,1 + γ ; 1+2u , −sign ( a) c(1 + u ) ) 2 2π Γ(γ )γ

where: c = a 2 + b 2 , u =

b 2 +b 2

and Kα ( x) and Φ (α , β , γ ; x, y ) 11 represent the modified Bessel

function of the second kind and the degenerate hypergeometric function of two variables respectively. This closed form has a disadvantage. The functions involved, expressible as power series, are computationally expensive. Calculating the call price using [7] is slower 12 than computing the price numerically. For it, Carr-Madan’98 transform that integral in such a way that option prices may be performed by FFT. It leads to a much faster solution than using the closed form [7]. The method has the disadvantage that it needs an external dampening factor to employ the FFT. The present article fills this gap by providing an alternative method to obtain the formulas. The pricing expression is obtained as an integral of an elementary function that can be performed by FFT, making unnecessary introducing a dampening factor. The VG processes and in general truncated Lévy flights slowly converge towards the gaussian process (Koponen’95). This is in connection to that BS prices better long-dated options.

2.- THE ALTERNATIVE METHOD Let us consider a VG gamma process given by X t = θγ (t ) + σWγ (t ) . The characteristic function is given in [3]: t /ν

  1  . φ VG (u ) =  2 2   1 − iθu + (σ ν / 2)u  It is well known that the VG density can be expressed as the inverse Fourier transform of φ VG (u ) : 1 f X t (x ) = 2π 11 12

∫

+∞

−∞

e

−iux

 1  2 2  1 − iθν + (σ ν / 2)u

Also represented as 1 F1 (α , β , γ ; x , y ) ≡ See Carr -Madan ’98.

  

t /ν

du ,

M (α , β , γ ; x, y ) , one of the confluent hypergeometric functions.

7

an alternative to integrating the gaussian characteristic function with respect the gamma-distributed variance [1]. This integral may be easily expressed as a real form using complex variable theory, obtaining the VGdensity function [6]. We first make the change of variable z = u − i θ / σ 2 , resulting: 1 f X t (x ) = 2π

∫

+∞

−∞

 1   2  σ ν / 2 

t /ν

1  1    2π  σ 2ν / 2 

t /ν

1  1    2π  σ 2ν / 2 

t /ν

1 2π

e

∫

−iux

  1   2 2   1 − iθν u + (σ ν / 2)u 

+∞

−∞

e

e θx / σ

−iux

2

θx / σ 2

e

t /ν

du =

  1  2  2 2   u − ( 2iθ / σ )u + 2 / σ v 

t /ν

du =

1  −izx  ∫−∞ −iθ /σ 2 e  z 2 + θ 2 / σ 4 + 2 / σ 2ν  + ∞ − iθ / σ 2

∫

+∞ − iθ / σ 2

− ∞ −i θ / σ

2

e

−izx

 1  2 2 z +β

  

t /ν

dz =

t /ν

dz , making

θ2 σ4

+ σ22ν = β 2

We now consider an infinite rectangle in C, consisting of the real axis, a parallel axis given by R − iθ / σ 2 and two symmetrical vertical segments closing the path at the infinite. The integrand is a complex function, and we now enquire where is it analytical. This may be easily checked it out by applying the Cauchy-Riemann conditions, resulting that: - If t/v is an integer, the integrand is analytical if z ≠ +i β , −i β . This case will have interesting consequences commented below. But generally, t must not be considered as an integer, because it represents the time parameter. - If t is real but not an integer, the function under consideration will be analytical if and only if z 2 + β 2 is positive. So the integrand is analytical in C excepting in the set {ix| x ∈ R, x ≥ β } θ2 σ4

As

< β 2 , the rectangle in consideration lies in the region where the integrand is analytical. By

application of the Cauchy’s therorem13 :

∫

+ ∞−

−∞ −

iθ

σ iθ

σ

2

2

e

− izx

 1  2 2 z +β

  

t /ν

+∞

dz = ∫ e −∞

−izx

 1   2  2  z +β 

t /ν

dz ,

taking into account that the other two contributions are zero .

