Some aspects of optimal switching and pricing Bermudan options
Abstract: This thesis consists of four papers that are all related to the Snell envelope. In the first paper, the Snell envelope is used as a formulation of a two-modes optimal switching problem. The obstacles are interconnected, take both profit and cost yields into account, and switching is based on both sides of the balance sheet. The main result is a proof of existence of a continuous minimal solution to a system of Snell envelopes, which fully characterizes the optimal switching strategy. A counter-example is provided to show that uniqueness does not hold.The second paper considers the problem of having a large number of production lines with two modes of production, high-production and low-production. As in the first paper, we consider both expected profit and cost yields and switching based on both sides of the balance sheet. The production lines are assumed to be interconnected through a coupling term, which is the average optimal expected yields. The corresponding system of Snell envelopes is highly complex, so we consider the aggregated yields where a mean-field approximation is used for the coupling term. The main result is a proof of existence of a continuous minimal solution to a system of Snell envelopes, which fully characterizes the optimal switching strategy. Furthermore, existence and uniqueness is proven for the mean-field reflected backward stochastic differential equations (MF-RBSDEs) we consider, a comparison theorem and a uniform bound for the MF-RBSDEs is provided.The third paper concerns pricing of Bermudan type options. The Snell envelope is used as a representation of the price, which is determined using Monte Carlo simulation combined with the dynamic programming principle. For this approach, it is necessary to estimate the conditional expectation of the future optimally exercised payoff. We formulate a projection on a grid which is ill-posed due to overfitting, and regularize with the PDE which characterizes the underlying process. The method is illustrated with numerical examples, where accurate results are demonstrated in one dimension.In the fourth paper, the idea of the third paper is extended to the multi-dimensional setting. This is necessary because in one dimension it is more efficient to solve the PDE than to use Monte Carlo simulation. We relax the use of a grid in the projection, and add local weights for stability. Using the multi-dimensional Black-Scholes model, the method is illustrated in settings ranging from one to 30 dimensions. The method is shown to produce accurate results in all examples, given a good choice of the regularization parameter.
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