# Most-likely-path in Asian option pricing under local volatility models

## TL;DR

This paper develops a novel approximation method for Asian option prices under local volatility models using a most-likely path approach, combining path integrals, asymptotic analysis, and large deviation theory.

## Contribution

It introduces a new framework for approximating Asian option prices via the most-likely path, integrating path integrals and asymptotics in local volatility models.

## Key findings

- Derived a path-integral expression for Asian option prices.
- Established the existence of the most-likely path using large deviation theory.
- Provided asymptotic formulas for small-time option pricing and implied volatility.

## Abstract

This article addresses the problem of approximating the price of options on discrete and continuous arithmetic average of the underlying, i.e. discretely and continuously monitored Asian options, in local volatility models. A path-integral-type expression for option prices is obtained using a Brownian bridge representation for the transition density between consecutive sampling times and a Laplace asymptotic formula. In the limit where the sampling time window approaches zero, the option price is found to be approximated by a constrained variational problem on paths in time-price space. We refer to the optimizing path as the most-likely path (MLP). Approximation for the implied normal volatility follows accordingly. The small-time asymptotics and the existence of the MLP are also recovered rigorously using large deviation theory.