Partial differential equations for Asian option prices

In this article, 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 paper, 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. Our analysis includes the case of a dividend yield; we also include the case of in progress Asian options with floating strike, whereby we discuss the new equation proposed by Vecer, which uses the average asset as numeraire. We perform an error analysis on the four special partial differential equations and Vecer’s new equation and find that their truncation errors are all of the same order. We also perform numerical comparisons of the five partial differential equations 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 as they perform numerically well or better than the others, must be considered the preferred choice.