"Continuous Mixed-Laplace Jump Diffusion Models for Stocks and Commodities"

This paper proposes two jump diffusion models with and without mean reversion,for stocks or commodities, capable to fit highly leptokurtic processes. The jump component is acontinuous mixture of independent point processes with Laplace jumps. As in financial markets,jumps are caused by the arrival of information and sparse information has usually more importancethan regular information, the frequencies of shocks are assumed inversely proportional to their averagesize. In this framework, we find analytical expressions for the density of jumps, for characteristicfunctions and moments of log-returns. Simple series developments of characteristic functions arealso proposed. Options prices or densities are retrieved by discrete Fourier transforms. An empirical study demonstrates the capacity of our models to fit time series with a high kurtosis. The ContinuousMixed-Laplace Jump Diffusion (CMLJD) is fitted to four major stocks indices (MSWorld, FTSE, S&Pand CAC 40), over a period of 10 year... Abstract This paper proposes two jump di(cid:27)usion models with and without mean reversion, for stocks or commodities, capable to (cid:28)t highly leptokurtic processes. The jump component is a continuous mixture of independent point processes with Laplace jumps. As in (cid:28)nancial markets, jumps are caused by the arrival of information and sparse information has usually more importance than regular information, the frequencies of shocks are assumed inversely proportional to their average size. In this framework, we (cid:28)nd analytical expressions for the density of jumps, for characteristic functions and moments of log-returns. Simple series developments of characteristic functions are also proposed. Options prices or densities are retrieved by discrete Fourier transforms. An empirical study demonstrates the capacity of our models to (cid:28)t time series with a high kurtosis. The Continuous Mixed-Laplace Jump Di(cid:27)usion (CMLJD) is (cid:28)tted to four major stocks indices (MS World, FTSE, S&P and CAC 40), over a period of 10 years. The mean reverting CMLJD is (cid:28)tted to four time series of commodity prices (Copper, Soy Beans, Crude Oil WTI and Wheat), observed on four years. Finally, examples of implied volatility surfaces for European Call options are presented. The sensitivity of this surface to each parameters is analyzed.


Introduction
The success of the geometric Brownian motion is directly related to its analytical tractability. Prices of European and most of exotic options are calculable without intensive numerical computations. However, there are many piece of evidences proving that stocks returns are slightly asymmetric and have especially heavier tails than these suggested by a Brownian motion. Furthermore, an analysis of past stocks or commodities prices contradicts the assumption of continuity, inherent to a Brownian motion. Since the eighties, 1 many alternatives have been developed to incorporate asymmetric and/or leptokurtic features in stocks dynamics. In a rst category, we nd models with an innite number of jumps, obtained e.g. by subordinating a Brownian motion with an independent increasing Lévy process. This approach has been studied by Madan and Seneta (1990), Madan et al. (1998), Heyde (2000), Barndor-Nielsen O.E., Shephard (2001) or more recently by Hainaut (2016 b). In a second category, called jump diusion models, the evolution of prices is driven by a diusion process, punctuated by jumps at random interval. The two most common jump-diusion models for stocks are Merton's model with Gaussian jumps (1976) and the double-exponential jump diusion (DEJD) model, such as presented by Kou (2002) or Lipton (2002). In this last model, the amplitude of jumps is distributed as a doubly exponential random variable. As characteristic functions and Laplace transforms have closed form expressions, Kou andWang (2003, 2004) priced path dependent options and obtained probabilities of hitting times. Boyarchenko and Levendorskii (2002) and Levendorskii (2004) appraised American, Barrier and Touch-and-out options for the same process, using expected present value operators. Hainaut and Le Courtois (2014), Hainaut (2016 a) studied a switching regime version of the DEJD process, for credit risk applications. Cai and Kou (2011) replaced doubly exponential distributions by mixed exponential jumps. But this model, being over-parameterized, is of limited interest for econometric applications. On another hand, jump diusion processes are not appropriate to represent commodities. Their prices tend indeed to revert to long run equilibrium prices as illustrated in Bessembinder et al. (1995). Mean reversion is mainly induced by convenience yields. To remedy to this issue, Gibson and Schwartz (1990), Cortazar and Schwartz (1994) and Schwartz (1997)  This work contributes to previous researches in several directions. Firstly, it proposes parsimonious models with and without mean reversion, for stocks and commodities, capable to t highly leptokurtic processes. To achieve this goal, the return is modeled by a diusion and a sum of compound Poisson processes. Jumps are Laplace random variables and their frequencies of occurrences are inversely proportional to their average size.
