Hedging American contingent claims with constrained portfolios

Abstract. The valuation theory for American Contingent Claims, due to Bensoussan (1984) and Karatzas (1988), is extended to deal with constraints on portfolio choice, including incomplete markets and borrowing/short-selling constraints, or with different interest rates for borrowing and lending. In the unconstrained case, the classical theory provides a single arbitrage-free price $u_0$; this is expressed as the supremum, over all stopping times, of the claim's expected discounted value under the equivalent martingale measure. In the presence of constraints, $\{u_0\}$ is replaced by an entire interval $[h_{\rm low}, h_{\rm up}]$ of arbitrage-free prices, with endpoints characterized as $h_{\rm low} = \inf_{\nu\in{\cal D}} u_\nu, h_{\rm up} = \sup_{\nu\in{\cal D}} u_\nu$. Here $u_\nu$ is the analogue of $u_0$, the arbitrage-free price with unconstrained portfolios, in an auxiliary market model ${\cal M}_\nu$; and the family $\{{\cal M}_\nu\}_{\nu\in{\cal D}}$ is suitably chosen, to contain the original model and to reflect the constraints on portfolios. For several such constraints, explicit computations of the endpoints are carried out in the case of the American call-option. The analysis involves novel results in martingale theory (including simultaneous Doob-Meyer decompositions), optimal stopping and stochastic control problems, stochastic games, and uses tools from convex analysis.