Markov Functional interest rate models with stochastic volatility
Dissertation -Markov functional models
With respect to modeling of the (forward) interest rate term structure under consideration of the market observed skew, stochastic volatility Libor Market Models (LMMs) have become predominant in recent years. A powerful representative of this class of models is Petersburg’s forward rate term structure of skew LMM (FL-TSS LMM). However, by construction market models are high-dimensional which is impediment to their efficient implementation.
The class of Markov functional models (MFMs) attempts to overcome this inconvenience by combining the strong points of market and short rate models, namely the exact replication of prices of calibration instruments and tractability. This is achieved by modeling the numeraire and terminal discount bond (and hence the entire term structure) as functions of a low-dimensional Markov process whose probability density is known.
This study deals with the incorporation of stochastic volatility into a MFM framework. For this sake an approximation of Petersburg’s FL-TSS LMM is devised and used as pre-model which serves as driver of the numeraire discount bond process. As a result the term structure is expressed as functional of this pre-model. The pre-model itself is modelled as function of a two-dimensional Markov process which is chosen to be a time-changed brownian motion. This approach ensures that the correlation structure of Petersburg’s FL-TSS is imposed onto the MFM; especially the stochastic volatility component is inherited.
As part of this thesis an algorithm for the calibration of Petersburg’s FL-TSS LMM to the swaption market and the calibration of a two-dimensional Libor MFM to the (digital) caplet market was implemented. Results of the obtained skew and volatility term structure (Petersburg parameters) and numeraire discount bond functional forms are presented.
