On Dynamic Measures of Risk

Abstract

In the context of complete continuous-time models for financial markets, we study dynamic measures for the risk asscociated with a given liability C: a random variable representing the payoff that has to be delivered at a future time T. The risk is defined as the supremum over a set of possible "real world" probability measures (corresponding to different mean return rates of the risky assets) of the minimal expected discounted loss at time T. By "loss" we mean the positive part of the difference between the liability C and the value of a dynamic admissible portfolio strategy. If the equivalent martingale measure is included in the set of possible subjective probability measures, and if the initial wealth x at our disposal is less than the Black-Scholes price C(0) of C, the risk value is equal to C(0)-x. This corresponds to borrowing C(0)-x at the initial time, and investing in risky assets according to the Black-Scholes portfolio for C. We also find explicit expressions for the optimal portfolio in the case we know the value of the mean return rates, as well as in a Bayesian framework in which we only have a prior distribution on the vector of the return rates. In the former case, and with only one risky asset, the optimal strategy depends only on the sign of the drift, and not on its value. Risk measures of this type were introduced by Artzner, Delbaen, Eber and Heath in a static setting, and were shown to possess certain desirable "coherence" properties, not all of which are shared by Value at Risk, or any other measures of risk.

JEL Classification: G11, G13