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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Abstract

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

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

Dates
First available in Project Euclid: 12 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015961161

Digital Object Identifier
doi:10.1214/aoap/1015961161

Mathematical Reviews number (MathSciNet)
MR1890062

Zentralblatt MATH identifier
1033.60051

Subjects
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

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

Citation

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. https://projecteuclid.org/euclid.aoap/1015961161


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References

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  • ROCQUENCOURT, BP 105 78153 LE CHESNAY CEDEX FRANCE E-MAIL: claude.martini@inria.fr