The distribution of the maximum of a variance gamma process and path-dependent option pricing

Although numerical procedures often supply a required accuracy, closed-form expressions allow one to escape any accumulation of errors. In this paper, we discuss the possibility of obtaining explicit results for a variance gamma process. We derive the exact distribution of the maximum of the variance gamma process over a finite interval of time and establish the prices of path-dependent options including digital barrier, fixed-strike lookback, and lookback options. The obtained formulas are based on values of hypergeometric functions.