Maturity-independent risk measures

Despite all the recent work in the area of risk measurement, there is still a number of theoretical, as well as practical, questions left unanswered. The one we focus on in the present paper deals with the problem one faces when the maturity (horizon, expiration date, etc.) associated with a particular risky position is not fixed. We take the view that the mechanism used to measure the risk content of a certain random variable should not depend on any a priori choice of the measurement horizon. This is, for example, the case in complete financial markets. Indeed, consider for simplicity the Samuelson (Black-Scholes) market model with zero interest rate and the procedure one would follow to price a contingent claim therein. The fundamental theorem of asset pricing tells us to simply compute the expectation of the discounted claim under the unique martingale measure. There is no explicit mention of the maturity date of the contingent claim in this algorithm, or, for that matter, any other prespecified horizon. Letting the claim’s payoff stay unexercised for any amount of time after its expiry would not change its arbitrage-free price in any way. It is exactly this property that, in our opinion, has not received sufficient attention in the literature. As one of the fundamental properties clearly exhibited under market completeness, it should be shared by any workable risk measurement and pricing procedure in arbitrary incomplete markets.