Short-maturity asymptotics for option prices with interest rates effects

We derive the short-maturity asymptotics for option prices in the local volatility model in a new short-maturity limit $T\to 0$ at fixed $\rho = (r-q) T$, where $r$ is the interest rate and $q$ is the dividend yield. In cases of practical relevance $\rho$ is small, however our result holds for any fixed $\rho$. The result is a generalization of the Berestycki-Busca-Florent formula 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 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 exact evaluation in the square-root model model $\sigma(S)=\sigma/\sqrt{S}$, which demonstrates that the new asymptotic result is in very good agreement with exact evaluation in a wide range of model parameters relevant for practical applications.