Abstracts

We revisit the optimal stopping problem using Heath-Jarrow-Morton (HJM) approach. The HJM method was originally introduced to model the fixed-income market by Heath et al. (1992). More recently, it was implemented in equity market models by Schweizer and Wissel (2008), and Carmona and Nadtochiy (2008, 2009). Prior work has mainly focused on European derivative pricing, while in this paper we apply the HJM philosophy to American derivative pricing with a focus toward solving optimal stopping problems in general. As a counterpart to forward rate for the fixed-income market and forward volatility for the equity market, we introduce forward drift for the optimal stopping problem. The standard results for HJM-type models are confirmed for the forward drift dynamics: the drift condition and the spot consistency condition. More interestingly, we discover a forward stopping rule that is fundamentally different from the classical stopping rule based on backward induction. Although simple, the binomial model enables us to clearly illustrate the calculation difference between the two approaches in finding the optimal stopping times and show that the binomial tree only needs to be built up to the optimal stopping time in the forward approach, i.e., the decision to stop does not depend on the future evolution of the tree. In the classical backward approach, it is usually much easier to obtain solutions for infinite-horizon problems than finite-horizon problems. With the Black-Scholes model we highlight another novelty for the forward approach: there is fundamentally no difference in terms of level of difficulty for the finite or infinite-horizon case. In addition to the usual minimal optimal stopping time, we characterize the maximal optimal stopping time in the forward approach.