General smile asymptotics with bounded maturity (Q2832614)

The authors study the problem of connecting the price of a European option to the Black-Scholes implied volatility. Since exact formulas of implied volatility for a given model are typically out of reach, it is natural to find its asymptotic expansions. A key problem is to link the implied volatility explicitly to the distribution of the risk-neutral log-return \(X_t\), because the latter can be computed or estimated for many models. The asymptotical behavior of implied volatility can be connected to the tail probabilities for the log-returns. Thus, the authors provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. They provide a unified approach, which includes as special cases both the regime of extreme strike with fixed maturity and that of small maturity with fixed strike. Mixed regimes, where strike and maturity vary simultaneously, are also allowed. This flexibility yields asymptotic formulas for the volatility surface in open regions of the plane. The results are illustrated through applications to popular models, such as the Carr-Wu finite moment logstable model and Merton's jump diffusion model. Heston's model is also discussed.