Probability-conservative simulation for Lévy financial model by a mixed finite element method

Abstract

We propose an expanded mixed finite element method (EMFEM) with sharp recognition for such intrinsic natures of a Markov process as the probability conservation, the Chapman-Kolmogorov equation, skewness and excess kurtosis for the fractional-order advection diffusion equation (FADE) oriented in financial engineering. We improve the result in [11] for purely fractional diffusion model by modifying the domain of one bilinear form so that it is well defined for the appearance of the advection term, and the result in [13] for second order elliptic models by using lower-order finite element spaces to reduce computation costs. The solvability and optimal convergence rates of the EMFEM are proved by showing the coerciveness and the inf-sup condition for the modified bilinear forms over the lower-order finite element spaces, and thus the systematically mathematical theory of the EMFEM for general FADEs are developed. Numerical experiments are conducted to confirm these theoretical findings.

Introduction

The well-known Black-Scholes (B-S) model plays a crucial rule in pricing and hedging derivative securities in financial market. It uses the normal distribution to fit the log returns of the underlying, and consequently its probability density function (pdf) of the log returns solves the second order diffusion equation in macroscopic sense. Unfortunately, the data suggest that the log returns of stocks/indices are skewed and have excess kurtosis, and thus, are not normally distributed and do not follow a second-order diffusion equation (see Section 4 of [38]).

As a remedy, Lévy process, a kind of more sophisticated infinitely divisible distributions, was employed to replace the normal to represent skewness and excess kurtosis [4], for examples, $VG,NIG$ and $CGMY$ models [1], [2], [3]. By an application of the compound Poisson approximation, Lévy representation and the semigroup theories, the generator of this kind of Lévy process, endowed with the Lévy measure as two-side α-Pareto distribution, is represented by a fractional differential operator of order α, and thus the pdf of the Lévy process solves a kind of fractional differential equations [27].

From this point of view, it is reasonable to develop a numerical algorithm with sharp recognition for the conservation of the probability, the Chapman-Kolmogorov equation, skewness and excess kurtosis, the intrinsic natures of the Markov model driven by a Lévy process. For this purpose, we apply the local-mass-conservation nature of mixed finite element method and propose an expanded mixed finite element method (EMFEM) for the Lévy financial model, which can recognizes sharply those intrinsic natures so as to understand better the mechanism of Lévy financial model macroscopically. Furthermore, the proposed EMFEM may use lower-order finite element spaces with the inf-sup condition, and thus reduces computation cost and improves the existing mathematical theories for EMFEMs.

The main objectives of this article are to: (1) define the expanded mixed finite element method (EMFEM) for the time dependent fractional advection-diffusion equation of order α. The proposed method can capture accurately the skewness and excess kurtosis, as well as preserve the probability and the Chapman-Kolmogorov equation which may be interpreted as the global and local mass conservation for general mass transport processes and may not be achieved by those traditional finite difference [9], [24], [36], [42], spectral method [21] and finite element methods [18], [20], [22], [23], [37], [41]. (2) improve the results in [11] for purely fractional diffusion models by modifying the domain of the bilinear form $\mathbb{A}(\cdot ,\cdot )$ so that it is well defined to adapt to the appearance of the advection term, and the results in [13] for second order elliptic models by using lower-order finite element spaces ${\mathbf{H}}_h$ to retrieve the commonly used piecewise constant space, and thus reduce computation costs. (3) prove the solvability and optimal convergence rates of the EMFEM by showing the coerciveness and the inf-sup condition for the modified bilinear forms over the lower-order finite element spaces, and thus the systematically mathematical theory of the EMFEM for general FADEs is developed. (4) conduct two numerical experiments to confirm these theoretical findings.

The outline of the remaining part is as follows: in Section 2, we revisit the B-S model, the fractional advection diffusion model driven by Lévy process and present the Chapman-Kolmogorov equation. In Section 3, we shall develop the expanded mixed formulation with two appropriately defined bilinear forms $\mathbb{A}(\cdot ,\cdot )$ and $\mathbb{B}(\cdot ,\cdot )$, and prove the solvability of the formulation and its equivalence to the fractional model by showing the coerciveness and the inf-sup condition of the two bilinear forms. Section 4 is devoted to develop the backward Euler discretization for the expanded mixed formulation and prove its solvability and the temporal convergence rate. Section 5 is to design the EMFEM and prove its solvability and convergence rates based on the newly defined finite element space ${\mathbf{H}}_h$. The numerical experiments are conducted in Section 6 and the concluding remark is given in the last section.