On Volatility of Prices in Arbitrage-Free Markets

This paper addresses the question of what one can learn from observing cross-sectional and time series variations in the volatility of prices in an arbitrage-free securities market. We introduce the notions of stochastic derivatives, marginal risk-adjusted growth rates, and marginal risk exposure in a single facotr economy. Using the martingale characterization of arbitrage-free prices, together with a martinagle representation formula due to Haussmann (18=978), we show that cross sectional variations in volatility can be linked to the marginal risk-adjusted growth rate in the economy and to changes in marginal risk exposure. We use prices ofoptions on the state variable to compute the stochastic derivative and hence the volatility of any state-contingent claim. We also study the information content of the term structure of volatility and apply the results to the Cox, Ingersoll and Ross (1985) model of interest rates.