Option Pricing in the Moderate Deviations Regime

TL;DR

This paper derives small-time asymptotic estimates for call option prices and implied volatility near expiry in diffusion models, bridging the gap between at-the-money and out-of-the-money regimes.

Contribution

It introduces a new asymptotic regime called 'moderately out of the money' and provides explicit formulas for option prices and implied volatility in this regime.

Findings

Explicit small-time asymptotic formulas for option prices.

Demonstrated accuracy of formulas with numerical examples in the Heston model.

Bridged the gap between at-the-money and out-of-the-money option regimes.

Abstract

We consider call option prices in diffusion models close to expiry, in an asymptotic regime ("moderately out of the money") that interpolates between the well-studied cases of at-the-money options and out-of-the-money fixed-strike options. First and higher order small-time moderate deviation estimates of call prices and implied volatility are obtained. The expansions involve only simple expressions of the model parameters, and we show in detail how to calculate them for generic local and stochastic volatility models. Some numerical examples for the Heston model illustrate the accuracy of our results.