The Annals of Applied Probability

Approximation of American Put Prices by European Prices via an Embedding Method

B. Jourdain and C. Martini

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In mathematical finance, the price of the so-called “American Put option” is given by the value function of the optimal-stopping problem with the option payoff $\psi: x \to (K - x)^+$ as a reward function. Even in the Black–Scholes model, no closed-formula is known and numerous numerical approximation methods have been specifically designed for this problem.

In this paper, as an application of the theoretical result of B. Jourdain and C. Martini [Ann. Inst. Henri Poincaré Anal. Nonlinear 18 (2001) 1–17], we explore a new approximation scheme: we look for payoffs as close as possible to $\psi$, the American price of which is given by the European price of another claim. We exhibit a family of payoffs $\hat{\varphi}_h$ indexed by a measure $h$, which are continuous, match with $(K - x)^+$ outside of the range $]K_*, K[$ (where $K_*$ is the perpetual Put strike), are analytic inside with the right derivative ( -1) at both ends. Moreover a numerical procedure to select the best $h$ in some sense yields nice results.

Article information

Ann. Appl. Probab., Volume 12, Number 1 (2002), 196-223.

First available in Project Euclid: 12 March 2002

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Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60G46: Martingales and classical analysis 65N21: Inverse problems 90A09 90C59: Approximation methods and heuristics

Optimal stopping free boundary problems inverse problems approximation methods American options European options


Jourdain, B.; Martini, C. Approximation of American Put Prices by European Prices via an Embedding Method. Ann. Appl. Probab. 12 (2002), no. 1, 196--223. doi:10.1214/aoap/1015961161.

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