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

TL;DR

This paper analyzes the short-maturity behavior of Asian options in local volatility models, deriving asymptotic formulas for prices and deltas using Gaussian approximations and Malliavin calculus, with convergence rates linked to payoff regularity.

Contribution

It introduces a Gaussian approximation approach for short-maturity Asian options in local volatility models, connecting their behavior to European options and quantifying convergence based on payoff smoothness.

Findings

Asian option prices approximate European counterparts at short maturity.

Convergence rate depends on the H"older exponent of the payoff.

Asymptotic behavior analyzed via large deviation principles.

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.