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.