13

∫

γ

If

f : G → C is an analytical function and γ is a closed rectifiable curve in G such γ is homotopic to zero, then

f = 0 (Conway’78)

8

This resulting integral expression can be written in terms of the modified Bessel function of the second kind as (see Gradshteyn and Ryzhik’65, pg.959): 1  − izx  ∫−∞ e  z 2 + β 2  +∞

t /ν

2 π x dz = t /ν  2 Γ( t / v )  β

t 1 −

v 2  K t 1 ( βx ) , which leads to [5] −  v 2

Now we are interested in obtaining an alternative expression for the European call price. We will perform the calculation following the same line of reasoning used above, obtaining a fairly simple integral expression. For the sake of simplicity and for reasons of economic relevance, we will restrict ourselves to the case θ < 0 . It does not imply a lost of generality, because a parallel derivation can be performed for θ > 0 . But θ < 0 corresponds to the case of positive risk aversion in options markets, the relevant case. It causes the smile asymmetry and is in agreement with empirical studies. All the cases studied in CarrGeman-Madan-Yor’00 relative to option markets are negatively skewed with negative θ . We have then to calculate CT = e − rT E RN ( S T − K ) , where RN denotes that the expectation is taken under the risk-neutral probability: CT = e − rT E( S T − K ) + = E( S 0 e ωT + Xt − e − rT K ) +

e

−ωT

being

Xt

the

risk-neutral

= φ (−i ) the convexity correction. ∞

∞

B

B

CT = S 0 eω T ∫ e x f X T ( x )dx − Ke −rT ∫ f X T ( x )dx and B = log( K / S 0 ) − rT − ωT . Both integrals belong to the same parametric class. Let us see it. ∞

Let us denote ∫ f XT ( x )dx by I (1 / v, θ , T ; B ). B

∞

Using [1], ∫ e x f XT ( x) dx may be expressed as B

∫

∞

B

dxe

x

∞

∞

B

0

∞

∞

B

0

∫ dx ∫ ∫ dx ∫

∫

∞

0

1 σ 2πτ

1 σ 2πτ

1 σ 2πτ

e

( x −θτ )2 2 σ 2τ

(θ +σ 2 / 2 )τ

−

e

e

−

τ T / v −1e −τ / v dτ = v T / v Γ( Tv ) −

e

[x − (θ +σ )τ ] 2

2σ τ 2

2

[x −(θ +σ )τ ] 2

2σ τ 2

2

τ T / v −1e −τ / v dτ = v T / v Γ( Tv )

τ T / v −1e − (1/ v −θ −σ v T / v Γ( Tv )

2

/ 2 )τ

dτ

9

VG

process

and

which expressed in terms of I(·) results 14 : T

 1 v 2 2 x ∫B e f X T (x )dx =  1 −θv− σ 2  I (1v − θ − σ2 ,θ + σ 2 , T ; B) = e −ωT I (1v − θ − σ2 , θ + σ 2 , T ; B) v 2  and then the call price will be, with the restrictions commented above: ∞

CT = S 0 I ( 1v − θ − σ2 ,θ + σ 2 , T ; B) − Ke −rT I ( 1v ,θ , T ; B) 2

[9a]

Now, we will obtain a simple integral expression for I (1 / v, θ , T ; B ). It will be based on the VG logprice density, before expressing it in closed form.

I (1 / v, θ , T ; B ) = ∫

∞

B

1  1  =   2π  σ 2ν / 2 

T /ν

1  1  f XT ( x) dx =   2π  σ 2ν / 2 

 1  ∫−∞  z 2 + β 2  +∞

T /ν

∞

dz ∫ e

T/ν

∫

∞

B

θ ( −iz + 2 ) x σ

B

θ

dxe

σ2

x

∫

+∞

−∞

e

−izx

 1  2 2 z +β

  

T/ν

dz =

dx

by reversing the order of integration. We will in principle consider positive risk-aversion in option markets, characterised by θ < 0 . The second integral may be easily computed by the Barrow’s theorem: 1  1  I (1 / v, θ , T ; B ) =   2π  σ 2ν / 2  1  1    2πi  σ 2ν / 2 

T /ν

θ

e

σ2

B

T /ν

  1 ∫− ∞  z 2 + β 2  +∞

T /ν

∞

dz ∫ e

( − iz +

θ σ2

)x

B

dx =

e − izB ∫−∞ ( z + i θ2 )( z 2 + β 2 )T /ν dz σ +∞

[10a]

And as a real valued integral: 1  1  I (1 / v, T ,θ , B) =   2π  σ 2ν / 2 

14

This requires

1 v

T /ν

θ

e

σ2

B

∫

+∞

−∞

− σθ2 cos Bz − zsinBz

( z 2 + ( σθ2 ) 2 )( z 2 + β 2 ) T /ν

dz

[10b]

− θ − σ2 > 0 , but as it was pointed out in the previous section, this is a restriction of the model 2

parameters. The cases studied in Carr-Geman-Madan-Yor ’99 are all very far from violating this condition.

