## A moving boundary approach to American option pricing.(English)Zbl 1181.91300

Summary: This paper describes a method to solve the free-boundary problem that arises in the pricing of American options. Most numerical methods for American option pricing exploit the representation of the option price as the expected pay-off under the risk-neutral measure and calculate the price for a given time to expiration and stock price. They do not solve the related free-boundary problem explicitly. The advantage of solving the free-boundary problem is that it provides the entire price function as well as the optimal exercise boundary explicitly. Our approach, which we term the moving boundary approach, is based on using a boundary guess and the value associated with the guess to construct an improved boundary. It is also shown that on iteration, the sequence of boundaries converge monotonically to the optimal exercise boundary. Examples illustrating the convergence behavior as well as discussions providing insight into the method are also presented. Finally, we compare runtimes and speeds with other methods that solve the free-boundary problem and compute the optimal boundaries explicitly, like the front-fixing method, penalty method, method based on the integral representations and the method by M. Brennan and E. Schwartz [“The valuation of American put options”, J. Finance 32 (2), 449–462 (1977)].

### MSC:

 91G10 Portfolio theory 91G80 Financial applications of other theories 91G60 Numerical methods (including Monte Carlo methods)
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### References:

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