Appl Math Optim 39:411–422 (1999)
1999 Springer-Verlag New York Inc.
American Options Exercise Boundary When the
Volatility Changes Randomly
CEREMADE, Universit´e Paris IX Dauphine,
Place du Mar´echal de Lattre de Tassigny,
75775 Paris Cedex 16, France
CREST, 15 bd Gabriel P´eri,
92240 Malakoff cedex, France
Communicated by A. Bensoussan
Abstract. The American put option exercise boundary has been studied exten-
sively as a function of time and the underlying asset price. In this paper we analyze
its dependence on the volatility, since the Black and Scholes model is used in prac-
tice via the (varying)implied volatility parameter. We consider a stochastic volatility
modelfor theunderlying assetprice.We provideanextensionof theregularityresults
of the American put option price function and we prove that the optimal exercise
boundary is a decreasing function of the current volatility process realization.
Key Words. Incomplete markets, Optimal stopping, Viscosity solutions.
AMS Classiﬁcation. 60G40, 90A09, 93E20.
The fundamental work by Black and Scholes (1973, BS hereafter) introduced a new
approach for contingent claims valuation,based on a portofolio strategy which duplicates
the future payments structure; the arbitrage price of the contingent claim coincides with
the duplicating portfolio price. BS assumed that the underlying asset price is generated
by a constant volatility diffusion process, and derived an explicit formula for a European