Forward Rate Curve Research Articles

Finite Difference Methods for Hull-White Pricing of Interest Rate Derivatives

We study a novel implementation of the explicit and the implicit Crank-Nicolson (CN) numerical schemes for solving time-dependent Parabolic Partial Differential Equations (PDEs) in one spatial dimension in a variety of applications in computational finance related with the the One-Factor Hull-White Model. Finite differences on uniform grids are used for both the spatial and time discretization of the Hull-White PDE. Both implementations allow the use of dynamical Consistent Forward Rate Curves in Bjork sense among others, as the traditional use of the Forward Induction technique is avoided. As examples, we apply these methods to price European and American-style Bond Options, European-style Interest-Rate Caps as well as Digital Caps with no early provision. Numerical results comparing the efficiency of these numerical implementations are provided.

Pricing Stock Options by Bivariate Binomial Lattices

Pricing stock options by using the binomial lattice method still poses a significant challenge in the finance literature. This is particularly true for the bivariate binomial lattice method that simultaneously allows for stock prices and interest rates to be stochastic. This article introduces a more efficient bivariate binomial lattice method to price stock options. The proposed approach simply replaces the drift term of the stock price process by the integration of the forward rate curve over the horizon of the option's maturity. The numerical simulation studies as well as the theoretical justification show that the option prices computed by the proposed lattice approach can effectively and accurately converge.

Compact Embeddings for Spaces of Forward Rate Curves

The goal of this paper is to prove a compact embedding result for spaces of forward rate curves. As a consequence of this result, we show that any forward rate evolution can be approximated by a sequence of finite dimensional processes in the larger state space.

Explaining the Forward Interest Rate Term Structure

We present compelling empirical evidence for a new interpretation of the Forward Rate Curve (FRC) term structure. We find that the average FRC follows a square-root law, with a prefactor related to the spot volatility, suggesting a Value-at-Risk like pricing. We find a striking correlation between the instantaneous FRC and the past spot trend over a certain time horizon. This confirms the idea of an anticipated trend mechanism proposed earlier and provides a natural explanation for the observed shape of the FRC volatility. We find that the one-factor Gaussian Heath-Jarrow-Morton model calibrated to the empirical volatility function fails to adequately describe these features.

Models of the yield curve and the curvature of the implied forward rate function

We examine several alternative models of the UK gilt yield curve using daily data for the period 12 July 1996–10 February 2010. We select the best models according to two criteria: low out of sample errors in pricing bonds and low curvature of the implied forward rate curve function. We suggest additions to some of the models that significantly improve their performance. Some of the new models out perform those typically used by the central banks. In particular this paper suggests that the model used by the Canadian Central Bank which both outperforms other models and is particularly easy to estimate, is well suited to the UK gilt market.

On the calibration of a Gaussian Heath–Jarrow–Morton model using consistent forward rate curves

Este articulo forma parte de un numero monografico titulado Special Issue on Rates and FX.

Interpolation of forward price from liquidly traded forwards application to energy models

This article investigates how the infinite dimensional forward price process can be interpolated from finite dimensional standardized liquidly traded forwards. Under suitable restrictions on the forward price volatility and the forward rate volatility, it is shown that the forward price is an affine function of a finite number of some liquidly traded forwards. As a consequence, we find that the forward price is independent of the fluctuations of the interest rate, but still dependent on the structure of the forward rate curve. The results developed in the paper are illustrated by the study of Oil and Electricity forward price model.

ON FINITE DIMENSIONAL REALIZATIONS OF TWO-COUNTRY INTEREST RATE MODELS

This paper explores how consistent two-dimensional families of forward rate curves can be constructed on an international market. Applying the approach in Björk and Christenssen (1999) and Björk and Svensson (2001), we study when a system of inherently infinite dimensional domestic and foreign forward rate processes in a two-country economy with spot (forward) exchange rate possesses finite dimensional realizations. In the system with the forward exchange rate, the forward interest rate equations are supplemented by a third infinite dimensional stochastic differential equation representing the forward exchange rate dynamics. We construct and fit consistent families to observed Euro and USD yields as well as the forward (spot) EUR/USD exchange rate.

Hedging in a HJM model

This note shows how to hedge in a HJM model when the term structure evolution is Markov in the entire forward rate curve.

