Asymptotic Analysis of Stock Price Densities and Implied Volatilities in Mixed Stochastic Models

In this paper, we obtain sharp asymptotic formulas with error estimates for the Mellin convolution of functions defined on (0,∞) and use these formulas to characterize the asymptotic behavior of marginal distribution densities of stock price processes in mixed stochastic models. Special examples of mixed models are jump-diffusion models and stochastic volatility models with jumps. We apply our general results to the Heston model with double exponential jumps and perform a detailed analysis of the asymptotic behavior of the stock price density, the call option pricing function, and the implied volatility in this model. We also obtain similar results for the Heston model with jumps distributed according to the normal inverse Gaussian law.