Closed form approximation methods for portfolio valuation and risk management

In this thesis we present three closed form approximation methods for portfolio valuation and risk management.The first chapter is titled Kernel methods for portfolio valuation and risk management'', and is a joint work with Damir Filipovi\'c (SFI and EPFL). We introduce a simulation method for portfolio valuation and risk management building on machine learning with kernels. We estimate the value process of a portfolio from a finite sample of its cumulative cash flow. The estimator of a portfolio value process is given in closed form thanks to a suitable choice of the kernel. We show asymptotic consistency and derive finite sample error bounds under conditions that are suitable for finance applications. Numerical experiments show good results for examples in dimensions 12 and 36. The second chapter is titled Ensemble learning for portfolio valuation and risk management'', and is a joint work with Damir Filipovi'c (SFI and EPFL). We introduce a second closed form estimator of the value process of a portfolio. This estimator is based on ensemble learning methods with regression trees. In contrast to the first estimator, this estimator is fast to construct, and readily scalable with sample size and path space dimension. We also show how this estimator can be applied to derive a closed form estimator of the value process of a Bermudan option. Numerical experiments show good results for examples in dimensions 12 and 36. The third chapter is titled ``Polynomial approximation for interest rate derivatives valuation'', and is a joint work with Damir Filipovi'c (SFI and EPFL). We model the risk-neutral discount factor using a linear-rational model with a polynomial diffusion. Thanks to this modelling, the zero-coupon bond price, the continuously compounded overnight rate, and the forward rate are given in closed form. For an interest rate derivative whose price is not given in closed form, such as caplets, futures, and futures options, we use Bernstein polynomials to derive a closed form approximation of such a price that satisfies an $L^\infty$-error bound.