Risk Premium Impact in the Perturbative Black Scholes Model

TL;DR

This paper examines how the risk premium affects option pricing within the Perturbative Black Scholes model, revealing that the implied volatility skew depends on the risk premium and underlying asset behavior.

Contribution

It provides a theoretical analysis of the impact of risk premium on implied volatility skew in the Perturbative Black Scholes model, extending previous models to include drift effects.

Findings

Implied volatility exhibits a skewed structure influenced by the risk premium.

The position of the minimum implied volatility depends on the risk premium parameter.

Under certain conditions, the asset price behaves as a sub-martingale, affecting volatility dynamics.

Abstract

We study the risk premium impact in the Perturbative Black Scholes model. The Perturbative Black Scholes model, developed by Scotti, is a subjective volatility model based on the classical Black Scholes one, where the volatility used by the trader is an estimation of the market one and contains measurement errors. In this article we analyze the correction to the pricing formulas due to the presence of an underlying drift different from the risk free return. We prove that, under some hypothesis on the parameters, if the asset price is a sub-martingale under historical probability, then the implied volatility presents a skewed structure, and the position of the minimum depends on the risk premium $\lambda$.