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Abstract Game contingent claims (GCCs), as introduced by Kifer (2000), are a generalization of American contingent claims where the writer has the opportunity to terminate the contract, and must then pay the intrinsic option value plus a penalty. In complete markets, GCCs are priced using no-arbitrage arguments as the value of a zero-sum stochastic game of the type described in Dynkin (1969). In incomplete markets, the neutral pricing approach of Kallsen and Kühn (2004) can be used. In Part I of this thesis, we introduce GCCs and their pricing, and also cover some basics of mathematical finance. In Part II, we present a new algorithm for valuing game contingent claims. This algorithm generalises the least-squares Monte-Carlo method for pricing American options of Longstaff and Schwartz (2001). Convergence proofs are obtained, and the algorithm is tested against certain GCCs. A more efficient algorithm is derived from the first one using the computational complexity analysis technique of Chen and Shen (2003). The algorithms were found to give good results with reasonable time requirements. Reference implementations of both algorithms are available for download from the author’s Github page https://github.com/del/ Game-option-valuation-library

This thesis treats a range of stochastic methods with various applications, most notably in finance. It is comprised of five articles, and a summary of the key concepts and results these are built on. The first two papers consider a jump-to-default model, which is a model where some quantity, e.g. the price of a financial asset, is represented by a stochastic process which has continuous sample paths except for the possibility of a sudden drop to zero. In Paper I prices of European-type options in this model are studied together with the partial integro-differential equation that characterizes the price. In Paper II the price of a perpetual American put option in the same model is found in terms of explicit formulas. Both papers also study the parameter monotonicity and convexity properties of the option prices. The third and fourth articles both deal with valuation problems in a jump-diffusion model. Paper III concerns the optimal level at which to exercise an American put option with finite time horizon. More specifically, the integral equation that characterizes the optimal boundary is studied. In Paper IV we consider a stochastic game between two players and determine the optimal value and exercise strategy using an iterative technique. Paper V employs a similar iterative method to solve the statistical problem of determining the unknown drift of a stochastic process, where not only running time but also each observation of the process is costly.

<p>This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.</p><p>In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.</p><p>In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.</p><p>Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.</p><p>A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.</p><p>In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.</p><p>Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model. </p>

This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing. In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility. In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied. Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary. A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility. In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion. Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.