## The Annals of Applied Probability

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

#### 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

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

Digital Object Identifier
doi:10.1214/aoap/1015961161

Mathematical Reviews number (MathSciNet)
MR1890062

Zentralblatt MATH identifier
1033.60051

#### 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

#### References

• [1] BARLES, G., BURDEAU, J., ROMANO, M. and SAMSON, N. (1993). Estimation de la frontière libre des options américaines au voisinage de l'échéance. C. R. Acad. Sci. Paris Sér. I Math. 316 171-174.
• [2] BRANDT, A. and CRYER, C. W. (1983). Multigrid algorithm for the solutions of linear complementarity problems arising from free boundary problems. SIAM J. Sci. Statist. Comput. 4 655-684.
• [3] BROADIE, M. and DETEMPLE, J. (1996). American option caluation: New bounds, approximations, and a comparison of existing methods. Review of Finance Studies 9 1211-1250.
• [4] CARR, P., JARROW, R. and MYNENI, R. (1992). Alternative characterizations of American put options. Math. Finance 2 87-106.
• [5] JOURDAIN, B. and MARTINI, C. (2001). American prices embedded in European prices. Ann. Inst. H. Poincaré Anal. Nonlinear 18 1-17.
• [6] JU, N. (1996). Pricing an American option by approximating its early exercise boundary as a multipiece exponential function. Review of Finance Studies 11 627-646.
• [7] LAMBERTON, D. (1995). Critical price for an American option near maturity. In Seminar on Stochastic Analysis, Random Fields and Applications 353-358. Birkhäuser, Basel.
• [8] LITTLE, T., PANT, V. and HOU, C. (2000). An integral representation of the early exercise boundary for American put options. J. Comput. Finance 3.
• ROCQUENCOURT, BP 105 78153 LE CHESNAY CEDEX FRANCE E-MAIL: claude.martini@inria.fr