Pricing American Stock Options by Linear Programming

Journal of Computational Finance, 2002

The variational inequality formulation provides a mechanism to determine both the option value and the early exercise curve implicitly . Standard finite difference approximation typically leads to linear complementarity problems with tridiagonal coefficient matrices. The second order upwind finite difference formulation gives rise to finite dimensional linear complementarity problems with nontridiagonal matrices, whereas the upstream weighting finite difference approach with the van Leer flux limiter for the convection term yields nonlinear complementarity problems.

The model for pricing of American option gives rise to a parabolic variational inequality. We first use penalty function approach to reformulate it as an equality problem. Since the problem is defined on an unbounded domain, we truncate it to a bounded domain and discuss error due to truncation and penalization. Finite element method is then applied to the penalized problem on the truncated domain. By coupling the penalty parameter and the discretization parameters, error estimates are established when the initial data in H 1 0. Finally, some numerical experiments are conducted to confirm the theoretical findings.

This paper presents a numerical scheme that avoids iterations to solve the nonlinear partial differential equation system for pricing American puts with constant dividend yields. Upon applying a frontfixing technique to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the discrete nonlinear scheme. In the comparison with the solutions from articles that cover zero dividend and constant dividend yields cases, our results are found accurate. The current method is conditionally stable since the Euler scheme is used, the convergency property of the scheme is shown by numerical experiments.

2024

The objective of this research work was to determine the price of American put option: - When the underlying price does not experience jumps, i.e., the price is assumed to follow the Black-Scholes model and - When the underlying price experiences jumps, i.e., the price is assumed to follow the Merton Jump Diffusion model. An American option is more difficult to price than a European option because the American option has the possibility of early exercise at every instant of its lifetime. It’s never optimal to exercise early an American call option with non-dividend stock. Since we were interested in an early exercise problem with a non-dividend stock option, we opted for an American put option. A brief introduction to options is given in chapter one. This chapter also includes brief description of concepts, such as arbitrage and risk free rate. The chapter finally ends with the introductory summary of the American options. The second chapter introduces the American option and stopping time. We discuss the Black-Scholes model in section 2.1. This section also mentions about the boundary conditions and free boundary problem for American put, which lead to a linear complementarity problem (LCP) under certain assumptions. In section 2.2, we discuss how the solutions of the discrete LCPs are approximated using the operator splitting method. We briefly describes various methods of discretization applied to the Black-Scholes Greeks and boundary conditions in section 2.3. The application of implicit Crank-Nicolson finite difference method on Black-Scholes model is described in section 2.4. The final section of the chapter describes the Thomas algorithm using implicit Crank-Nicolson scheme to price an American put option. We have implemented this method using Matlab to compute the American put price in Black Scholes model. The chapter 3 deals with the pricing model for the American put option when the price of the underlying follows the jumps. In our thesis, we have considered the jump diffusion model proposed by Merton. The section 3.1 describes various integrals associated with the jump process. The section 3.2 introduces the Itô-Doeblin formula for a jump process. We discuss the application of Itô lemma to a jump diffusion process in section 3.3. The section 3.4 mentions the derivation of partial integro-differential equation for a jump diffusion process of an American put option. The section 3.5 describes the discretization approach to a partial integro-differential equation for jump diffusion process. The last section of this chapter describes the generalized implicit-explicit finite difference method to price an American put option under the Merton jump diffusion model. We have also implemented this method using Matlab to compute the American put price with jumps.

2001

In this paper we introduce two methods for the efficient and accurate numerical solution of Black-Scholes models of American options: a penalty method and a front-fixing scheme.

2016

In this manuscript we present two-grid algorithms for the American option pricing problem with a smooth penalty method where the variational inequality, associated with the optimal stopping time problem, is approximated with a nonlinear Black-Scholes equation. In order to compute the numerical solution of the latter unconstrained problem we must solve a system of nonlinear algebraic equations resulting from the discretization by e.g. the finite difference or the finite element method. We propose two-grid algorithms as we first solve the nonlinear system on a coarse grid with mesh size h c and further a linearized system on a fine grid with mesh size h f , satisfying h f = O((h c) 2 k), k = 1, 2,. . ., where k is the number of Newton iterations. Numerical experiments illustrate the computational efficiency of the algorithms.

This paper presents a numerical scheme that avoids iterations to solve the nonlinear partial differential equation system for pricing American puts with constant dividend yields. Upon applying a frontfixing technique to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the discrete nonlinear scheme. In the comparison with the solutions from articles that cover zero dividend and constant dividend yields cases, our results are found accurate. The current method is conditionally stable since the Euler scheme is used, the convergency property of the scheme is shown by numerical experiments.

Applied Mathematics and Computation, 2011

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black–Scholes partial differential equation, a predictor–corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution

SIAM Journal on Scientific Computing, 2009

The deterministic numerical valuation of American options under Heston's stochastic volatility model is considered. The prices are given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. A new truncation of the domain is described for small asset values while for large asset values and variance a standard truncation is used. The finite difference discretization is constructed by numerically solving quadratic optimization problem aiming to minimize the truncation error at each grid point. A Lagrange approach is used to treat the linear complementarity problems. Numerical examples demonstrate the accuracy and effectiveness of the proposed approach.

Computers & Mathematics with Applications, 2016

We consider the problem of pricing American options in the framework of a well-known stochastic volatility model with jumps, the Bates model. According to this model, the price of an American option can be obtained as the solution of a linear complementarity problem governed by a partial integro-differential equation. In this paper, a numerical method for solving such a problem is proposed. In particular, first of all, using a Bermudan approximation and a Richardson extrapolation technique, the linear complementarity problem is reduced to a set of standard linear partial differential problems (see, for example, Ballestra and Sgarra, 2010; Chang et al. 2007, 2012). Then, these problems are solved using an ad hoc pseudospectral method which efficiently combines the Chebyshev polynomial approximation, an implicit/explicit time stepping and an operator splitting technique. Numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing American options under the Bates model.