Computational Finance lies at the intersection of numerical analysis and stochastic calculus. An important aspect of research in this field is to further increase the performance of pricing and risk measurement methods.
Of particular interest to our group is the efficient computation of the risk measures widely used in credit and market risk such as the Value-at-Risk (VaR) and the Expected Shortfall. The accurate estimation of the individual risk contributions is an important issue as well. We develop numerical methods capable to calculate these measures in a short CPU time, allowing to rebalance very large portfolios frequently and avoiding this way the time-consuming Monte Carlo simulations. We are also interested in the valuation of credit derivatives such as Collateralized Debt Obligations, which are typically used to transfer the risk associated to a certain underlying portfolio. So far, the machinery to carry out this work is mainly based on Haar wavelets.
Another important research line of our group is option pricing. The robust and efficient valuation of options is an interesting recent field in applied mathematics and scientific computing. The best known option pricing partial differential equation (PDE) is without any doubt the Black-Scholes equation, pricing a European option under geometric Brownian motion asset price dynamics. When considering more realistic asset dynamics, other option pricing PDEs, or even partial integro-differential equations, will be encountered. Option pricing is often done by the discounted expected pay-off approach, and its connection with the solution of the option pricing PDEs is the Feynman-Kac theorem. In many cases of interest, we do not have the conditional probability density function for the asset prices available, but we do have its Fourier transform. The application of Fourier and wavelets-based techniques to recover a density function from its Fourier transform is subject of our interest.