Implied volatility of basket options at extreme strikes

TL;DR

This paper analyzes the asymptotic behavior of implied volatility for basket call options at extreme strikes across various models, providing formulas and bounds that enhance understanding of tail risk in complex financial settings.

Contribution

It introduces new asymptotic formulas for implied volatility at extreme strikes in multidimensional models, including model-free results linked to copula dependence structures.

Findings

Asymptotic formula with error bounds for Black-Scholes model

Leading term asymptotics with stochastic time change

Model-free tail-wing formula relating to copula dependence

Abstract

In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a model-free tail-wing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function.