In this thesis two novel approaches to pricing of barrier and American options are developed in the setting of local volatility models with jumps: the moments method and the Markov chain method.

The moments method is a valuation approach for barrier options that is based on a characterisation of the exit location measure and the expected occupation measure of the price process of the underlying in terms of the corresponding moments. It is shown how the value of barrier-type derivatives can be expressed using these moments, which are in turn shown to be characterised by an infinite-dimensional linear system. By solving finite-dimensional linear programming problems, which are obtained by restricting to moments of a finite degree, upper
and lower-bounds are found for the values of the options in question.

The Markov chain method for the valuation of American options is based on an approximation of the underlying price process by a continuous-time Markov chain. The value-function of the American option driven by the approximating chain is identified by solving the associated optimal stopping problem. In particular, a novel explicit characterisation of the optimal exercise boundary is derived in terms of the generator of the Markov chain. Using this characterisation it is shown that the optimal exercise boundary and the corresponding value-function can be evaluated efficiently.

For both of the presented methods convergence results are established. The methods are implemented for a range of local volatility models with jumps, and a number of numerical examples are discussed in detail to illustrate the scope of the methods.