The Performance of Multi-Factor Term Structure Models for Pricing and Hedging Caps and Swaptions

On the Information in the Interest Rate Term Structure and Option Prices

One-factor no-arbitrage term structure models where the instantaneous interest rate follows either the process proposed by Vasicek (1977) or by Cox, Ingersoll, and Ross (1985), commonly known as CIR, are parsimonious and analytically tractable. Models based on the original CIR process have the important characteristic of allowing for a time-varying conditional interest rate volatility but are undefined in negative interest rate environments. A Shifted-CIR no-arbitrage term structure model, where the instantaneous interest rate is given by the sum of a constant lower bound and a non-negative CIR-like process, allows for negative yields and benefits from similar tractability of the original CIR model. Based on the U.S. and German yield curve data, the Vasicek and Shifted-CIR specifications, both considering constant and time-varying risk premia, are compared in terms of information criteria and forecasting ability. Information criteria prefer the Shifted-CIR specification to models based on the Vasicek process. It also provides similar or better in-sample and out-of-sample forecasting ability of future yield curve movements. Introducing a time variation of the interest rate risk premium in no-arbitrage one-factor term structure models is instead not recommended, as it provides worse information criteria and forecasting performance.

Recent events have highlighted the potential importance of nonlinear efiects of flscal variables (notably debt and deflcits) on interest rates: While in times when government solvency is not a concern the standard crowding-out efiects are of moderate magnitude, in times when default risk becomes an issue the interest rate efiects can become very large. This paper provides new evidence on the magnitude of these efiects. For the case when default risk is not a concern, it uses an arbitrage-free term structure model to estimate the dynamic efiects of flscal policy shocks on interest rates along the entire maturity spectrum. For the case when default risk becomes a concern (thereby violating a central assumption of the term structure model), I present evidence based on EMU government bond spread regressions on time-varying efiects of national flscal policies on spreads as well as the time-varying sensitivity of yield spreads to international risk aversion as a function of the state of flscal policy. JEL classiflcation: E6, H6.

Bermudan swaptions are options on interest rate swaps which can be exercised on one or more dates before the final maturity of the swap. Because the exercise boundary between the continuation area and stopping area is inherently complex and multi-dimensional for interest rate products, there is an inherent “tug of war” between the pursuit of calibration and pricing precision, tractability, and implementation efficiency. After reviewing the main ideas and implementation techniques underlying both single- and multi-factor models, we offer our own approach based on dimension reduction via Markovian projection. Specifically, on the theoretical side, we provide a reinterpretation and extension of the classic result due to Gyöngy which covers non-probabilistic, discounted, distributions relevant in option pricing. Thus, we show that for purposes of swaption pricing, a potentially complex and multidimensional process for the underlying swap rate can be collapsed to a one-dimensional one. The empirical contribution of the paper consists in demonstrating that even though we only match the marginal distributions of the two processes, Bermudan swaptions prices calculated using such an approach appear well-behaved and closely aligned to counterparts from more sophisticated models.

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability density. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the density functions associated with a given family of term structures to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. It follows that the theory of interest rate dynamics can be represented by a class of processes in Hilbert space. We also derive the dynamics for the central moments associated with the distribution determined by the yield curve.

Pricing and Hedging Interest Rate Options: Evidence from Cap-Floor Markets

With weekly data of repo rates in the SSE, it is found that the expectations hypothesis fails to explain the repo rates, and risk premiums are significant and time-varying. One-factor and two-factor Guassian essential affine models are estimated to model the time-varying risk premiums, and the likelihood ratio test shows that the two-factor model has no significant improvement over the one-factor model. Although one-factor Guassian essential affine model fits the risk premiums very well, the model doesn't fit the means and standard deviations of the repo rates.

EMPIRICAL STUDIES ON INTEREST RATE DERIVATIVES by Xudong Sun Dr. Hongtao Yang, Examination Committee Chair Associate Professor of Mathematics University of Nevada, Las Vegas, USA Interest rate models are the building blocks of nancial market and the interest rate derivatives market is the largest derivatives market in the world. In this dissertation, we shall focus on numerical pricing of interest rate derivatives, estimating model parameters by Kalman lter, and studying various models empirically. We shall propose a frontxing nite element method to price the American put option under the quadratic term structure framework and compare it with a trinomial tree method and common nite element method. Numerical test results show the superiority of our frontxing nite element method in the aspects of computing the option and free boundary simultaneously with high accuracy. We shall also employ the Kalman lter and its variant techniques to estimate parameters of the a ne term structure models as well as quadratic term structure models. Various comparisons of di erent Kalman lter performance and both the in-sample t and out-sample t for Monte Carlo simulations as well as real treasury yield data are presented. In addition, we shall propose a general one-factor interest rate model and apply a homotopy perturbation method to valuate bond prices. One of the attractive qualities of the approximated solution of homotopy perturbation method is its fast speed of achieving the same accuracy compared to the tree method.