This assumption is based on the observation that sparse information has a bigger impact on stocks or commodities prices than regular information. This model, called Continuous Mixed-Laplace Jump Diusion (CMLJD) duplicates a wide variety of leptokurtic distribution. It is adjustable to time series by likelihood maximization. A second appealing feature of CMLJD is that the number of compound Poisson processes is uncountable. CMLJD is in this sense an extension to continuous time of the Mixed Exponential Jump Diusion model. The CMLJD converges weakly to a diusion process punctuated by single jumps, distributed as a continuous mixture of Laplace random variables. In this setting, we infer closed form expressions for the density of jumps, for characteristic functions and moments of log-returns, both for CMLJD with and without mean reversion. Approached formulas of characteristic functions are available and can be used to speed up calculations. The last contribution is empirical. To illustrate its eciency, the CMLJD model is tted to four stocks indices (MS World, FTSE, S&P and CAC 40), over a period of 10 years. Whereas the mean reverting CMLJD model is calibrated to four commodities (Copper, Soy Beans, Crude Oil WTI and Wheat), observed over four years. These empirical tests conrm that the CMLJD outperforms the DEJD and Brownian processes. Probability density functions and European options prices are retrieved by a discrete Fourier's transform (DFT). Finally, we study the sensitivity of options implied volatility to parameters.
The rest of the paper is organized as follows. Section 2 introduces the Continuous Mixed-Laplace Jump Diusion (CMLJD) process and its properties. Section 3 develops the mean reverting CMLJD. Sections 4 and 5 review the DFT methods to compute the probability density functions and options prices. Finally, the section 6 presents an empirical study.

The Continuous Mixed Laplace Jump Diusion model
This work extends the mixed double exponential jump diusion model of Cai and Kou (2011) by considering an uncountable number of jump processes. The construction of this model proceeds with the following steps. Firstly, we present a process for asset log-returns with a nite mixture of Laplace jumps. So as to propose a parsimonious model, parameters are replaced by functions. Secondly, we nd the moment generating function of this process when the number of jump processes tends to innity and show that it converges weakly (or in distribution) toward a jump diusion process with a single jump component.
The asset price is a process denoted by (A t ) t≥0 and is dened on a probability space Ω, endowed with its natural ltration (F t ) t≥0 and a probability measure P . P is indierently the real historical measure or a risk neutral measure used for pricing purposes. The log return of A t noted X n k t , is such that where n k is a parameter that points out the number of jump processes involved in the dynamics of log-return. We assume that X n k t is driven by the following jump-diusion: where µ , σ are respectively the return, and volatility of the Brownian motion W t . Whereas K is constant and strictly above one (K > 1). The n k = K ∆k processes L k t , are compound Poisson processes parameterized by k, dened as the sum of N k t independent and identically distributed jumps noted J k : The counting processes N k t , have intensities equal to λ k ∆k for k = 1 : ∆k : K. The most popular distributions for jumps are either the Gaussian as in Merton (1976) or the double exponential distributions. However, as emphasized in Kou (2002) or in Kou andWang (2003, 2004), adding a single double exponential jump process to a diusion considerably improves the explanatory power of the model. Furthermore, the process remains analytically tractable for options pricing and ts relatively well the surface of implied volatility.
From an econometric point of view, the popularity of the double exponential jump diusion (DEJD) comes from its ability to exhibit reasonable leptokurticity and asymmetry.
Cai and Kou (2011) extend the DEJD by considering a mixture of double exponential jumps and study a dynamics similar to the one of equation (2.2). But the over-parameterization of this model constitutes a serious drawback. As our purpose is to extend their model with an uncountable number of jump processes to t processes with a high kurtosis, we remedy to this problem by doing two assumptions. Firstly, jumps J k are Laplace random variables. The Laplace law is a double exponential distribution, with symmetric positive and negative exponential jumps. Secondly, parameters are replaced by functions of the index k. The process obtained by this way is parsimonious: it counts the same number of parameters as the DEJD model of Kou (2002). We lose the asymmetry of the Cai and Kou process but our model exhibits a wider range of kurtosis, which is an important driver of option prices. Furthermore, empirical investigations concluding this work emphasizes that our approach outperforms the DEJD model. On the other hand, this specication entails that the jump part in the equation (2.2) is a martingale. The expectation of dX n k t is equal to the drift, µdt and we don't need to introduce a compensator for the jump processes.
