On volatility smile and an investment strategy with out-of-the-money calls

TL;DR

This paper examines a simple out-of-the-money call trading strategy, revealing its apparent profitability in the Black-Scholes-Merton model and showing how model adjustments introduce volatility smiles, highlighting structural inconsistencies.

Contribution

It demonstrates the incompatibility of the BSM model with CAPM when applying certain trading strategies and links the emergence of volatility smiles to model adjustments.

Findings

The strategy appears highly profitable under BSM assumptions.

Adjusting models to align with CAPM induces volatility smiles.

Structural considerations explain the emergence of volatility smiles.

Abstract

A motivating question in this paper is whether a sensible investment strategy may systematically contain long positions in out-of-the-money European calls with short expiry. Here we consider a very simple trading strategy for calls. The main points of this note are the following. First, the presented trading strategy appears very lucrative in the Black-Scholes-Merton (BSM) framework. In fact, it is such even to the extent that the BSM model turns out to be, in a sense, incompatible with the CAPM. Second, if one wishes to adapt these models together, then the adjustment of the consistent pricing rule (i.e. modifying state price densities) inevitably leads to some form of volatility smile and this is the main point of the paper. Moreover, these observations arise from purely structural considerations.