Alessandro Iania, "Basket option pricing for processes with jumps using sparse grids and Fourier transforms"
The Common Clock Variance Gamma (CCVG) model, introduced by Madan and Seneta in 1990 , permits to overcome some of the shortcomings that affect the typical assumption of log-normal distribution for stock prices made in the Black-Scholes model . CCVG model has been therefore considered in the literature to model a set of correlated assets' prices. E. Luciano and W. Schoutens  calibrated this model in the particular case in which Σ is a diagonal matrix. We depart from the fundamental theorem of option pricing, which states that the price of a derivative is the expected income under a particular measure, the so-called risk-neutral measure. Assuming a risk-neutral measure given, both algorithms compute this expected income by computing the corresponding multidimensional integral using sparse grids. The use of sparse grids allows us to overcome the curse of dimensionality to a certain extent. As the sparse grid integration requires a special class of integrand functions, we manipulate our d-dimensional integral using the law of total expectation to split it into two parts. Once we obtain a problem formulation that can be solved using sparse grids, we present two numerical algorithms to compute our multidimensional integrals. The first algorithm uses the Fast Fourier Transform (FFT) to evaluate the (d − 1)-dimensional joint probability density function. The second algorithm exploits the fact that the CCVG's pdf conditioned to the random time change (called the clock) has a normal distribution. We describe the two algorithms and we report the numerical results obtained running them to compute integrals in different dimensions. We conclude the thesis by commenting on the numerical results and reporting suggestions for further works.
In this thesis, we present two algorithms based on sparse grids to compute basket options' price under the CCVG model. We introduce the fundamental concepts needed to fully understand the dissertation, but the reader is still supposed to be competent in basic probability and measure theory. A Matlab implementation of the two methods is included respectively in Appendix A and in Appendix B.