Meshfree methods in financial engineering

In this thesis I use the radial basis function (RBF) interpolation, a meshfree method, to solve problems in financial engineering. They involve partial differen- tial equations whose closed-form solutions do not exist or are difficult to compute, cumbersome or time-consuming. The thesis consists of three major studies. In the first one, I extend the existent literature on meshfree methods applied to option pricing. The RBF interpolation is performed for pricing American options adopting the constant elasticity of variance model (Cox and Ross (1976)) and the Heston model (Heston (1993)). Several experiments are run to evaluate the performance ofthis approach. The results are compared with solutions given by the Monte Carlo simulation (MCS) and the finite difference method (FDM). In the second study, I employ the RBF interpolation to approximate zero- coupon bond prices and survival probabilities to price credit default swap (CDS) contracts. The default intensity is assumed to follow an Exponential-Vasicek process (Brigo and Mercurio (2006)) while the interest rate is modelled with a Cox- Ingersoll-Ross (CIR) process (Cox et al. (1985)). Numerical experiments are run for one- and two-factor models. The results are compared with the approximations obtained by the FDM and the analytical solution if it exists. Finally, in the third study I perform a nonlinear filter to infer the default risk implicit in the term structure of CDS spreads. In fact, I carry out a sequential joint estimation of both the default intensity and the CIR model parameters. The filter is based on the numerical solution of the Fokker-Planck equation by the RBF v interpolation method. The filter is applied on daily CDS spreads of 27 companies of the Dow Jones index between 2005 and 2010. The results in the thesis provide evidence of the high accuracy and computa- , tional efficiency of the RBF interpolation. Moreover, its performance is outstand- ing compared with traditional techniques in finance such as the standard FDM and the MCS.EThOS - Electronic Theses Online ServiceGBUnited Kingdo