On the Snell envelope approach to optimal switching and pricing Bermudan options

Abstract: This thesis consists of two papers related to systems of Snell envelopes. The first paper uses a system of Snell envelopes to formulate the problem of two-modes optimal switching for the full balance sheet in finite horizon. This means that the switching problem is formulated in terms of trade-off strategies between expected profit and cost yields, which act as obstacles to each other. Existence of a minimal solution of this system is obtained by using an approximation scheme. Furthermore, the optimal switching strategies are fully characterized. The second paper uses the Snell envelope to formulate the fair price of Bermudan options. To evaluate this formulation of the price, the optimal stopping strategy for such a contract must be estimated. This may be done recursively if some method of estimating conditional expectations is available. The paper focuses on nonparametric estimation of such expectations, by using regularization of a least-squares minimization, with a Tikhonov-type smoothing put on the partial diferential equation which characterizes the underlying price processes. This approach can hence be viewed as a combination of the Monte Carlo method and the PDE method for the estimation of conditional expectations. The estimation method turns out to be robust with regard tothe size of the smoothing parameter.