An Asian option is a financial contract with payoff depending on the average of an asset price over a predetermined period. It is known to be one of the hard computational problems in mathematical finance as the PDE derived for its price is two-dimensional degenerate, and creates many numerical problems. Monte Carlo simulation methods work, but they are slow. We exploit a recently proposed formulation and use the simple form of the derived PDE to custom-design a fast and accurate numerical procedure. The PDE admits an explicit, closed-form solution when we replace the non-smooth terminal condition by a polynomial, but the solution becomes prohibitively complex as the degree of the polynomial increases. We prove the existence of a weak solution to the terminal value problem and use Hormander's Hypoellipticity Theorem to prove that the solution is smooth. Through a priori bounds on the solution and artifacts of the financial model, we show that the solution is given in closed form to the right of the degeneracy curve and we justify the boundary conditions that we use in our numerical solutions. Moreover, through comparison principles and numerical investigation, we examine the sensitivity of the resulting prices as we vary model parameters and contrast the results with those for vanilla options. We demonstrate that our method's results are consistent with those in the literature, but that our method performs better and faster than other PDE pricing methods.Ph.D.FinanceMathematicsPure SciencesSocial SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/124042/2/3121914.pd
