Fitting the term structure of interest rates for Taiwanese government bonds

The term structure of interest rates provides a basis for pricing fixed-income securities and interest rate derivative securities as well as other capital assets. Unfortunately, the term structure is not always directly observable because most of the substitutes for default-free bonds are not pure discount bonds. We use curve fitting techniques with the observed government coupon bond prices to estimate the term structure. In this paper, the B-spline approximation is used to estimate the Taiwanese Government Bond (TGB) term structure. We apply the B-spline functions to approximate the discount function, spot yield curve, and forward yield curve respectively. Among the three approaches, the discount fitting approach and the spot fitting approach are reasonable and reliable, but the spot fitting approach achieves the most suitable fit. Using this methodology, we can investigate term structure fitting problems, identify coupon effects, and analyze factors which drive term structure fluctuations in the TGB market.     

Estimating yield curves of the U.S. Treasury securities: An interpolation approach

Following the approach of interpolation, this paper proposes the multiple exponential decay model to fit yield curves for both the U.S. TIPS market and the conventional Treasury security market. Several estimation methods, including the unconstrained/constrained nonlinear minimization, quadratic programming, and the iterative linear least squares, are applied to estimate the unknown parameters according to different curve‐fitting purposes. Comparisons between the proposed model and the alternatives show that the multiple exponential decay successfully (1) adapts to a variety of shapes associated with yield curves, (2) (partially) keeps in line with the economic interpretations of Nelson–Siegel summarized by Diebold and Li ( 2006 ), and (3) dominates the competing models in curve‐fitting performance measured by mean fitted‐price errors over the sample period. In addition, the exact specification of a nonparametric interpolation model is pinned down by applying three statistical tools, which enable us to jointly take into account validity, optimality, and parsimoniousness of the proposed model.     

Bond pricing with a surface of zero coupon yields

We present a new method for consistent cross‐sectional pricing of all traded bonds in the fixed income market. By applying thin plate regression splines ( Wood, 2003 ) to bootstrapped zero coupon bond yields ( Hagan and West, 2006 ), the method decomposes traded yields into a risk‐free component plus premia for credit and liquidity risks, where the decomposition is consistent with the market valuations and underlying cash flows of the bonds. We apply the framework to end of quarter yield data from 2008 to 2011 on Australian dollar denominated semi‐government, supranational and agency (SSA) bonds, and find that the surface provides an excellent fit to the underlying zero coupon yield curves. Further, the decomposition of selected yield time series and cross‐sections demonstrates how credit premia increased for Australian SSA bonds through the Global Financial Crisis (GFC), but were counterbalanced by liquidity discounts as investors sought safe haven securities.     

Nonparametric Smoothing of Yield Curves

This paper proposes a new nonparametric approach to the problem of inferring term structure estimates using coupon bond prices. The nonparametric estimator is defined on the basis of a penalized least squares criterion. The solution is a natural cubic spline, and the paper presents an iterative procedure for solving the non-linear first-order conditions. Besides smoothness, there are no a priori restrictions on the yield curve, and the position of the knots and the optimal smoothness can be determined from data. For these reasons the smoothing procedure is said to be completely data driven. The paper also demonstrates that smoothing a simple transformation of the yield curve greatly improves the stability of longer-term yield curve estimates.     

EXACT FORMULAS FOR PRICING BONDS AND OPTIONS WHEN INTEREST RATE DIFFUSIONS CONTAIN JUMPS

I develop Heath‐Jarrow‐Morton extensions of the Vasicek and Jamshidian pure‐diffusion models, extend these models to incorporate Poisson‐Gaussian interest rate jumps, and obtain closed‐form models for valuing default‐free, zero‐coupon bonds and European call and put options on default‐free, zero‐coupon bonds in a market where interest rates can experience discontinuous information shocks. The jump‐diffusion pricing models value the instrument as the probability‐weighted average of the pure‐diffusion model prices, each conditional on a specific number of jumps occurring during the life of the instrument. I extend the models to coupon‐bearing instruments by applying Jamshidian's serial‐decomposition technique.     

基于贝塞尔曲线的中、美、日利率期限结构静态研究

利率期限结构曲线的静态拟合是指,使用不同类型的数学函数近似地描述整条利率期限结构曲线。当前最流行的静态拟合方法是利用B样条曲线来拟合利率曲线。然而,该方法往往受制于阶数的限制,而仅仅停留在3阶。本文通过利用B样条曲线的特殊形式——Bezier曲线拟合了中国、美国、日本国债利率的期限结构曲线,获得了一种可以升阶的拟合方法。同时,将复杂的曲线拟合计算,简化为对散点的聚类分析,取得了中国利率期限结构的模型。     

Presentation of the English translation of Ettore Majorana's paper: The value of statistical laws in physics and social sciences

We consider a consistent pricing model of government bonds, interest-rate swaps and basis swaps in one currency within the no-arbitrage framework. To this end, we propose a three yield-curve model, one for discounting cash flows, one for calculating LIBOR deposit rates and one for calculating coupon rates of government bonds. The derivation of the yield curves from observed data is presented, and the option prices on a swap or a government bond are studied. A one-factor quadratic Gaussian model is proposed as a specific model, and is shown to provide a very good fit to the current Japanese low-interest-rate environment.     

