Fractional diffusion models of option prices in markets with jumps

Abstract

Most of the recent literature dealing with the modeling of financial assets assumes that the underlying dynamics of equity prices follow a jump process or a Lévy process. This is done to incorporate rare or extreme events not captured by Gaussian models. Of those financial models proposed, the most interesting include the CGMY, KoBoL and FMLS. All of these capture some of the most important characteristics of the dynamics of stock prices. In this article we show that for these particular Lévy processes, the prices of financial derivatives, such as European-style options, satisfy a fractional partial differential equation (FPDE). As an application, we use numerical techniques to price exotic options, in particular barrier options, by solving the corresponding FPDEs derived.

A predictor–corrector approach for pricing American options under the finite moment log-stable model

“…Following the work of Cartea and del-Castillo-Negrete [5], it is not difficult to show that V (x, t; α) should be governed by…”

“…As a result, −∞D α x U is not well defined for the current case; a truncation/regularization for the fractional operator is thus needed. It should be pointed out that the truncation approach outlined in [5] cannot be directly applied to the current case because U does not satisfy the homogeneous boundary condition at x = −∞. Alternatively, an approach very similar to the one proposed in [7,8] will be adopted.…”

“…Regarding the option pricing under the FMLS model, Carr and Wu [3] introduced this model to the literature, and showed its superior performance against several widely used alternatives. Cartea and del-Castillo-Negrete [5] considered the pricing of barrier options under the FMLS model purely numerically, by using a finite difference method. Recently, Chen et al [9] derived closed-form analytical solutions for European-style options under the same model, and analyzed quantitatively the impact of different tail indexes.…”

Numerical Solution of Fractional Black-Scholes Equation by Using the Multivariate Padé Approximation

“…Many powerful numerical and analytical methods have been presented in literature on finance. Among them, homotopy perturbation method with Sumudu transform and Laplace transform [7][8][9], homotopy analysis method [10], fractional variational iteration method [11], variational iteration method with Sumudu transform, finite difference method [12] and fractional diffusion models [13,14] are relatively new approaches providing an analytical and numerical approximation to Black-Scholes option pricing equation. Methods described in [15,16] are the other numerical methods, used in order to solve dynamic problems in elastic media and generalized semi-infinite programming.…”

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

“…2, 3, 4, 5, 6, 7, 8, 9, 10, applications and enhancements of these techniques were presented. The relevance of fractional calculus in the phenomenological description of anomalous diffusion has been discussed within applications of statistical mechanics in physics, chemistry and biology [11,12,13,14,15,16,17] as well as finance [18,19,20,21,22]; even human travel and the spreading of epidemics were modeled with fractional diffusion [23]. A direct Monte Carlo approach to fractional Fokker-Planck dynamics through the underlying CTRW requires random numbers drawn from the Mittag-Leffler distribution.…”