Dissertation Writing Help - Markov Functional interest rate models with stochastic volatility
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With respect to modelling 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 Piterbarg's forward rate term structure of skew LMM (FL-TSS LMM). However, by construction market models are high- dimensional which is an 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 that is the exact replication of prices of calibration instruments and tractability. This is attained by modelling 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 estimation of Piterbarg'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 confirms that the correlation structure of Piterbarg'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 Piterbarg'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 (Piterbarg parameters) and numeraire discount bond functional forms are presented.