Discrete Dynamics in Nature and Society

Volume 2018, Article ID 5908646, 12 pages

https://doi.org/10.1155/2018/5908646

## Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries

School of Mathematics and Finance, Xiangnan University, Chenzhou 423000, China

Correspondence should be addressed to Zhiqiang Zhou; nc.ude.efuws.4102@uohzqz

Received 24 March 2018; Accepted 13 August 2018; Published 4 September 2018

Academic Editor: Jorge E. Macias-Diaz

Copyright © 2018 Zhiqiang Zhou and Hongying Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Laplace transform method (LTM) has a lot of applications in the evaluation of European-style options and exotic options without early exercise features. However the Laplace transform methods for pricing American options have unsatisfactory accuracy and suffer from the instability. The aim of this paper is to develop a Laplace transform method for pricing American Strangles options with the underlying asset price following the constant elasticity volatility (CEV) models. By approximating the free boundaries, the Laplace transform is taken on a fixed space region to replace the moving boundaries space. After solving the linear system in Laplace space, Gaver-Stehfest formula (GSF) and hyperbola contour integral method (HCIM) are applied to compute the Laplace inversion. Numerical results show that the LTM-HCIM outperform the LTM-GSF in regard to the accuracy and stability for the option values.

#### 1. Introduction

The Laplace transform methods for option pricing originate from the idea of randomizing the maturity in [1]. The Laplace transform methods are applied in the pricing options without early exercise features: Pelsser [2] for pricing double barrier options, Davydov and Linetsky [3] for pricing and hedging path dependent options under constant elasticity of variance (CEV) models, Sepp [4] for pricing double barrier options under double-exponential jump diffusion models, Cai and Kou [5] for pricing European options under mixed-exponential jump diffusion models, and Cai and Kou [6] for pricing Asian options under hyperexponential jump diffusion models.

This technique is known to work well for options without early exercise features, but, for American-style options, one difficulty has been perceived that the Black-Scholes-Merton PDE only holds where it is optimal to retain the option. Mallier and Alobaidi [7] develop a partial Laplace transform method for pricing American options in which the location of the free boundary for all values of the transform variable has to be determined by solving nonlinear integral equations. The approach is only conceptually appealing but practically ineffective in numerical implementations as for each transform variable one has to solve the nonlinear integral equations. In fact Mallier and Alobaidi [7] do not give the numerical computations.

A simple framework for Laplace transform methods is introduced by Zhu [8] for pricing American options under GBM models. This framework is later developed for evaluation of finite-lived Russian options by Kimura [9], pricing American options under CEV models by Wong and Zhao [10] and by Pun and Wong [11], pricing American options under hyperexponential jump-diffusion model by Leippold and Vasiljević [12], pricing stock loan (essentially an American option with time-dependent strike) by Lu and Putri [13], and pricing American options with regime switching by Ma et al. [14]. But this framework suffers from a drawback that numerical Laplace inversion such as Gaver-Stehfest (GS) and Gaver-Wynn-Rho (GWR) algorithm are not stable. Pun and Wong [11] explain the complex calculation of special functions as one of the possible reasons causing instability.

Zhou et al. [15] develop a new Laplace transform method for solving the free-boundary fractional diffusion equations arising in the American option pricing. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace the moving boundaries space. Then the hyperbola contour integral method is exploited to restore the option values. The coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation and computational stability of the fast Laplace transform method are guaranteed.

In this paper, we develop the new Laplace transform method for solving American Strangles option pricing under CEV model. The Black-Scholes-Merton PDE of Strangles option is defined on the moving region , where is put side and is call side. The idea of the new method is modifying PDE into a new form on certain fixed region . Then performing the Laplace transform leads to a ODE which involves the inverse functions and of and , respectively. Using the finite difference method combined with the approximation of functions and , where the optimal parameters of the approximation are obtained by minimizing the prescribed residual error, we obtain the numerical solution of the ODE in the Laplace space. In the final step, both the inversion Laplace Gaver-Stehfest formula (GSF) and the hyperbola contour integral method (HCIM) are used to recover the option value and the free boundary.

Numerical examples show the inverse Laplace GSF is unstable if the number of discrete integral nodes is great than 20, so HCIM is an effective algorithm for computing Laplace inversion. However, the technics of spectral analysis in [15] cannot be applied over here, because the coefficients of PDEs arising from CEV Strangles option are not constant. This paper gives a convergence theorem by analysing the so-called Laplace transform iteration algorithm.

The remaining parts of this paper are arranged as follows: In Section 2, we develop the Laplace transform method for pricing American CEV Strangles option; in Section 3, we discuss the hyperbola contour integral method to compute inverse Laplace and give some stable conditions for HCIM. In Section 4, some examples are taken to conform the theoretical results of HCIM. Conclusions are given in the final section.

#### 2. Evaluation of American Strangles under CEV Model

Assume that the underlying asset price is governed by CEV (see, e.g., [16]):where is the risk-free interest rate, is the dividend yield, and is standard Brownian motion. represents the local volatility function and can be interpreted as the elasticity of . is the scale parameter fixing the initial instantaneous volatility at time , i.e., . If , the SDE (1) becomes the standard log-normal diffusion model or so-called geometric Brownian motion models (GBM). If , the SDE (1) nests the Cox-Ingersoll-Ross (CIR) model.

Let be the price of an American Strangles position written on an underlying asset with price at time and the payoffThis position is formed using a long put with strike and a long call with strike . Note that . Let , , and the early exercise boundary on the put side be denoted by , and the early exercise boundary on the call side be denoted by . It is known that satisfies the Black-Scholes PDE:with initial conditionand free boundary conditionsChiarella and Ziogas [17] apply Fourier transform technique to derive a coupled integral equation system for the Strangles free boundaries, and then a numerical algorithm is provided to solve this system.

To establish the new Laplace transform method proposed by Zhou et al. [15], we first note some properties of early exercise boundary for American Strangle option under the CEV model (1). It is known from [18–20] that the free-boundary functions and of American Strangles are continuously differentiable on the interval . We call the fact that is a decreasing function of variable , while is an increasing function of variable . DenoteThen we know, and satisfy the following ODE (similar to perpetual American option):

The values of and could be computed by secant method. Define uniform mesh and take larger enough number such that . The discrete form of (10) is as follows:for with . Assuming and , we have the pseudocode as Algorithm 1.