Zeit- und Volatilitätsstruktur von Zinssätzen - Modellierung, Implementierung, Kalibrierung

Abstract

The thesis traces the modelling developments of the term structure of interest rates from the beginnings in the early seventies of the last century up to the multifactorial cross-currency stochastic volatility Libor Market Models in modern times. We follow the general classification in traditional approaches, which emphasise the equilibrium model by Vasicek and the no arbitrage model by Hull/White (HW) as crucial cornerstones of interest rate modelling, and in modern market approaches, of which the most prominent examples include the Heath/Jarrow/Morton (HJM) framework and the Libor Market Model (LMM). In spite of deriving a general partial differential equation (PDE) for arbitrary payoffs, the thesis casts the traditional models in the mould of the martingale pricing theory and steers clear from the application of PDE techniques. Furthermore, an option pricing model based on a functional dependence of the volatility structure on the interest rate is developed. The equivalence between the HJM family and the HW is shown for a specific volatility function and it is argued that the modern approach defines no universal, self-contained modelling technique. The modern approach rather provides a more flexible vocabulary to describe the characteristics of an already existing traditional model. The second part of the thesis implements the LMM based on a deterministic volatility function and stages a full-blown calibration procedure under the aspect of perfect reproduction of the market observable Black volatilities and the implications for the neighbouring swaption markets. The results from the attempt at a simultaneous calibration to both caplets and swaptions at the same time indicate the possible lack of congruence between both markets. The concluding chapter of the thesis is concerned with stochastic volatility models. It proposes a cross-currency LMM based on a separate variance process for the exchange rate. The model is capable of controlling the skew (vanna risk) and the smile (volga risk) of the implied volatilty surface. In addition, suggestions are made how to extend the model in order to incorporate as much market information as possible.