More precisely, Laplace densities of J k depend on a parameter α k as follows: This is a double exponential distribution for which the probability of observing an upward or downward shock is 1 2 , with respective averages 1 α k and − 1 α k . With a such distribution, the expected jump is null, E J k = 0. The characteristic function of J k is also equal to the following quotient: On the other hand, jump processes L k t have a null mean and a variance given by: Furthermore L k t are martingales, given that increments of L k t are independent and such that: In order to limit the degrees of freedom, α k and λ k are parameterized with the following reasoning. Jumps are related to the ow of information. Good news or bad news, of dierent importance, arrive according to Poisson processes and prices change in response, according to an exponential jump. If we assume that sparse information has a bigger impact on prices than regular information, intensities λ k , and average absolute values of jumps 1 α k , respectively increase and decrease with the index k. The next functions for α k and λ k satisfy these features: where λ 0 , α 0 , β 1 and β 2 are positive constants. The positivity of β 1 and β 2 ensures that intensities λ k are inversely proportional to average jumps, 1 α k . Before any further developments, let us dene N t , a Poisson process with an intensity λ equal to Let us denote by B a random variable on the interval [1, K] and dened by the measure µ B (k) as follows: Let J be a random variable distributed as a continuous mixture of Laplace random variables: Then using nested conditional expectations, the characteristic function of J is such that: When the number of jumps components tends to innity, n k → ∞, the process X n k t converges in distribution (weak convergence) toward X t that is a jump diusion process. As stated in the next proposition, the jump component of X t is a unique compound Poisson process, with jumps distributed as a nite mixture of Laplace's random variables.
which is a process dened by: (2.10) where L t := Nt k=1 J is a compound Poisson process. J is dened by equation (2.8) and N t is a point process with an intensity given by equation (2.7). The characteristic function of X t is equal to: . (2.11) Proof. To prove this result, we show that characteristic functions of jump processes (2.2) converge to the one of equation (2.10). The L k and M j k (z) is the characteristic function of J k , such as dened by equation (2.4). Then, M L k t (z) is equal to: f or k = 1 : ∆k : K.
Given that L k t 's are independent, the sum of all jumps components till time t, L t := lim ∆k→0 k=1:∆k:K L k t , has the following characteristic function: Wherein, the product in this limit is equal to: If λ ∆k = k=1:∆k:K (λ k ∆k), this characteristic function becomes: where B ∆k denotes here a random variable dened on the interval [1, K] by a discrete measure µ B ∆k (k): is then the characteristic function of a jump process, of intensity λ ∆k , with jumps distributed as a nite mixture of Laplace random variables. Taking the limit of (2.12) when ∆k → 0, and according to the denition of λ, we get that lim ∆k→0 k=1:∆k:K On another hand, we have that The characteristic function of L t is then equal to: As there is an unequivocal correspondence between the moment generating function of a random variable and its probability density function, we have proven that lim ∆k→0 P k=1:∆k:K X n k t converges then weakly or in distribution toward X t .

7
This proposition reveals an interesting feature of our model and shared with all Mixed Exponential Models. Whatever the number of jumps components, the dynamics of the asset return always converges in a weak sense toward a jump diusion model, with a single compound Poisson process for which jumps are distributed as a mixture of distributions.
The next proposition presents a closed form expression for the density of the mixture of Laplace jumps.
Proposition 2.2. Let us dene a constant γ by: the probability density function of jumps J dened in equation (2.8) is given by the following expression: (2.15) where Γ (a, x) is the incomplete Gamma function dened as: 16) Proof. By construction, the probability density function of jumps is equal to Substituting k = α 0 k β 2 |x| to the integration variable k leads to the following relations k = 1 and to the next expression for the density: The integral in this last equation is the dierence between two incomplete Gamma functions, such as dened by equation (2.16).