Explosion in the quasi-Gaussian HJM model

We study the explosion of the solutions of the SDE in the quasi-Gaussian HJM model with a CEV-type volatility. The quasi-Gaussian HJM models are a popular approach for modeling the dynamics of the yield curve. This is due to their low-dimensional Markovian representation which simplifies their numerical implementation and simulation. We show rigorously that the short rate in these models explodes in finite time with positive probability, under certain assumptions for the model parameters, and that the explosion occurs in finite time with probability one under some stronger assumptions. We discuss the implications of these results for the pricing of the zero coupon bonds and Eurodollar futures under this model.     

A Two-Factor Model for Low Interest Rate Regimes

This paper derives a two-factor model for the term structure of interest rates that segments the yield curve in a natural way. The first factor involves modelling a non-negative short rate process that primarily determines the early part of the yield curve and is obtained as a truncated Gaussian short rate. The second factor mainly influences the later part of the yield curve via the market index. The market index proxies the growth optimal portfolio (GOP) and is modelled as a squared Bessel process of dimension four. Although this setup can be applied to any interest rate environment, this study focuses on the difficult but important case where the short rate stays close to zero for a prolonged period of time. For the proposed model, an equivalent risk neutral martingale measure is neither possible nor required. Hence we use the benchmark approach where the GOP is chosen as numeraire. Fair derivative prices are then calculated via conditional expectations under the real world probability measure. Using this methodology we derive pricing functions for zero coupon bonds and options on zero coupon bonds. The proposed model naturally generates yield curve shapes commonly observed in the market. More importantly, the model replicates the key features of the interest rate cap market for economies with low interest rate regimes. In particular, the implied volatility term structure displays a consistent downward slope from extremely high levels of volatility together with a distinct negative skew. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Forward-neutral valuation relationships for options on zero coupon bonds

This paper extends the literature on Risk-Neutral Valuation Relationships (RNVRs) to derive valuation formulae for options on zero coupon bonds when interest rates are stochastic. We develop Forward-Neutral Valuation Relationships (FNVRs) for the transformed-bounded random walk class. Our transformed-bounded random walk family of forward bond price processes implies that (i) the prices of the zero coupon bonds are bounded below at zero and above at one, and (ii) negative continuously compounded interest rates are ruled out. FNVRs are frameworks for option pricing, where the forward prices of the options are martingales independent of the market prices of risk. We illustrate the generality and flexibility of our approach with models that yield several new closed-form solutions for call and put options on discount bonds.     

VaR limits for pension funds: an evaluation

In a series of papers during the last ten years an interest rate theory with models which are driven by Lévy or more general processes has been developed. In this paper we derive explicit formulas for the correlations of interest rates as well as zero coupon bonds with different maturities. The models considered in this general setting are the forward rate (HJM), the forward process and the LIBOR model as well as the multicurrency extension of the latter. Specific subclasses of the class of generalized hyperbolic Lévy motions are studied as driving processes. Based on a data set of parametrized yield curves derived from German government bond prices we estimate correlations. In a second step the empirical correlations are used to calibrate the Lévy forward rate model. The superior performance of the Lévy driven models becomes obvious from the graphs.     

MODELING THE TERM STRUCTURE FROM THE ON‐THE‐RUN TREASURY YIELD CURVE

We propose a new model to estimate the term structure of interest rates using observed on‐the‐run Treasury yields. The new model is an improvement over models that require a priori knowledge of the shape of the yield curve to estimate the term structure. The general form of the model is an exponential function that depends on the estimation of four parameters fit by nonlinear least squares and has straightforward interpretations. In comparing the proposed model with current yield‐curve‐smoothing models, we find that, for the data used, the proposed model does best overall in terms of pricing accuracy both in sample and out of sample. JEL classification: E43, G12     

The Slope of the Credit Yield Curve for Speculative-Grade Issuers

Many theoretical bond pricing models predict that the credit yield curve facing risky bond issuers is downward-sloping. Previous empirical research (Sarig and Warga (1989), Fons (1994)) supports these models. Our study examines sets of bonds issued by the same firm with equal priority in the liability structure, but with different maturities, thus holding credit quality constant. We find, counter to prior research, that risky bonds typically have upward-sloping credit yield curves. Moreover, when we combine our matched sets of bonds (no longer controlling credit quality), the estimated slope is negative, indicating a sample selection bias problem associated with maturity.     

