Dynamic Asset Pricing Theory with Uncertain Time-Horizon

An investment horizon is in practice not frequently known with certainty at the initial investment date. This paper addresses the problem of pricing and hedging a random cash‡ow received at a random date in a general stochastic environment. We ...rst argue that speci...c timing risk is induced by the presence of an uncertain time-horizon if and only if the random time under consideration is not a stopping time of the ...ltration generated by prices of traded assets. In that context, we provide an explicit characterization of the set of equivalent martingale measures, as well as a necessary and su¢cient condition for a convenient separation between adjustment for market risk and timing risk. We also present price bounds consistent with perfect replication in the absence of arbitrage for an asset paying o¤ a random amount at a random time. As is often the case, such bounds are actually too wide to be of any practical use and we also consider several choices (minimal martingale measure, minimum entropy measure) for narrowing down to one the number of equivalent martingale measures.