Single-factor Interest Rate Models Research Articles

A Unified Willow Tree Framework for One-Factor Short-Rate Models

The familiar binomial model allows the stock to move from the current node to just two possible prices in the next time step, which seriously limits the model’s ability to handle many realistic problems efficiently. The trinomial adds a branch and gains a lot more flexibility. The “willow tree” structure allows many branches at each node, and recent work has shown that it can produce significant improvements in performance for some classes of problems. Modeling interest rate behavior is an extremely important case. A viable short-rate model must allow stochastic volatility, and for practical use it must be arbitrage-free, meaning it can be calibrated to match the observed term structure of forward rates in the market. In this article, Wang and Xu show how to construct willow tree lattice models for eleven of the most common single-factor interest rate models, including those designed to be arbitrage-free. Unlike some of these models, the willow tree approach can handle interest rates that go negative, which is essential in the current environment.

Ultra-fast scenario analysis of mortgage prepayment risk

Stochastic scenario analysis of mortgage hedging strategies using single-CPU core machines is often too time consuming. In order to achieve a large practical speedup, we present two methods implemented on a many-core system consisting of graphical processing units (GPUs). The first method is based on Monte Carlo simulations, which are widely used in risk management. The second method relies on a parallel implicit finite difference (FD) discretization of a forward Kolmogorov equation. To estimate the speedup that can be achieved in practice, we compared the performance of both methods with an existing serial trinomial tree implementation on a single CPU core currently in use in our department. For both methods, a large speedup of roughly two orders of magnitude is achieved for realistic workloads. We show that the FD method is approximately four times faster than the Monte Carlo method when implemented on GPUs. On the other hand we argue that the Monte Carlo method is more adaptable to accommodate generic models, while the FD method is typically suitable to low dimensional models, such as single-factor interest rate models. To our knowledge, the application of GPUs for mortgage hedge calculations is new, as is the implementation of the FD method on GPUs.

The Role of the Risk-Neutral Jump Size Distribution in Single-Factor Interest Rate Models

We obtain a result that relates the risk-neutral jump size of interest rates with yield curve data. This function is unobservable; therefore, this result opens a way to estimate the jump size directly from data in the markets together with the risk-neutral drift and jump intensity estimations. Then, we investigate the finite sample performance of this approach with a test problem. Moreover, we analyze the effect of estimating the risk-neutral jump size instead of assuming that it is artificially absorbed by the jump intensity, as usual in the interest rate literature. Finally, an application to US Treasury Bill data is also illustrated.

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely, the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

Arbitrary Initial Term Structure within the CIR Model: A Perturbative Solution

Single factor interest rate models with constant coefficients are not consistent with arbitrary initial term structures. An extension which allows both arbitrary initial term structure and analytical tractability has been provided only in the Gaussian case. In this paper, within the context of the HJM methodology, we provide an extension of the CIR model which admits arbitrary initial term structure. We show how to calculate bond prices via a perturbative approach, and we provide closed formulas at every order. Since the parameter we select for the expansion is typically estimated to be small, the perturbative approach turns out to be adequate to our purpose. Using results on affine models, we estimate the extended CIR model via maximum likelihood on a time series of daily interest rate yields. Our results show that the CIR model has to be rejected with respect to the proposed extension, and point out that the extended CIR model provides a more flexible characterization of the link between risk neutral and natural probability.

Valuation of derivatives based on single-factor interest rate models

The CKLS [Chan, K.C., Karolyi, G.A., Longstaff, F.A., & Sanders, A.B. (1992). An empirical comparison of the short-term interest rate, Journal of Finance, 1(7), 1209–1227] short-term risk-free interest rate process leads to a valuation model for both default-free bonds and contingent claims that can only be solved numerically for the general case. Valuation equations of this nature in the past have been solved using the Crank Nicolson scheme. In this paper, we introduce a new numerical scheme — the Box method, and compare it with the traditional Crank Nicolson scheme. We find that in specific cases of the CKLS process where analytical prices are available, the new scheme leads to more accurate results than the Crank Nicolson scheme.

After-tax term structures of real interest rates: Inferences from the UK linked and non-linked gilt markets

This study estimates the after-tax term structure of real interest rates using the prices of UK linked and non-linked gilts over the period from 25 January 1986 until 25 October 1993. The impact of differential taxation and the existence of “noise” in observed market prices is found to produce a significant impact on the parameter estimation of term structure models when compared to methods, such as Brown and Schaefer (Brown, R., Schaefer, S., 1994. Journal of Financial Economics 35, 1–42), that did not. Two major observations can be made regarding the estimates for spot real interest rates. Firstly, the volatility of the short-term rate is much lower than that found by Brown and Schaefer (1994) which provides a better fit with the predictions of the CIR single factor model for interest rates. Secondly, consistent with Rumsey (Rumsey, J., 1993. An impact of the assumptions about taxes on the estimation of the properties of interest rates. Working Paper), there appears to be some evidence to suggest that single factor interest rate models produce a better fit to interest rates on an after-tax basis than on a before tax basis.

Factor Dependence of Bermudan Swaption Prices: Fact or Fiction?

This paper investigates the effect of interest rate correlation in the pricing of Bermudan swaptions. Investigating both Gaussian Markov models and Libor Market models, we find that Bermudan swaption prices depend only weakly on the number of factors in the underlying interest rate model. Moreover, we find that prices of standard Bermudan swaptions typically decrease slightly in the number of factors, primarily a consequence of effects on the time evolution of volatility induced by calibration of the model dynamics. Our findings are markedly different from those of Longstaff, Schwarz, and Santa-Clara (1999) who conclude that single-factor interest rate models significantly undervalue Bermudan swaptions. We argue that the conclusions of Longstaff, Schwarz, and Santa-Clara are due to non-standard choices of model dynamics and calibration methodology. Our study highlights the importance of using a reasonable set of calibration instruments when applying and comparing interest rate models.