Optimal Stopping and Model Robustness in Mathematical Finance
Abstract: Optimal stopping and mathematical finance are intimately connected since the value of an American option is given as the solution to an optimal stopping problem. Such a problem can be viewed as a game in which we are trying to maximize an expected reward. The solution involves finding the best possible strategy, or equivalently, an optimal stopping time for the game. Moreover, the reward corresponding to this optimal time should be determined. It is also of interest to know how the solution depends on the model parameters. For example, when pricing and hedging an American option, the volatility needs to be estimated and it is of great practical importance to know how the price and hedging portfolio are affected by a possible misspecification.The first paper of this thesis investigates the performance of the delta hedging strategy for a class of American options with non-convex payoffs. It turns out that an option writer who overestimates the volatility will obtain a superhedge for the option when using the misspecified hedging portfolio.In the second paper we consider the valuation of a so-called stock loan when the lender is allowed to issue a margin call. We show that the price of such an instrument is equivalent to that of an American down-and-out barrier option with a rebate. The value of this option is determined explicitly together with the optimal repayment strategy of the stock loan.The third paper considers the problem of how to optimally stop a Brownian bridge. A finite horizon optimal stopping problem like this can rarely be solved explicitly. However, one expects the value function and the optimal stopping boundary to satisfy a time-dependent free boundary problem. By assuming a special form of the boundary, we are able to transform this problem into one which does not depend on time and solving this we obtain candidates for the value function and the boundary. Using stochastic calculus we then verify that these indeed satisfy our original problem.In the fourth paper we consider an investor wanting to take advantage of a mispricing in the market by purchasing a bull spread, which is liquidated in case of a market downturn. We show that this can be formulated as an optimal stopping problem which we then, using similar techniques as in the third paper, solve explicitly.In the fifth and final paper we study convexity preservation of option prices in a model with jumps. This is done by finding a sufficient condition for the no-crossing property to hold in a jump-diffusion setting.
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