The subject of this thesis are selected interest rate models and valuation of interest rate derivatives, especially interest rate options. Time-homogeneous one-factor short rate models, Vasicek and Cox-Ingersoll-Ross, and time-inhomogeneous short rate model, Hull{White, are treated. Heath-Jarrow-Morton framework is introduced as an alternative to short rate models, evolving the entire term structure of interest rates. The short rate models are shown to be special cases of models within the framework. The models are derived using the risk-neutral pricing methodology.

This thesis established the cubature method developed by Gyurkó & Lyons (2010) and Lyons & Victor (2004) for the Heath–Jarrow–Morton (HJM) model. The HJM model was first proposed by Heath, Jarrow, and Morton (1992) to model the evolution of interest rates through the dynamics of the forward rate curve. These dynamics are described by an infinite-dimensional stochastic equation with the whole forward rate curve as a state variable. To construct the cubature method, we first discretize the infinite dimensional HJM equation and thereafter apply stochastic Taylor expansion to obtain cubature formulae. We further used their results to construct cubature formulae to degree 3, 5, 7 and 9 in 1-dimensional space. We give, a considerable step by step calculation regarding construction of cubature formulae on Wiener space.

A popular class of yield curve models is based on the Nelson and Siegel (1987) (hereafter NS) approach of fitting yield curve data with simple functions of maturity. However, NS models are not theoretically consistent and they also lack an economic foundation, which limits their wider application in finance and economics. This thesis derives an intertemporally-consistent and arbitrage-free version of the NS model, and provides an explicit macroeconomic foundation for that augmented NS (ANS) model. To illustrate the general applicability of the ANS model, it is then applied to four distinct topics spanning finance and economics, each of which are active areas of research in their own right: i.e (1) forecasting the yield curve; (2) investigating relationships between the yield curve and the macroeconomy; (3) fixed interest portfolio management; and (4) investigating the uncovered interest parity hypothesis (UIPH). In each application, the ANS model allows the formal derivation of a parsimonious theoretical framework that captures the essence of the topic under investigation and is readily applicable in practice. Respectively: (1) the intertemporal consistency embedded in the ANS model results in a vector-autoregressive equation that projects the future yield curve from the current yield curve, and forecasts from that model outperform the random-walk benchmark; (2) the economic foundation for the ANS model leads to a single-equation relationship between the current shape of the yield curve and the magnitude and timing of future output growth, and empirical estimations confirm that the theoretical relationship holds in practice; (3) the ANS model provides a theoretically-consistent framework for quantifying risk and returns in fixed interest portfolios, and portfolios optimised ex-ante using that framework outperform a passive benchmark; and (4) the ANS model allows interest rates to be decomposed into a component related to economic fundamentals in the underlying economy, and a component related to cyclical influences. Empirical tests based on the fundamental interest rate components do not reject the UIPH, while the UIPH is rejected based on the cyclical interest rate components. This provides empirical support for suggestions in the theoretical literature that interest rate and exchange rate dynamics associated with cyclical interlinkages between the economy and financial markets under rational expectations may contribute materially to the UIPH puzzle.

<p>In this paper the Heath, Jarrow and Morton (HJM) framework is applied in the programming language Java for the estimation of the future spot rate. The subcase of an exponential model for the diffusion coefficient (volatility) is used for the pricing of interest rate derivatives (caps).</p>

In this paper the Heath, Jarrow and Morton (HJM) framework is applied in the programming language Java for the estimation of the future spot rate. The subcase of an exponential model for the diffusion coefficient (volatility) is used for the pricing of interest rate derivatives (caps).

Thesis (MSc)--Stellenbosch University, 2015. / ENGLISH ABSTRACT : The standard-Heath–Jarrow–Morton (HJM) framework is well-known for its application to pricing and hedging interest rate derivatives. This study implemented the extended HJM framework introduced by Eberlein and Raible (1999), in which a Brownian motion (BM) is replaced by a wide class of processes with jumps. In particular, the HJM driven by the generalised hyperbolic processes was studied. This approach was motivated by empirical evidence proving that models driven by a Brownian motion have several shortcomings, such as inability to incorporate jumps and leptokurticity into the price dynamics. Non-homogeneous Lévy processes and the change of measure techniques necessary for simplification and derivation of pricing formulae were also investigated. For robustness in numerical valuation, several transform methods were investigated and compared in terms of speed and accuracy. The models were calibrated to liquid South African data (ATM) interest rate caps using two methods of optimisation, namely the simulated annealing and secant-Levenberg–Marquardt methods. Two numerical valuation approaches had been implemented in this study, the COS method and the fractional fast Fourier transform (FrFT), and were compared to the existing methods in the context. Our numerical results showed that these two methods are quite efficient and very competitive. We have chose the COS method for calibration due to its rapidly speed and we have suggested a suitable approach for truncating the integration range to address the problems it has with short-maturity options. Our calibration results provided a nearly perfect fit, such that it was difficult to decide which model has a better fit to the current market state. Finally, all the implementations were done in MATLAB and the codes included in appendices. / AFRIKAANSE OPSOMMING : Die standaard-Heath–Jarrow–Morton-raamwerk (kortom die HJM-raamwerk) is daarvoor bekend dat dit op die prysbepaling en verskansing van afgeleide finansiële instrumente vir rentekoerse toegepas kan word. Hierdie studie het die uitgebreide HJM-raamwerk geïmplementeer wat deur Eberlein en Raible (1999) bekendgestel is en waarin ’n Brown-beweging deur ’n breë klas prosesse met spronge vervang word. In die besonder is die HJM wat deur veralgemeende hiperboliese prosesse gedryf word ondersoek. Hierdie benadering is gemotiveer deur empiriese bewyse dat modelle wat deur ’n Brown-beweging gedryf word verskeie tekortkominge het, soos die onvermoë om spronge en leptokurtose in prysdinamika te inkorporeer. Nie-homogene Lévy-prosesse en die maatveranderingstegnieke wat vir die vereenvoudiging en afleiding van prysbepalingsformules nodig is, is ook ondersoek. Vir robuustheid in numeriese waardasie is verskeie transformmetodes ondersoek en ten opsigte van spoed en akkuraatheid vergelyk. Die modelle is vir likiede Suid-Afrikaanse data vir boperke van rentekoerse sonder intrinsieke waarde gekalibreer deur twee optimiseringsmetodes te gebruik, naamlik die gesimuleerde uitgloeimetode en die sekans-Levenberg–Marquardt-metode. Twee benaderings tot numeriese waardasie is in hierdie studie gebruik, naamlik die kosinusmetode en die fraksionele vinnige Fourier-transform, en met bestaande metodes in die konteks vergelyk. Die numeriese resultate het getoon dat hierdie twee metodes redelik doeltreffend en uiters mededingend is. Ons het op grond van die motiveringspoed van die kosinus-metode daardie metode vir kalibrering gekies en ’n geskikte benadering tot die trunkering van die integrasiereeks voorgestel ten einde die probleem ten opsigte van opsies met kort uitkeringstermyne op te los. Die kalibreringsresultate het ’n byna perfekte passing gelewer, sodat dit moeilik was om te besluit watter model die huidige marksituasie die beste pas. Ten slotte is alle implementerings in MATLAB gedoen en die kodes in bylaes ingesluit.

Diss. (sammanfattning) Stockholm : Handelshögskolan, 2008 Sammanfattning jämte 3 uppsatser