Application of Homotopy Analysis Method to Option Pricing Under Lévy Processes

Abstract

Option pricing under the Lévy process has been considered an important research direction in the field of financial engineering, where a closed-form expression for the standard European option is available due to the existence of analytically tractable characteristic function according to the Lévy-Khinchin representation. However this approach cannot be applied to exotic derivatives (such as barrier options) directly, although a large volume of exotic derivatives are actively traded in the current options market. An alternative approach is to solve the corresponding partial integrodifferential equation (PIDE) numerically, which is, in fact, time-consuming and is not computationally tractable in general. In this paper, we apply the so-called homotopy analysis method (HAM) to solve the corresponding PIDE in a semi analytic form, being obtained from the following three steps: (1) Apply the Fourier transform to convert the PIDE to an ordinal differential equitation (ODE), and construct a differential system of ODEs. (2) Solve the system of ODEs, where each differential equation is shown to have an analytical solution. (3) Express the option price using the sum of infinite series, where each term may be expressed analytically and derived by applying Steps (1) and (2) recursively. To illustrate our technique more precisely, we take the variance gamma model as an example and provide the semi-analytic form. Numerical examples demonstrate a fast convergence of our proposed method to the prices of European and down-and-out call options with a few number of terms. Note that this method is easy to implement and can be applied to other types of options under general Lévy processes.

Modelling Crop Insurance Based on Weather Index Using The Homotopy Analysis for American Put Option

“…The implementation of homotopy method is common in sector such as science, finance, and engineering. The homotopy analysis is also can be used to do pricing under option using Levy process (Sakuma & Yamada, 2014) and stochastic volatility (Park & Kim, 2011).…”

Path Averaged Option Value Criteria for Selecting Better Options

“…Beginning in 1973, it was described that a mathematical framework for finding the fair price of a European option by Black and Scholes [1,2], several numerical methods have been presented for the cases where analytic solutions are neither available nor easily computable. See more details about numerical methods such as finite difference method (FDM) [3,4,5,6,7,8,9,10,11,12,13], finite element method [14,15,16], finite volume method [17,18,19], and a fast Fourier transform [20,21,22,23,24]. For convenience, we use the closed-form of the Black-Scholes equation in this work.…”

Accuracy, Robustness, and Efficiency of the Linear Boundary Condition for the Black-Scholes Equations

“…Therefore we need to use a numerical approximation. To obtain an approximation of the option value, one can compute a solution of the BS equations (1) and (2) using a finite difference method (FDM) [2][3][4][5][6][7][8], finite element method [9][10][11], finite volume method [12][13][14], a fast Fourier transform [15][16][17], and also their optimal BC [18].…”