On volatility of prices in arbitrage-free markets

Summary This paper addresses the question of what one can learn about the dynamics of an economy 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 factor economy. We show that future variations in the state of the economy are due to two independent sources: the marginal risk-adjusted growth rate and the changes in marginal risk exposure. Using the martingale characterization of arbitrage-free prices, together with a martingale representation formula due to Haussmann (1978), we show that cross sectional variations in price volatility of assets with linear payoffs can be used to identify the sum of these two sources. Measurements of price volatility for assets with linear payoffs are not sufficient for complete identification of the independent determinants of possible future variations in the economy. However, using the volatility of prices of options on the state variable, we can identify the stochastic derivative and hence compute the price volatility of any path independent contingent claim.I am grateful to David Kreps and Kenneth Singleton for helpful conversations and to an anonymous referee for useful remarks. Financial support from Stanford Graduate School of Business Faculty Fellowship is gratefully acknowledged.