## The Annals of Applied Probability

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

### 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

**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