Trolle-Schwartz HJM interest rate model
Master Thesis
2016
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University of Cape Town
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Abstract
The Trolle and Schwartz (2009) interest rate model prices interest rate derivatives in a generalised stochastic volatility framework. It is a reformulation of the multifactor Heath, Jarrow and Morton (1992) framework with stochastic volatility terms presented in an analogous fashion to the seminal Heston (1993) model. The Trolle and Schwartz (2009) model provides semi-analytical pricing formulas for zerocoupon bonds and zero-coupon bond options. These formulas are extended to price interest rate caplets, and therefore caps, as well as swaptions. These formulas are described as semi-analytical because of the use of numerical methods as well as their dependency on unobserved state variables. These state variables are estimated by applying an extended Kalman filter on a dataset of interest rates and interest rate derivative prices. Although Trolle and Schwartz (2009) confirm the accuracy of their model when testing against empirical prices, they do not provide an analysis of the consistency between the semi-analytical formulas and Monte Carlo pricing. Presenting this test for consistency seeks to confirm the validity of these pricing formulas. The aim of this dissertation is to implement the Trolle and Schwartz (2009) model and discuss the performance of the semi-analytical pricing formulas against a Monte Carlo simulation. Emphasis will be placed firstly on reviewing the derivations outlined in Trolle and Schwartz (2009) and secondly, building a Monte Carlo framework capable of comparing prices with the semi-analytical pricing formulas. Simulated data will be considered for the purpose of confirming that the estimation of the state vector is sufficiently accurate. Thereafter, an analysis on an empirical dataset can determine whether the results hold across different sets of data.
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Schumann, G. 2016. Trolle-Schwartz HJM interest rate model. University of Cape Town.