Remark that X t , being a jump diusion process, belongs to the family of Lévy processes. Its innitesimal generator is then equal to for any function u(x) that is twice continuously dierentiable and where µ J (.) is given by (2.15). This generator is the key used later, to build the Feynman-Kac equation, satised by option prices. This equation is solved numerically by inverting the Fourier transform of option prices. But this approach requires to know the characteristic function of X t . The next proposition provides us this important result: where G(b, x) is the hypergeometric function: Proof. As mentioned in the proof of proposition 2.1, the characteristic function To calculate the integral´s 0 Given that the hypergeometric function, 2 F 1 (a, b, c, x) , is dened by The moments of X t are obtained by dierentiating the characteristic function, as stated in the next proposition. The skewness is null as by construction the distribution of X t is symmetric. But the kurtosis is always above 3. X t has then fatter tails than a normal distribution.
Proposition 2.4. The mean, variance, skewness and kurtosis of X t are respectively given by: Proof. The moments of X t are obtained by deriving the characteristic function with respect to z, In particular, The skewness and kurtosis are inferred from following relations: A helpful feature of the hypergeometric function for numerical purposes, is that it can be rewritten as an innite sum. In this case, the characteristic exponent admits the following alternative expression: Corollary 2.5. The characteristic exponent ψ(z) is equal to the sum: has the property to be equivalent to the innite series: This feature allows us to develop G(b, x) as follows: x 3 + . . . (2.25) and the dierence present in the characteristic exponent, becomes The mean reverting CMLJD model As mentioned in the introduction, a simple jump diusion is not appropriate to represent commodities as their prices revert to long run equilibrium prices. To insert this feature in assets dynamics, the following mean reversion mechanism is studied  b is the constant mean reversion level whereas a is the speed of mean reversion. On the other hand, Y t is non Gaussian Ornstein-Uhlenbeck process with Y 0 = X 0 , driven by the next SDE: where dZ t is a Lévy process, sum of a Brownian component and of a jump process: As previously, L t = Nt j=1 J j where N t is a Poisson process and J is a continuous mixture of Laplace's law. L t is the limit in the weak sense of the sum of processes L k t when ∆ k tends to zero. In this setting, Applying the Lévy Ito formula to e at Y t leads to the following expression for Y t , The statistical distribution of Y s is unknown but may be inferred from its characteristic function in numerical applications. The asset value at time t, conditionally to the available information at time s is given by: Given that Y 0 = X 0 , the characteristic function of the asset return is: The expectation is valued by the following result, proposed by Eberlein and Raible (1999): Proposition 3.1. Let Z t be a Lévy process having a cumulant transform dened as follows ψ(u) = log E(exp(uZ 1 )) , and let f : In particular, if Z t is a mixed Laplace process, its cumulant transform is equal to: where the integral´t 0ψ (ize −a(t−u) ) du is given by the next expression: and where G(b, x) is again the hypergeometric function: 13 Proof. A direct calculation leads to the following development: and the integral in the second term is equal tô Then the integral in equation (3.8) becomes: For any constant κ, several changes of variables similar to these done in proposition 2.3, lead to the following expression for the integral: . (3.10) Combining equations (3.8) (3.9) and (3.10) allows us to conclude the proof.
Notice that the dynamics of X t can be described by the following SDE We deduce from this last relation, its innitesimal generator is equal to for any function u(x) that is twice continuously derivable and where µ J (.) is given by (2.15). This generator is used for pricing purposes in appendix A.
The moments of X t are then obtained by dierentiating the characteristic function. Again, the skewness is null and the kurtosis is always above 3. Proposition 3.3. The mean, variance, skewness and kurtosis of X t are respectively given by: Proof. The moments of X t are obtained by dierentiating the characteristic function with respect to z, In particular, if g(t) denotes the following function, If the hypergeometric function is developed as an innite serie, the following result speeds up the numerical calculation of the moment generating function: Corollary 3.4. The integral´t 0ψ (ize −a(t−u) ) du is equal to the following sum: where ψ(z) is dened by equation (2.19). If X t is CMLJD with mean reversion, the function f Xt (.) is approached by where ψ(t, z) is dened by equation (3.5). This last relation can be computed by any DFT procedure.
Proof. By denition, the characteristic function is the inverse Fourier transform of the If X t is CMLJD, the density is retrieved by calculating the Fourier transform of M Xt (z) as The last equality comes from the fact that ψ(z) and ψ(−z) are complex conjugate. Equation (4.1) is deduced by approaching this last integral with the trapezoid rule´b is proven in the same way.
