Abstracts Mathematics

Polynomial Multi-Curve Models And Extensions In Mathematical Finance

by Thomas Du Toit

This thesis is organized into three chapters: In the first chapter, we introduce the changes that have occurred in the fixed-income market due to the credit crisis in 2007–2008. We then discuss the impact of this crisis on the pre-crisis relation between zero-coupon bonds and forward rate agreements written on the Libor/Euribor rates. This particularly, this includes describing the dynamics of spread rates in addition to interest rates, which to date had not been part of pre-crisis models. Following this, we introduce a multicurve model set-up in order to compute the non-arbitrage prices of a forward rate agreement. This is done by assuming that the underlying factor processes, which describe the dynamics of the interest and spread rates are given by a diffusion process such that bond prices can be expressed as polynomials and the forward Libor rates as rational functions. At the end of the first chapter, we consider a multi-curve model set-up in the discrete time setting. As in the continuous case, the bond prices can be expressed as polynomial and the forward Libor rates as rational functions. In this case we consider linear functions and give a calibration method. In the second chapter, we extend from diffusion processes to jump processes, allowing the factor process to have jumps. Our main result is the classification of such models for which the bond prices can be expressed as polynomials. These models are arbitrage-free in the sense that the discounted zero-coupon bond prices are local martingales. In the third chapter, we consider a specific type of volatility models, which all have the characteristic that the moments of the stock can be expressed as a polynomial. Further, we show that there is a relation between the k-th moment and the degree of the polynomial.