10

As θ + σ 2 might be positive, the integrals in [9a] may have different parameter restrictions. We will calculate I (even for the relevant case where θ < 0 ) considering that θ can be positive. To take into account this fact, we will use that ∞

∫e B

x

(

∫

∞

B

f XT ( x) dx = 1 − ∫

B

−∞

f X T ( x ) dx . The resultant expression is:

f X T ( x )dx = e −ωT 1{θ +σ 2 > 0} + I ( 1v − θ − σ2 ,θ + σ 2 , T ; B) 2

)

and the call price will be, if θ < 0 :

{

}

CT = S 0 1{θ + σ 2 > 0} + I ( 1v − θ − σ2 ,θ + σ 2 , T ; B ) − Ke −rT I ( 1v ,θ , T ; B) 2

[9b]

being 1{a> 0} =1 if a>0 and zero otherwise. If θ + σ 2 =0 , we can obtain a closed form using Gradshteyn-Ryzhik’65 ([6.561.4]). If we accept negative risk-aversion, the second term will include a function like 1{a> 0} too. [10 a,b] can be computed by means of the Fast Fourier Transform (FFT). This is not possible expressing the call price by Fourier-inversion of [3], a method generalised in Bakshi-Madan’98. Given the great computational advance that the FFT implies, Carr-Madan’98 propose an alternative way to perform it. It consists of introducing an external factor, making possible direct application of FFT. Obviously, the solution must be independent of the dampening factor, so the method carries a computational cost. Analytically, [10a] is simpler than its two former alternatives. Moreover, the call price can be expressed as combination of real valued integrals [10b].

3.- PRICING OPTIONS BY INTERPOLATION The following procedure will enable us to compute option prices even more efficiently than using the FFT. With such purpose in mind, we will enquire about the complex nature of [10a]. As it was established, the integrand of [10a] is an analytic function in C (excepting at some points) if time to maturity is an integer multiple of the kurtosis parameter v. For those maturity dates, the integrand is analytical excepting at three 15 pure imaginary values. Two of them are symmetric located respect the origin and the y are poles of order T/v. Between them there exists a simple pole at − i θ / σ 2 . This expression is easily computable using the Residue theorem. It becomes (but constants) the sum of the residues of [10a] at the poles situated above the real axis if B 0 

k CGMY

C,G,M>0, 00, and M − θ − σ 2 / 2 plays the role of M in the first term. Moreover, we need such inequality to apply Cauchy’s theorem and substitute the complex axis of integration by the real one. We may proceed like that since M − θ − σ 2 / 2 > 0 , because φ CMY ( −i ) must be real. Finally, I(·) can be written as: θ

I=

{

1 σ 2 B +∞ e −ivB 2 2 e ∫ exp − TCΓ( −Y ) M − iuθ + σ u / 2 θ − ∞ 2π iv − σ 2

(

)

Y

−M

Y

}dv

also expressible in real terms. Exponential price shocks: We now consider a process that represents the building block of the completely monotone processes and which does not belong to the gamma convolution. We will see that the proposed method is not valid for such process, because the terms P1 , P2 are not in the same parametric class. It is given by a Poisson process with exponential prevailing price order size, instead of gaussian (Madan-Geman-Yor’98). Its Lévy density is given by k ε ( x )dx = e − ax dx .

This also can be written as respectively:

X t = θ T (t ) + σWT ( t) . The characteristic functions of X t and T (t ) are

19

It can also be computed by conditioning the Brownian characteristic function on the random time, CMY distributed. Its density may be expressed Fourier-inverting its characteristic function. The resultant double integral is then reduced to a single one by reversing the order of integration and carrying out the second integral. The remaining integral may be very easily calculated by applying the Residue theorem.

