| Summary: | This thesis consists of three separate articles. In the first article we extend a fast algorithm to price European options on underlying assets which pay discrete dividends to the two-dimensional case. Firstly, by using convexity, we formulate upper and lower bounds for the price of a classical (univariate) European option written on a dividend-paying Black-Scholes asset in closed form, and show that those bounds converge to the true option price. The errors introduced by the method decrease with the square of the discretisation step used and scale with the gamma of the option. Secondly, the procedure is extended to obtain similar bounds for the price of a bivariate European call on the maximum of two underlying assets. Prices of other bivariate European options can then be found through put-call/min-max parity relations. The second article concerns the derivation of analytical expressions for the future Expected Exposure in several Inflation Indexed Swaps under a stochastic model for inflation. These can be used to find a closed form solution for the Credit Value Adjustment (CVA). The CVA of a Zero-Coupon Inflation Indexed Swap is obtained analytically under this framework. For the Expected Exposure of a Year-on-Year Inflation Indexed Swap and for a portfolio of many Zero-Coupon Inflation Indexed Swap instruments, semi-analytical solutions are derived which are based on moment matching approximations. Extensive tests of the algorithms using Monte Carlo simulations show that the approximating formulae provide very fast and accurate methods to determine the CVA for different products. In the third article we show that an equilibrium bid-ask spread for European derivatives may arise in dry markets for the underlying asset, even under symmetric information and absence of transaction costs. By dry markets we mean that the underlying asset may not be traded at all points in time, generating a particular form of market incompleteness. Using a partial equilibrium analysis in a one period model, we show two results. For monopolistic risk-neutral market makers, we fully characterise the bid-ask spread within the no-arbitrage bounds. For oligopolistic risk-neutral market makers, we prove that there is no pure symmetric Nash equilibrium of the game and that a bid-ask spread can only exist under a mixed strategy equilibrium.
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