Out-of-the-Money Calls, Volatility Smiles, and BSM-CAPM Consistency

Summary

This note considers a simple strategy that holds short-expiry, out-of-the-money European calls. It argues that the strategy appears unusually attractive under the Black-Scholes-Merton framework, raising a potential inconsistency between that model and the Capital Asset Pricing Model. The discussion uses this tension to examine how option pricing assumptions relate to investment returns.

The proposed resolution is structural: combining the models consistently requires adjusting the pricing rule, expressed through changes to state price densities. The note says such an adjustment necessarily produces some form of volatility smile. It presents this as a theoretical implication, rather than relying on a reported empirical test. The available description gives no trade construction details, return estimates, calibration procedure, or market data, so it does not establish that buying these calls is profitable in practice. Its main lesson is about consistency among pricing frameworks and the implications for implied volatility across strikes.

Key ideas

• The note examines a strategy holding short-dated, out-of-the-money European calls.
• The strategy appears unusually profitable within the Black-Scholes-Merton framework.
• That implication is presented as potentially inconsistent with the Capital Asset Pricing Model.
• Consistent pricing adjustments involve changing state price densities.
• The note argues that such changes imply a volatility smile, but gives no empirical trading results.

Tags

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


# 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 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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.