Approximation of American put prices by European prices via an embedding method. (English) Zbl 1033.60051

Jourdain, B.; Martini, C.

Ann. Appl. Probab. 12, No. 1, 196-223 (2002).

American options differ from their European counterparts since the exercise date is at the holder’s disposal and not fixed in advance. This feature makes American options more valuable and more difficult to price since no closed-form solutions are available and one has to rely on approximate solutions obtained by solving numerically a free boundary problem.
The authors apply the theoretical result of the authors [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18, No. 1, 1–17 (2001; Zbl 0982.60028)] to propose a numerical scheme to price an American put option in the Black-Scholes model. The method, named amusingly YAAAP (Yet Another Approximation for the American Put), fares quite well when compared with a finite-difference method, and the LUBA and Ju methods. The main contribution of the scheme is the procedure itself. According to the theoretical result mentioned above, the American price of certain payoffs coincides with the European price of an associated claim. But, even for an American put with strike \(K\), the corresponding European payoff is unknown. Nevertheless, the authors define a parametrized family of claims that match the payoff of the American put at least outside \((k^*,k)\), where \(k^*\) is the perpetual strike, for which the European price can be computed. Finally, they propose a procedure to choose the “best” member of that family to approximate the American put value and delta. Therefore this approach also provides an approximate hedging strategy and could be used to price other options.

Reviewer: Miguel Ángel Mirás Calvo (Vigo)

MSC:

60G40	Stopping times; optimal stopping problems; gambling theory
60G46	Martingales and classical analysis
91B28	Finance etc. (MSC2000)

Keywords:

option pricing; American options; free boundary problems; approximation methods; optimal stopping

Citations:

Zbl 0982.60028

References:

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[9]	ROCQUENCOURT, BP 105 78153 LE CHESNAY CEDEX FRANCE E-MAIL: [e-mail]