Pricing and hedging short-maturity Asian options in local volatility models

Abstract

This paper discusses the short-maturity behavior of Asian option prices and hedging portfolios. We consider the risk-neutral valuation and the delta value of the Asian option having a H\"older continuous payoff function in a local volatility model. The main idea of this analysis is that the local volatility model can be approximated by a Gaussian process at short maturity $T.$ By combining this approximation argument with Malliavin calculus, we conclude that the short-maturity behaviors of Asian option prices and the delta values are approximately expressed as those of their European counterparts with volatility $$\sigma_{A}(T):=\sqrt{\frac{1}{T^3}\int_0^T\sigma^2(t,S_0)(T-t)^2\,dt}\,,$$ where $\sigma(\cdot,\cdot)$ is the local volatility function and $S_0$ is the initial value of the stock. In addition, we show that the convergence rate of the approximation is determined by the H\"older exponent of the payoff function. Finally, the short-maturity asymptotics of Asian call and put options are discussed from the viewpoint of the large deviation principle.