The American put option in a one-dimensional diffusion model with level-dependent volatility

Abstract

The well-known results on the American put option problem obtained for the classical Black–Scholes model are generalized to the case of a diffusion model with level-dependent volatility. An early exercise premium representation of the value function of the American put option is established. The proof of this result is based on the properties of a stochastic integral with respect to an arbitrary continuous semimartingale over the predictable subsets of its zeros.

A nonlinear integral equation is derived for the optimal exercise boundary of the American put option. The uniqueness of the solution of this integral equation is established by introducing the corresponding Cauchy problem with discontinuous right-hand side and with a unique solution in the second order Sobolev space.

Keywords:

• Amercian put option
• Snell envelope
• Optimal stopping
• Zeros of semimartingales
• Sobolev space

2000 Mathematics Subject Classification::

• 60G48
• 60G40
• 91B28

Acknowledgements

We're grateful to an anonymous referee for pointing out the important preprint [18], in which the regularity properties of the value function are established based on the powerful methods of the theory of partial differential equations.