Abstract
This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
Highlights
American call and put options are one of the most traded products in financial markets
Let us restrict our attention to the American option pricing problem in the Black Scholes framework
We introduce an auxiliary parameter in the American option pricing problem and we deduce power series expansions in this parameter of the option price and of the corresponding free boundary
Summary
American call and put options are one of the most traded products in financial markets. They determine an approximation of the free boundary solving numerically a nonlinear equation that depends from the time variable as a parameter This approximate solution of the American option pricing problem is called Barone-Adesi, Whaley formula and is widely used in the financial markets by practitioners. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter These series expansions are a formal solution of the American option pricing problem. For example the improvement obtained with the higher order terms of the expansions is significant when we compare the approximations of the free boundary of the American option pricing problem obtained using the Barone-Adesi, Whaley formula with those obtained using the n-th order truncated power series expansions, n = 1, 2 (see Section 4, Table 1, Table 4 and Figure 2). These results are compared with those discussed in the scientific literature obtained with some alternative methods to solve the American option pricing problem
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