Short-maturity asymptotics for option prices with interest rate effects

We derive the short-maturity asymptotics for option prices in the local volatility model in a new short-maturity limit T -> 0 at fixed rho=(r-q)T, where r is the interest rate and q is the dividend yield. In the case of practical relevance rho being small, however, our result holds for any fixed rho. The result is a generalization of the Berestycki-Busca-Florent formula (Berestycki et al., 2002) [Asymptotics and calibration of local volatility models, Quantitative Finance 2, 61-69] for the short-maturity asymptotics of the implied volatility which includes interest rates and dividend yield effects of O(((r-q)T)n) to all orders in n. We obtain the analytical results for the ATM volatility and skew in this asymptotic limit. Explicit results are derived for the CEV model. The asymptotic result is tested numerically against the exact evaluation in the square-root model sigma(S)=sigma/S, which demonstrates that the new asymptotic result is in very good agreement with the exact evaluation in a wide range of model parameters relevant for practical applications.