In this talk, we consider fixed and floating strike European style Asian call and put options. For such options, there is no convenient closed-form formula for the prices. Previously, Rogers and Shi, Vecer, and Dubois and Lelièvre have derived partial differential equations with one state variable, with the stock price as numeraire, for the option prices. In this talk, we derive a whole family of partial differential equations, each with one state variable with the stock price as numeraire, from which Asian options can be priced. Any one of these partial differential equations can be transformed into any other. This family includes four partial differential equations which have a particularly simple form including the three found by Rogers and Shi, Vecer, and Dubois and Lelièvre. Recently, Vecer derive a new PDE using the average asset as numeraire. We perform numerical comparisons of the five partial differential equations by Crank-Nicolson method and conclude, as expected, that Vecer’s equations and that of Dubois and Lelièvre do better when the volatility is low but that with higher volatilities the performance of all five equations is similar. Vecer’s equations are unique in possessing a certain martingale property and they perform numerically well or better than the others.
Joint work with C. BROWN (1), J. C. HANDLEY (2) and K. J. PALMER (3)
(1) Monash University, Caulfield, Victoria, Australia,
(2) University of Melbourne, Parkville, Victoria, Australia,
(3) National Taiwan University, Taiwan, retired