Hedging Longevity Risk When Interest Rates are Uncertain

Abstract

This paper proposes an asset liability management strategy to hedge the aggregate risk of annuity providers under the assumption that both the interest rate and mortality rate are stochastic. We assume that annuity providers can invest in longevity bonds, long-term coupon bonds, and shortterm zero-coupon bonds to immunize themselves from the risks of the annuity for the equity holders subject to a required profit. We demonstrate that the optimal allocation strategy can lead to the lowest risk under different yield curves and mortality rate assumptions. The longevity bond can also be regarded as an effective hedging vehicle that significantly reduces the aggregate risk of the annuity providers.     

Stochastic durations,the convexity effect,and the impact of interest rate changes

This article shows that the equilibrium models of bond pricing do not preclude arbitrage opportunities caused by convexity. Consequently, stochastic durations derived from these models are limited in their ability to act as interest rate risk measures. The research of the present article makes use of an intertemporal utility maximization framework to determine the conditions under which duration is an adequate interest rate risk measure. Additionally, we show that zero coupon bonds satisfy those equilibrium conditions, whereas coupon bonds or bond portfolios do not as a result of the convexity effect. The results are supported by empirical evidence, which confirms the influence of convexity on the deviation of coupon bond returns from equilibrium.     

Empirically Effective Bond Pricing Model and Analysis on Term Structures of Implied Interest Rates in Financial Crisis

In his book (1993) Kariya proposed a government bond (GB) pricing model that simultaneously values individual fixed-coupon (non-defaultable) bonds of different coupon rates and maturities via a discount function approach, and Kariya and Tsuda (Financ Eng Japanese Mark 1:1–20, 1994) verified its empirical effectiveness of the model as a pricing model for Japanese Government bonds (JGBs) though the empirical setting was limited to a simple case. In this paper we first clarify the theoretical relation between our stochastic discount function approach and the spot rate or forward rate approach in mathematical finance. Then we make a comprehensive empirical study on the capacity of the model in view of its pricing capability for individual GBs with different attributes and in view of its capacity of describing the movements of term structures of interest rates that JGBs imply as yield curves. Based on various tests of validity in a GLS (Generalized Least Squares) framework we propose a specific formulation with a polynomial of order 6 for the mean discount function that depends on maturity and coupon as attributes and a specific covariance structure. It is shown that even in the middle of the Financial Crisis, the cross-sectional model we propose is shown to be very effective for simultaneously pricing all the existing JGBs and deriving and describing zero yields.     

Identification of Maximal Affine Term Structure Models

Building on Duffie and Kan (1996) , we propose a new representation of affine models in which the state vector comprises infinitesimal maturity yields and their quadratic covariations. Because these variables possess unambiguous economic interpretations, they generate a representation that is globally identifiable. Further, this representation has more identifiable parameters than the “maximal” model of Dai and Singleton (2000) . We implement this new representation for select three‐factor models and find that model‐independent estimates for the state vector can be estimated directly from yield curve data, which present advantages for the estimation and interpretation of multifactor models.     

CREDIT SPREADS AND THE ZERO‐COUPON TREASURY SPOT CURVE

We examine the relation between credit spreads on industrial bonds and the underlying Treasury term structure. We use zero‐coupon spot rates to eliminate the coupon bias and to allow for a consistent study both within and across the different credit ratings. Our results indicate that the level and slope of the Treasury term structure are negatively correlated with changes in the credit spread on investment‐grade corporate bonds. We also find that the relation between credit spreads and the Treasury term structure is relatively stable through time. This is good news for value‐at‐risk calculations, as this suggests that the correlations among assets of different credit classes are stable; therefore use of historic correlations to model spread relations can be valid.     

OPTIONS ON U.S. TREASURY COUPON ISSUES

Pricing models for options on default-free coupon bonds are developed and tested under the assumption that the bond prices, rather than interest rates, are the underlying stochastic factors. Under the assumption that coupon bond prices, excluding accrued interest, follow a generalized Brownian bridge process, preference-free, continuous-time pricing models are developed for European put and call options, and a discrete-time model is developed for American puts and calls. The empirical validity of the models is assessed using a six-moth sample of daily closing prices.     

A constrained least square approach to the estimation of the term structure of interest rates

This paper proposes a new practical method for estimating forward rate curves using bond prices available in the market. It is intended to improve the least square estimation method proposed by Carleton and Cooper by imposing additional constraints to guarantee the smoothness of the forward rate curves. The resulting problem is a nonconvex minimization problem, for which we will propose an efficient algorithm for calculating an approximate optimal solution. Computational experiments show that this method can efficiently generate smooth forward rate curves without increasing the residual errors in terms of least square fitting. Also, we will compare this result with an alternative and more efficient constrained least absolute deviation method.