Estimation and inference in the yield curve model with an instantaneous error term

Many variations exist of yield curve modeling based on the exponential components framework, but most do not consider the generating process of the error term. In this paper, we propose a method of yield curve estimation using an instantaneous error term generated with a standard Brownian motion. First, we add an instantaneous error term to Nelson and Siegel’s instantaneous forward rate model [C.R. Nelson, A.F. Siegel, Parsimonious modeling of yield curves, Journal of Business 60 (1987) 473–489]. Second, after differencing multiperiod spot rate models transformed using Nelson and Siegel’s instantaneous forward rate model [C.R. Nelson, A.F. Siegel, Parsimonious modeling of yield curves, Journal of Business 60 (1987) 473–489], we obtain a model with serially uncorrelated error terms because of independent increment properties of Brownian motion. As the error term in this model is heteroskedastic and not serially correlated, we can apply weighted least squares estimation techniques. That is, this specification of the error term does not lead to incorrect estimation methods. In an empirical analysis, we compare the instantaneous forward rate curves estimated by the proposed method and an existing method. We find that the shape from the proposed estimation equation differ from the latter method when fluctuations in the interest rate data used for the estimation are volatile.

Swap Market Model: Theory and Empirical Evidence

This paper tests the co-terminal swap market model (SMM) pricing and hedging performance on Bermudan swaptions. To our knowledge, the drift for SMM is derived explicitly for the first time here, and the procedures for calibration and simulation using a collection of forward swap rates are also shown in detail. The Longstaff-Schwartz least square method is used to approximate the early exercise decision in Bermudan swaption. By introducing individual parameters for volatility of each co-terminal forward swap rate, the model can match the market quoted European swaption price perfectly. It is noted that, for the SMM, one particular volatility formula may not be enough to capture the term structure of different markets. Hedging performance of the model is tested on Euro and USD European co-terminal swaption using a set of swaps as hedge instrument. Principle component analysis (PCA) is adopted to capture the trend of the forward rates??? movement and absolute mean of the PCA factors is used for the bumping of forward rate curve. Hedge ratio is calculated based on the delta ratios with respect to PCA factors. P&L of the hedge portfolio generated by co-terminal SMM is examined on Euro and USD market. The result shows that more factors may be needed in order to improve the hedge performance of the model. Nevertheless, the SMM is very similar to LMM in terms of implementation and model performance, and the SMM is more convenient when dealing with exotic interest rate derivatives where swap rate is the underlying.

Finite‐dimensional Realizations of Regime‐switching HJM Models

This paper studies Heath–Jarrow–Morton‐type models with regime‐switching stochastic volatility. In this setting the forward rate volatility is allowed to depend on the current forward rate curve as well as on a continuous time Markov chain y with finitely many states. Employing the framework developed by Björk and Svensson we find necessary and sufficient conditions on the volatility guaranteeing the representation of the forward rate process by a finite‐dimensional Markovian state space model. These conditions allow us to investigate regime‐switching generalizations of some well‐known models such as those by Ho–Lee, Hull–White, and Cox–Ingersoll–Ross.

A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives

We develop a tractable and flexible stochastic volatility multifactor model of the term structure of interest rates. It features unspanned stochastic volatility factors, correlation between innovations to forward rates and their volatilities, quasi-analytical prices of zerocoupon bond options, and dynamics of the forward rate curve, under both the actual and risk-neutral measures, in terms of a finite-dimensional affine state vector. The model has a very good fit to an extensive panel dataset of interest rates, swaptions, and caps. In particular, the model matches the implied cap skews and the dynamics of implied volatilities. (JEL E43, G13)

The Monetary Policy Decision-Making Process and the Term Structure of Interest Rates

This paper presents a theoretical model of the term structure of interest rates based on the monetary policy decision-making process at modern central banks. Evaluations of explicit expressions for the spot and forward rate curve render several important results: (i) Spot and forward rates are explicit functions of the number of policy meetings during the time to maturity rather than the time to maturity itself. Consequently, the forward rate curve is step-shaped. (ii) In addition, there are calendar time effects, i.e. the position within the policy cycle is also of importance, especially for short term interest rates. (iii) The forward rate curve exhibits hump-shaped responses to economic shocks and a modified version of the Nelson-Siegel model can be obtained as a special case.

