Pricing American Options
Background
The publication of the celebrated work of Black and Scholes in 1973,
has revolutionised the world's financial markets. By considering a
simple model for the price of a financial asset, they were able to
obtain a simple formula for the fair price of a European call option
on this asset. The European call option is a simple financial
derivative which gives the holder the right, but not the obligation, to
buy a unit of asset at a fixed time T, for a fixed price K. The
holder of such an option will receive max(value at time T - K,0) at maturity and
the fundamental question answered by Black and Scholes was, what was
this option worth? By assuming that there were no arbitrage in the
market they were able to obtain a unique price for the option which
would allow a bank to take this money and, by using a hedging strategy,
guarantee to pay the option at maturity.
An American option gives the holder the right to exercise it at any
time up until maturity. This makes for a more complicated pricing
problem. The American call on a non-dividend paying asset has the same
price as its European counterpart but in general there is no closed
form pricing formula for an American option. The techniques employed
for evaluation of prices are PDE or binomial tree techniques.
However, such techniques are useless when the dimension of the
underlying is large. For instance an American put option on the minimum of a
basket of more than 4 assets presents a challenge to compute an
accurate numerical price. The usual approach in the case of a European
option is to use a Monte Carlo technique. This simulation technique is
not well suited to American options where the price is determined
through a backwards induction approach.
Recent Work
There are a number of attempts to develop Monte Carlo algorithms for
American options some of which are more successful than others. Some
simple algorithms can be shown to converge to the wrong solution and
other approaches only yield bounds. The aim of the project is to
consider the convergence problems with the available techniques.