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.