Asymptotic analysis of European and American options with jumps in the underlying

In a jump-diffusion model for a single-asset market, we present an asymptotic analysis of European and American call options where the volatility is small compared with the drift terms. We precisely derive in the limit where volatility is negligible, relative to drifts, asymptotic expansion formulas for European, American and perpetual American call prices. As in the Black-Scholes model, we find that at leading order, the American call behaves in the same manner as a perpetual call option, except in a boundary layer about the option's expiry date. The derived expansion formulas contribute to the option pricing theory and provide a powerful tool to approximate call prices with a good accuracy, which allows to avoid the use of any numerical method.