Pricing and Evaluating a Bond Portfolio Using a Regime Switching Markov Model

We apply a regime switching Markov chain model to determine bond prices. Our work builds upon the work of Thomas, Allen & Morkel-Kingsbury (2002) and Jarrow, Lando & Turnbull (1997). The interest rate process and credit rating migration process are considered. Our aim is to study the price evolution of a portfolio of defaultable bonds. We are interested in determining the density process to compute the Value at Risk (VaR) and Conditional Value at Risk (CVaR) a year ahead. Whereas Thomas et al. describe a model under the pricing measure only, our model takes into consideration both the physical and pricing measures. We also describe the whole (risk free) forward rate curve, with appropriate conditions for the absence of arbitrage, while Thomas et al. describe the spot rate process. Most bond models make use of zero coupon bond prices. By using bond stripping we are able to discover zero coupon bond prices for bonds of difierent ratings. We use a mathematical programming approach to strip coupons and minimise squared error, subject to no arbitrage constraints.

Random field forward interest rate models, market price of risk and their statistics

In this paper we consider discrete time forward interest rate models. In our approach, unlike in the classical Heath–Jarrow–Morton framework, the forward rate curves are driven by a random field. Hence we get a general interest rate structure. Our aim is to give an overview of our results in such a model on the following questions: no-arbitrage conditions, maximum likelihood estimation of the volatility, as well as the joint estimation of the parameters and the asymptotic behaviour of the estimators, relationship with continuous models. Finally we give discussion on the practical problems of the estimation and we show several numerical results on the statistics of such models.

Mean Reversion Level Extensions of Time‐Homogeneous Affine Term Structure Models

It is well‐known that time‐homogeneous affine term structure models are incompatible with most observed initial forward rate curves. For the Vasiček (1977) and Cox et al. (1985) models, time‐inhomogeneous extensions capable of fitting any given initial forward rate curve were introduced in Hull and White (1990), and similar extensions, for short rate models in general, were introduced in Björk and Hyll (2000), Brigo and Mercurio (2001), and Kwon (2004). In this paper, we introduce a general and systematic method for obtaining time‐inhomogeneous extensions of affine term structure models that are compatible with any observed initial forward rate curve. These extensions are minimal in the sense that the system of Riccati equations determining the bond prices remain essentially unchanged under the extension. Moreover, the extensions considered in Björk and Hyll (2000), Brigo and Mercurio (2001), and Kwon (2004), for time‐homogeneous affine term structure models, are all special cases of the extensions introduced in this paper.

Model misspecification analysis for bond options and Markovian hedging strategies

In this paper, we analyse the model misspecification risk of Markovian hedging strategies for discount bond options. We show how to decompose the Profit and Loss that results from model misspecification, and emphasize the importance of the position’s gamma in order to control it. We further provide mathematical results on the distribution of the forward Profit and Loss function for specific univariate term structure models. Finally, we run numerical simulations for options’ hedging strategies in order to examine the sensitivity of the forward Profit and Loss function with respect to the volatility of the forward rate curve, the frequency of the position rebalancing and the characteristics of the position being hedged.

A General Stochastic Volatility Model for the Pricing of Interest Rate Derivatives

We develop a tractable and flexible stochastic volatility multi-factor model of the term structure of interest rates. It features unspanned stochastic volatility factors, correlation between innovations to forward rates and their volatilities, quasi-analytical prices of zero-coupon bond options, and dynamics of the forward rate curve, under both the actual and risk-neutral measure, in terms of a finitedimensional affine state vector. The model has a very good fit to an extensive panel data set of interest rates, swaptions and caps. In particular, the model matches the implied cap skews and the dynamics of implied volatilities.

Duration, factor sensitivities, and interest rate Greeks

In this paper, we introduce for interest rate sensitive assets the natural analogs of delta and gamma for equity options by considering the derivatives of asset prices with respect to the directions along which the forward rate curve may evolve. Macaulay duration and convexity, as well as stochastic duration considered in Cox et al. (J Business 52:51–61, 1979) and Munk (Rev Derivat Res 3:157–181, 1999), are easily obtained as special cases of these in which the derivatives are computed along parallel shifts and the direction of the forward rate volatilities, respectively. Moreover, we demonstrate using the example of the Ritchken and Sankarasubramanian (Math Financ 5:55–72, 1995) model that the hedging strategy based on these sensitivity measures provides a superior performance in comparison to the traditional duration based hedging approaches.