17

 2t  a2  φ log S = X t (u ) = exp   2 − 1 2 2 2  a  a + γ (u − 2iθ / σ )   2t  a 2  φ T ( t) (u ) = exp   2 − 1  . So  a  a − iu   ∞

∫e

x

B

∞

f X T ( x) dx = ∫ dxe B

∞

∞

B

0

= ∫ dx∫ 1 = 2π

x

∫

∞

0

1 σ 2πτ

− 2 1 e (θ +σ / 2 )τ e σ 2πτ

∞

∞

B

0

∫ dx∫

1 σ 2πτ

e

−

e

−

( x−θτ )2 2 σ 2τ

[x −(θ +σ )τ ]

2

2

2σ τ

2

2σ τ 2

f (τ )dτ = ∫ dxe

1 2π

∫

+∞

−∞

e − iuτ e

2

∫

+∞

−∞

x

B

2

[x −(θ +σ )τ ]

∞

e

( −iu +θ +σ 2 / 2 )τ

e

∫

0

 2 t  a 2 −1  a  a 2 −iu 

 2t  a2  −1  a  a 2 −iu 

∞

1 σ 2πτ

e

−

( x −θτ )2 2σ 2τ

same parametric class as e

 2t  a 2  −1  a  a 2 −iu 

∫

+∞

−∞

e −iuτ φτ (u)dτ =

dτ

dτ

Making − iu + θ + σ / 2 = -iv, we observe that the resultant expression e 2

1 2π

 2t  a2  −1  a  a 2 −θ −σ 2 / 2 −iv 

is not in the

.

The same conclusion holds for a logprice driven by X t = θ N (t ) + σW N (t ) , where N(t) represents a Poisson process. This case corresponds to a gaussian Lévy density, which is not completely monotone, and it is related to the Merton’76 model. As it was pointed out in Geman-Madan-Yor’98, the time change N(t) is akin to the number of trades, independently of the magnitude of them20 . This is the time change observed to be relevant by Ané-Geman’00 in their empirical study of high frequency returns on the FTSE100 futures index. Although the last two examples have probability elements of different parametric classes, these probability terms can be computed by FFT using the method given above.

5.- CONCLUSIONS A method of inverting the distribution transform in order to price options is proposed. It is based on complex variable theory and it permits to obtain prices for a wide class of processes as simple integrals. This offers numerous advantages. Firstly, such integrals can be directly computed invoking the Fast Fourier Transform. Moreover, they can be expressed as real valued integrals. It means an advance from previous works. In such former

20

Empirical support is provided by Jones -Kaul-Lipson’94

18

formulations and in order to apply the FFT, it was introduced an external factor which complicated unnecessarily the expressions. Restricting ourselves to the VG model, another interesting feature is that option prices can be calculated by interpolation, because the probability elements admit elementary functional expressions for some maturity dates. This can also be applied to American put pricing, employing the results of Carr’98. The method is also generalised to the wide class of processes in the generalised gamma convolution, obtaining similar expressions to the VG case. Finally, relevant models in option pricing like the CGMY or the CMY are considered.

Appendix CGMY model call price: Making the change iu = iv +

∫

I (C , G, M , Y ; B) =

∫

∞

∫

∞

B

B

dx 21π

∫

∞

B

dx 21π

∫

+∞

−∞

M −G 2

,

e − ( M − G) x / 2 dx 21π

∫

]}

[

{

+ ∞ + i ( M −G ) / 2

−∞ +i ( M − G) / 2

[

{

e −iux exp TCΓ( −Y ) ( M − iu ) Y − M Y + (G + iu ) Y − GY du =

]}

e −(( M −G ) / 2+ iv) x exp TCΓ( −Y ) ( M2+ G − iv ) Y − M Y + ( M2+ G + iv ) Y − GY dv =

−∞ +i ( M − G) / 2

[

{

+ ∞ + i ( M −G ) / 2

]}

e −ivx exp TCΓ(−Y ) ( M 2+G − iv ) Y − M Y + ( M 2+G + iv ) Y − G Y dv

The integrand is analytic between the path of integration and the real axis. This justifies to write: I (C , G , M , Y , B ) =

1 2π

∫

B

1 2π

∫  ∫

=

1 2π

e −B ( M −G ) / 2 ∫ e −iBv

+∞

∞

B

[

{

+∞

]}

e − x( M −G ) / 2 dx ∫ e −ivx exp TCΓ (−Y ) ( M 2+G − iv ) Y − M Y + ( M 2+G + iv ) Y − G Y dv = −∞

[

{

=

−∞

∞

]}

e −(( M −G ) / 2 +iv) x dx  exp TCΓ (−Y ) ( M 2+G − iv ) Y + ( M 2+G + iv )Y − 2 ( M 2+G ) Y dv =  +∞