The CMLJD process without and with mean reversion are respectively identied by 7 parameters (µ, σ,α 0 , λ 0 , β 1 , β 2 , K) and 8 parameters (a, b , σ,α 0 , λ 0 , β 1 , β 2 , K). Both processes cannot be tted to time series by the method of moments matching, without setting a priori some parameters. An alternative consists to calibrate the process by loglikelihood maximization. We adopt this approach in numerical illustrations to t CMLJD processes. The matlab code implementing the equation (

Options pricing
Firstly, we consider that log-returns are ruled by a CMLJD process without mean reversion.
The pricing of nancial securities is done under a risk neutral measure. Under this measure, the discounted price process is a martingale and the expected return is equal to the risk free rate, r to avoid any arbitrage. X t is then dened by parameters (r, σ, α 0 , β 1 , λ 0 , β 2 , K).
The most common methods used for pricing derivatives are based on Fourier and Inverse Fourier transforms. We denote them respectively by: The Fourier transform maps spatial derivatives ∂ ∂x into multiplications in the frequency domain. As shown in the next result, this feature allows us to price any European derivatives.
Proposition 5.1. In the CMLJD model, the price at time t and when X t = x, of an European derivatives delivering a payo V (T, X T ) at maturity T, is given by Proposition 5.2. In the CMLJD model with mean reversion, the price at time t and when Y t = y, of an European derivatives delivering a payo V (T, Y T ) at maturity T, is given by ..N , is approached by the following DFTs sum: Proof. The Fourier transform is approached by the following sum The function being known at point ω k = (k − 1)∆ ω , approaching this last integral with the trapezoid rule, leads to: which is equivalent to (5.3).
Proposition 5.4. Let N and ∆ y = 2ymax N −1 be respectively the number of steps used in the Discrete Fourier Transforms (DFT) and the step of discretization. Let us dene ∆ ω = 2π N ∆y (5.2), at points y j = − N 2 ∆ x + (j − 1)∆ y , for j = 1...N , is approached by the following DFTs sum: Proof. By a change of variable y = e a(T −t) y and if ∆ y = ∆ y , the Fourier transform of the terminal is approached by a DFT as follows As ∆ ω ∆ y = 2π The function being known at point ω k = (k − 1)∆ ω , if we approach this last integral with the trapezoidal rule, we infer that: and we can conclude. . The CMLJD is also compared to the DEJD model of Kou (2002) that postulates the following dynamics for the log-return: where L DEJ As the CMLJD, the DEJD does not admit analytical expression for its probability density function. The same DFT algorithm is used to compute it. Fitted parameters, standard er-    . It is well known that commodities exhibit cointegration in prices. Like equity prices they are also exposed to sudden jumps of price. However, unlike equities, commodities tend to revert to long run equilibrium prices. Parameters are tted by maximization of the log-likelihood. However the calculation of this log-likelihood requires a pre-treatment of data to reduce the computation time. Firstly, we calculate the log-return where P j is the price of commodity on the j th day. The process Y j is next obtained by subtracting from X j the function ϕ(t j ) as dened by equation (3.2), If ∆ t represents one day of trading, according to equation (3.3), the process V j is distributed as follows The distribution of V j is nally retrieved by inverting numerically the characteristic function of´∆ t 0 e −a(t−u) dZ u , that is reported in proposition 3.     This article proposes two parsimonious models for stocks or commodities, driven by a diffusion and a mixture of Laplace jumps processes. These dynamics aim to replicate time series for which increments have a distribution with a high kurtosis. Despite the somewhat lengthy expressions of characteristic functions, the CMLJD remains analytically tractable and its density can be retrieved numerically by a discrete Fourier transform. Its ability to duplicate leptokurtic processes makes it eligible for option pricing or risk management.
As underlined by the empirical study, the CMLJD processes t stocks and commodities better than a Brownian motion or a DEJD. Furthermore, the parameters estimated by log-likelihood maximization show stability and consistency over a variety of assets. On the other side, CMLJD processes generate realistic surfaces of implied volatilities. In the CMLJD model with mean reversion, we observe a strong asymmetry in this surface, due to the reversion of prices.
There exist many possibilities for further researches. Firstly, the main criticism about the CMLJD is that it does not capture the asymmetry of returns. It is possible to remedy to this problem by considering double exponential jumps. However, this increases the number of parameters and requires additional parameterization. Secondly, we only consider power functions for α k and λ k . It is probably possible to nd another type of functions for which we can obtain a closed form expression for the limit of the characteristic function of