Essays on derivatives pricing

In the first part, by using convexity, we employ a fast algorithm to obtain upper and lower price bounds for a classical (univariate) European option written on a discrete-dividend-paying Black-Scholes asset in closed form, and show that those bounds converge to the true option price. The errors introduced decrease with the square of the discretisation step used and scale with the option’s gamma. Extension to price bounds for a bivariate European call-on-the-maximum of two underlying assets is presented. Prices of other bivariate European options can then be found through put-call/min-max parity relations. The second part derives the future Expected Exposure expressions for several Inflation-Indexed-Swaps under a stochastic model for inflation, 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. For a Year-on-Year-Inflation-Indexed-Swap and for a portfolio of Zero-Coupon-Inflation-Indexed-Swaps, semi-analytical solutions based on moment-matching-approximations are derived. Extensive tests using Monte Carlo simulations show that the formulas provide very fast and accurate methods. Third part shows how equilibrium bid-ask spread for European derivatives arises in dry markets (the underlying asset may not be traded at all points in time, generating market incompleteness), even under symmetric information and absence of transaction costs. In a one period model, for monopolistic risk-neutral market-makers we fully characterise the bid-ask spread within the no-arbitrage bounds, whereas 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.