Multidimensional Markov-Functional and Stochastic Volatiliy Interest Rate Modelling

Abstract: This thesis consists of three papers in the area of interest rate derivatives modelling. The pricing and hedging of (exotic) interest rate derivatives is one of the most demanding and complex problems in option pricing theory and is of great practical importance in the market. Models used in production at various banks can broadly be divided in three groups: 1- or 2-factor instantaneous short/forward rate models (such as HullXX1White (1990) or Cheyette (1996)), LIBOR/swap market models (introduced by Brace, GatarekXX1Musiela (1997), Miltersen, SandmannXX1Sondermannn (1997) and Jamshidian (1997)) and the one or two-dimensional Markov-functional models of Hunt, KennedyXX1Pelsser (2000)). In brief and general terms the main characters of the above mentioned three modelling frameworks can be summarised as follows. Short/forward rate models are by nature computationally efficient (implementations may be done using PDE or lattice methods) but less flexible in terms of fitting of implied volatility smiles and correlations between various rates. Calibration is hence typically performed in a ‘local’ (product by product based) sense. LIBOR market models on the other hand may be calibrated in a ‘global’ sense (i.e. fitting close to everything implying that one calibration may in principle be used for all products) but are of high dimension and an accurate implementation has to be done using the Monte Carlo method. Finally, Markov-functional models can be viewed as designed to combine the computational efficiency of short/forward rate models with flexible calibration properties. The defining property of a Markov-functional model is that each rate and discount factor at all times can be written as functionals of some (preferably computationally simple) Markovian driving process. While this is a property of most commonly used interest rate models Hunt et al. (2000) introduced a technique to numerically determine a set of functional forms consistent with market prices of vanilla options across strikes and expiries. The term a ‘Markov-functional model’ is typically referring to this type of model as opposed to the more general meaning, a terminology that is adopted also in this thesis. Although Markov-functional models are indeed a popular choice in practice there are a few outstanding points on the practitioners’ wish list. From a conceptual point of view there is still work to be done in order to fully understand the implications of various modelling choices and how to efficiently calibrate and use the model. Part of the reason for this is that while the properties of the short/forward rate and the LIBOR market models may be understood from their defining SDEs this is less clear for a Markov-functional model. To aid the understanding of the Markov-functional model BennettXX1Kennedy (2005) compares one-dimensional LIBOR and swap Markov-functional models with the one-factor separable LIBOR and swap market models and concludes that the models are similar distributionally across a wide range of viable market conditions. Although this provides good intuition there is still more work to be done in order to fully understand the implications of various modelling choices, in particular in a two or higher dimensional setting. The first two papers in this thesis treat extensions of the standard Markov-functional model to be able to use a higher dimensional driving process. This allows a more general understanding of the Markov-functional modelling framework and enables comparisons with multi-factor LIBOR market models. From a practical point of view it provides more powerful modelling of correlations among rates and hence a better examination and control of some types of exotic products. Another desire among practitioners is to develop an efficient way of using a process of stochastic volatility type as a driver in a Markov-functional model. A stochastic volatility Markov-functional model has the virtue of both being able to fit current market prices across strikes and to provide better control over the future evolution of rates and volatilities, something which is important both for pricing of certain products and for risk management. Although there are some technical challenges to be solved in order to develop an efficient stochastic volatility Markov-functional model there are also many (more practical) considerations to take into account when choosing which type of driver to use. To shed light on this the third paper in the thesis performs a data driven study in order to motivate and develop a suitable two-dimensional stochastic volatility process for the level of interest rates. While the main part of the paper is general and not directly linked to any complete interest rate model for exotic derivatives, particular care is taken to examine and equip the process with properties that will aid use as a driver for a stochastic volatility Markov-functional model.