We derive a form of the partial integro-differential equation (PIDE) for pricing American options under variance gamma (VG) process. We then develop a numerical algorithm to solve for values of American options under variance gamma model. In this study, we compare the exercise boundary and early exercise premia between geometric VG law and geometric Brownian motion (GBM). We find that GBM premia are understated and hence we conclude that further work is necessary in developing fast efficient algorithms for solving PIDE's with a view to calibrating stochastic processes to a surface of American option prices.
The payoffs of exotic options (e.g., up‐and‐out call options) are dependent on the time‐path of asset prices rather than the price of the asset at a fixed point in time. The authors of this article compare various models for calibrating volatility surfaces in order to price up‐and‐out call options.
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