Abstract
This thesis draws its inspiration from the existence of Guaranteed Minimum Death Benefits
(GMDB) within various deferred annuities, prompting an investigation into the computation
of the expected discounted value of a payment upon death. This payment is contingent on
the stock price at that moment, potentially influenced by the stock price’s historical trend.
In instances where the payment mirrors an option’s payoff, the agreement for such a payment
is termed a (life) contingent option. Historically, the estimation of each time-to-death distribution
involved a linear combination of exponential distributions. This study delves into
scenarios where the time until death follows a Phase-type (PH) distribution for an individual
life and adheres to a bivariate Sarmanov distribution for a joint life or the last surviving
individual. The time until death stochastic variable is presumed to be unrelated to the stock
price evolution, modeled as a geometric Brownian motion. The primary analytical tool utilized
is a discounted joint density function. A comprehensive set of analytical expressions
is derived for contingent call and put options, lookback options, as well as barrier options.
Within a segment focusing on multiple stocks, the technique of Esscher transforms emerges
as advantageous in determining, among other things, a precise valuation for contingent Margrabe
options or exchange options.