Interesting paper by Joseph L. McCauley and Gemunu H. Gunaratne (PDF) out of Houston.

Abstract

In their path-finding 1973 paper Black and Scholes presented two separate derivations of their famous option pricing partial differential equation (pde). The second derivation was from the standpoint that was Black’s original motivation, namely, the capital asset pricing model (CAPM). We show here, in contrast, that the option valuation is not uniquely determined; in particular, strategies based on the delta-hedge and CAPM provide different valuations of an option although both hedges are instantaneously riskfree. Second, we show explicitly that CAPM is not, as economists claim, an equilibrium theory.

But wait, there’s more!

The notion of increased expected return via increased risk is not present in the delta-hedge strategy, which tries to eliminate risk and to minimize return. We see now that the way that options are priced (even in the riskfree Gaussian returns case) is strategy dependent, which may be closer to the notion that psychology plays a role in trading. The CAPM option pricing equation depends on the expected returns for both stock and option, and so differs from the original Black-Scholes equation of the delta-hedge strategy. There is no such thing as a universal option pricing equation independent of the chosen strategy, even if that strategy is reflected in this era by the market. Economics is not like physics (non-thinking nature), but depends on human behavior and expectations.

In a paper to follow we will show how to use the empirical distribution of returns, which is far from Gaussian, to construct a stochastic differential equation and corresponding Fokker-Planck equation that not only reproduces the empirical distribution but
also prices options correctly without the use of ‘implied volatility’.

On CAPM and Black-Scholes: Differing risk-return strategies (PDF)