We consider a finite state Markov chain which models uncertainties in a financial market.
A stochastic discount function is considered and prices of perpetual American options and
optimal exercise times are initially investigated using a stationary variational inequality.
Our next topic uses backward and reflected backward stochastic differential equations
as tools for pricing. To begin with, comparison results for backward stochastic differential
equations with Lipschitz driver are introduced.
We price European options in a market where the randomness is modelled by the finite
state Markov chain. A hedging strategy for a European option is shown to be a solution of
a backward stochastic differential equation whose driver is continuous and the fair price of
the option is derived as the minimal solution of such an equation. The existence of solutions
and the minimal solution of the backward stochastic differential equation with continuous
driver are established.
We extend the backward stochastic differential approach to the so-called reflected backward
stochastic differential equation, again with the Markov chain noise. This is used to find
a superhedging strategy for an American option in the presence of the stochastic discount
function mentioned above. Existence and uniqueness results for the solution of a reflected
backward stochastic differential equation are obtained.