Valuation of Employee Stock Options (ESOs) by means of Mean-Variance Hedging

We consider the problem of ESO valuation in continuous time. In particular, we consider models that assume that an appropriate random time serves as a proxy for anything that causes the ESO's holder to exercise the option early, namely, reflects the ESO holder's job termination risk as well as early exercise behaviour. In this context, we study the problem of ESO valuation by means of mean-variance hedging. Our analysis is based on dynamic programming and uses PDE techniques. We also express the ESO's value that we derive as the expected discounted payoff that the ESO yields with respect to an equivalent martingale measure, which does not coincide with the minimal martingale measure or the variance-optimal measure. Furthermore, we present a numerical study that illustrates aspects or our theoretical results

On the Structure of General Mean-Variance Hedging Strategies

We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure $P^{\star}$ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to $P^{\star}$ coincides with the variance-optimal martingale measure relative to the original probability measure $P$.Comment: Published at http://dx.doi.org/10.1214/009117906000000872 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Mean Variance Hedging in a General Jump Market

Abstract. We consider a financial market in which the discounted price process S is an R d -valued semimartingale with bounded jumps, and the variance-optimal martingale measure (VOMM) Q opt is only known to be a signed measure. We give a backward semimartingale equation (BSE) and show that the density process Z opt of Q opt with respect to P is a stochastic exponential which may be negative if and only if this BSE has a solution. For a general contingent claim H, we consider the following generalized version of the classical mean-variance hedging problem We represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward martingale equation (BME) and an appropriate predictable process δ both with a straightforward intuitive interpretation

Existence and uniqueness results for BSDEs with jumps: the whole nine yards

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.Comment: 48 pages, final version, forthcoming in the Electronic Journal of Probabilit