−∞

{

[

]}

exp TCΓ(−Y ) ( M 2+G − iv )Y + ( M 2+G + iv ) Y − 2( M 2+G ) Y dv M −G 2 + iv

19

References Ané, T. and H. Geman (2000): “Order flow, transactions clock and normality of assets returns”, Journal of Finance, 55, 5, 2259 –2284. Bakshi, G., C.Cao and Z. Chen (1997): “Empirical performance of alternative option prices models”, Journal of Finance, 52, 2003-2049. Bakshi, G. and D. Madan, (1998): “What is the probability of a stock crash market?”, available on www.rhsmith.umd.edu/Finance/dmadan Bakshi, G. and D. Madan, (1999): “Spanning and derivative-security valua tion”, available on www.rhsmith.umd.edu/Finance/dmadan Bates, D. (1996): “Jumps and stochastic volatility: exchange rates processes implicit in Deutsch Mark options”, Journal of Financial Studies, 9, 69-107. Black, F. and M. Scholes (1973): “The pricing of options and corporate liabilities”, Journal of Political Economy, 81, 635-654. Bondesson, L (1992): Generalized gamma convolutions and related classes of distributions and densities, Lectures Notes in Statistics 76, Springer-Verlag, Heidelberg. Carr, P. (1998): “Randomization and the American put”, Review of Financial Studies, 11, 597-626. Carr, P. and D. Madan (1998): “Option valuation using the Fast Fourier Transform”, Journal of Computational Finance, 2, 61-73. Carr, P. and D. Madan (1999): “Optimal positioning in derivative securities”, available on www.rhsmith.umd.edu/Finance/dmadan Carr, P., H. Geman, D. Madan and M. Yor (2000): “The fine structure of asset returns: an empirical investigation”, available on www.rhsmith.umd.edu/Finance/dmadan Conway, J.B.(1978): Functions of one complex variable, 2nd edition, Springer Verlag, New York. Cox, J. and S. Ross (1976): “The valuation of options for alternative stochastic processes”, Journal of Financial Economics, 3, 145-160. Das, S.R. (1994): “Jump-diffusion processes and the bond markets”, Unpublished manuscript. Delbaen, F. and W. Schachermayer (1994): “A general version of the Fundamental Theorem of asset pricing”, Mathematische Annalen, 300, 520-563. Eberlein, E., U. Keller and K. Prause (1998): “New insights into smile, mispricing and Value at Risk: The hyperbolic model”, Journal of Business, 71, 3, 371-405.

20

Geman, H., D. Madan and M. Yor (1998): “Asset prices are Brownian motion: only in business time”, available on www.rhsmith.umd.edu/Finance/dmadan Geman, H., D. Madan and M. Yor (1999): “Time changes hidden in Brownian subordination”, available on www.rhsmith.umd.edu/Finance/dmadan Gradshteyn, I. and I. Ryzhik (1965): Tables of integrals, series and products, Academic Press, New York. Heston, S. (1993a): “A closed form solution for options with stochastic volatility with applications to bond and currency options”, Review of Financial Studies, 6, 327-343. Heston, S. (1993b): “Invisible parameters in option prices”, Journal of Finance, 48, 3, 933-947. Hull, J.C. and A. White (1987): “The pricing of options on assets with stochastic volatility”, Journal of Finance, 42, 281-300. Jones, C., G. Kaul and M.L. Lipson (1994): “Transaction, volumes and volatility”, Review of Financial Studies, 7, 631-651. Koponen, I (1995): “Analytical approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process”, Physical Review E, 52, 1, 1197-1199. Madan, D. and E. Seneta (1990): “The Variance Gamma model for share market returns”, Journal of Business, 63, 4, 511-524. Madan, D. and F. Milne (1991): “Option pricing with VG martingale components”, Mathematical Finance, 1, 39-55. Madan, D., P. Carr and E. Chang (1998): “The Variance Gamma process and option pricing model”, European Finance Review, 2, 79-105. Madan, D. (1999): “Purely discontinuous assets price processes”, available on www.rhsmith.umd.edu/Finance/dmadan McCulloch, J.H. (1978): “Continuous time processes with stable increments”, Journal of Business,51, 4, 601-620. Merton, R. (1976): “Option pricing when underlying stock returns are discontinuous”, Journal of Financial Economics, 3, 125-144. Monroe, I. (1978): “Processes that can be embedded in Brownian motion”, The Annals of Probability, 6, 1, 42-56.

21

Rubinstein, M. (1985): “Non-parametric tests of alternative option prices models”, Journal of Finance, 40, 455-480. Rubinstein, M. (1994): “Implied binomial trees”, Journal of Finance, 49, 3, 771-818. Schroder, M (1989): “Computing the Constant Elasticity of Variance option pricing formula”, Journal of Finance, 44, 1, 211-219.

22