Measuring risk with the Bodie put when stocks exhibit mean reversion

Abstract

A Bodie put is an option that pays the amount a stock investment underperforms a risk-free investment. Bodie (1995, Financial Analysts Journal, 51(3), 18-22) argued that the price of such a put measures the risk of the stock it insures. Since the price of this put increases as the time to expiration is lengthened, it follows that stocks do not become safe in the long run. Critics challenged Bodie's conclusions, reasoning that mean reversion, by reducing long-run volatility, would make long-run Bodie puts worth less than shorter ones. In this paper, the author examines the suitability of the Bodie put option price as a measure of risk and the pricing of the Bodie put in a general binomial model that accommodates mean reversion. It is shown that, under any binomial stock process, a long-term Bodie put must be worth more than a shorter one. Examples are presented in which the stock appears to become safer over a longer horizon, but the Bodie put nonetheless becomes more expensive. Consequently, it is concluded that Bodie's argument stands.