Option Pricing and Performance Evaluation

This dissertation deals with two aspects of mathematical finance. The first is the pricing of options in a jump-diffusion setting, with lognormal jumps, according to the model proposed by Merton in 1976. The American option pricing procedure by Hilliard and Schwartz, that has been modified by Dai et al. reducing the computational complexity from O(n^3) to O(n^(2.5)), is here further improved to a computational complexity of O(n^2 log n), by trimming of the bivariate tree while keeping the error in check, and then to an O(n^2) unidimensional procedure. These results are discussed in the joint works by Gaudenzi, Spangaro and Stucchi. The other issue addressed in this dissertation is the performance evaluation of investments under different reward to risk ratios. Different portfolio performance measures have been compared applied to asset class indexes with a distribution far from normality: Omega, Sortino, Reward-to-VaR, STARR, Rachev ratio are correlated. Even though the values the various performance measures attribute to each investment differ, both in the cases analysed by Eling and Schuhmacher and in that by Spangaro and Stucchi many of them express good rank correlation with Sharpe ratio. The results draw heavily on the joint work Spangaro and Stucchi.