Discrete-Time Pricing and Optimal Exercise of American Perpetual Warrants
in the Geometric Random Walk Model
Authors: R.J. Vanderbei, M.C. Pinar,
An American option (or, warrant) is the right but not the obligation
to purchase or sell an underlying equity at any time up to a
predetermined expiration date for a predetermined amount.
A perpetual American option differs from a plain American option in
that it does not expire. In this study, we solve the optimal stopping problem of
a perpetual American option (both call and put) in discrete time using
linear programming duality. Under the assumption that the underlying stock price
follows a discrete time and discrete state Markov process, namely a geometric random walk,
we formulate the pricing problem as
an infinite dimensional linear program using the excessive and majorant
properties of the value function. This formulation allows us to solve complementary
slackness conditions in closed-form, revealing an optimal stopping strategy
which highlights the set of stock-prices where the option should be exercised.
The analysis for
the call option reveals that such a critical value exists only in some cases,
depending on a combination of state-transition probabilities and
the economic discount factor (i.e., the prevailing interest rate) whereas
it ceases to be an issue for the put.
Full paper available on request.