Volatility smile and stochastic arbitrage returns

TL;DR

This paper investigates how stochastic arbitrage opportunities influence option pricing, proposing a model that explains the volatility smile through random arbitrage effects and deriving pricing bands independent of detailed arbitrage statistics.

Contribution

It introduces an asymptotic pricing theory that accounts for stochastic arbitrage, providing a novel explanation for the volatility smile in option markets.

Findings

Pricing bands are independent of arbitrage return statistics.

Volatility smile can be explained by random arbitrage opportunities.

Asymptotic theory leverages different time scales of option prices and arbitrage returns.

Abstract

The purpose of this work is to explore the role that random arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a stationary ergodic random process rapidly varying in time. We exploit the fact that option price and random arbitrage returns change on different time scales which allows us to develop an asymptotic pricing theory involving the central limit theorem for random processes. We restrict ourselves to finding pricing bands for options rather than exact prices. The resulting pricing bands are shown to be independent of the detailed statistical characteristics of the arbitrage return. We find that the volatility ``smile'' can also be explained in terms of random arbitrage opportunities.