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

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 a short expiry. Here we consider a very simple trading strategy for calls. The main points of this paper are the following. First, the presented trading strategy appears very lucrative in the Black–Scholes–Merton (BSM) framework. In fact, to such an 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, which is the main point of the paper. Moreover, these observations arise from purely structural considerations and only one hypothetical equity or equity index with many European options on it (such as S&P500) is required.