Pricing of index options when interest rates are stochastic: An empirical test

Short‐term interest rate processes determine the term structure of interest rates in an arbitrage‐free market and are central to the valuation of interest rate derivatives. We obtain parameter estimates and compare the empirical fit of alternative one‐factor continuous‐time processes for the South African short‐term interest rate (and hence of arbitrage‐free term structure models) using Gaussian estimation methods. We find support only for diffusions where the interest rate volatility is moderately sensitive to the level of the interest rate. Other common models with restrictions that either preclude this effect, or restrict it to be too high, do not fit the data. Differences in the specification of the drift function have no evident effect on model performance.

Correlating Market Models

This paper investigates short-term interest rate models using daily data for both the US and Japan over the five years October 1989 to January 1994. A nonparametric method is used to estimate the volatility of the short-term interest rate and the results are compared with those from a parametric method. Three important features are found. First, a two-factor model can capture the behaviour of the interest rate better than a one-factor model. Second, although the US interest rate does not exhibit the mean reverting property, the Japanese interest rate does. Third, in contrast to the Japanese interest rate, the conditional variance of US interest rate changes is found to depend on the level of the interest rate.

We present a parallel algorithm and its implementation that computes lower and upper bounds for prices of Bermudan swaptions. The evolving of the underlying forward rates is assumed to follow the extended multi-factor LIBOR market model. We follow the Longstaff-Schwartz least-squares approach in computing a lower bound and the Andersen-Broadie duality-based procedure in computing an upper bound. Parallelisation in the implementation is achieved through POSIX threading. High-performance Intel MKL functions are used for regression and linear algebra operations. The parallel implementation was tested using Bermudan swaptions with different parameters on Intel multi-core machines. In all the tests the parallel program produced close results to those reported in the previous studies. Significant speedups were observed against an efficient sequential implementation built for comparison.

In this paper, we propose a new exogenous model to address the problem of negative interest rates that preserves the analytical tractability of the original Cox–Ingersoll–Ross (CIR) model with a perfect fit to the observed term-structure. We use the difference between two independent CIR processes and apply the deterministic-shift extension technique. To allow for a fast calibration to the market swaption surface, we apply the Gram–Charlier expansion to calculate the swaption prices in our model. We run several numerical tests to demonstrate the strengths of this model by using Monte-Carlo techniques. In particular, the model produces close Bermudan swaption prices compared to Bloomberg’s Hull–White one-factor model. Moreover, it finds constant maturity swap (CMS) rates very close to Bloomberg’s CMS rates.

This study has two purposes. First, it estimates the market, interest rate, and exchange rate sensitivities (betas) of the Japanese banking institutions. Second, it investigates the relationship between the market-based measures of risk and accounting-based financial ratios. We extend the literature in three important ways. First, we employ a multi-factor GARCH model to estimate the betas. This framework incorporates non-linearities in the bank stock return modeling and allows for time-varying risk premia. Second, we investigate the determinants of market and exchange rate risk in terms of bank financial ratios. To this end, we regress the beta measures derived from the GARCH model against the corporate decision variables to determine the direction and the magnitude of the impact of the latter on the market and exchange rate risk exposures. Third, by using data on the Japanese banking institutions, we provide a comparison of the bank interest rate and exchange rate sensitivities and the strength of the links between the risk measure and the corporate decision variables between the U.S. and the Japanese banking institutions. This comparison sheds light on the robustness of the results concerning interest rate and exchange rate risk, and their determinants, across the two countries. Several interesting results are obtained. First, empirical results indicate that interest rate is only occasionally significant while market and exchange rate variables are significant for all the banks in the sample. Second, market and exchange rate risk measures do impound information in the financial ratios with the explanatory power of the market beta model being higher than that of the exchange rate beta model. Third, the association of the market-based risk measures and the financial ratios is weaker for the Japanese banks than those found for their U.S. counterparts in the existing literature.