Narrow Results

Term structure is a useful curve describing some financial asset as a function of time to maturity or expiration. In this paper, we propose to use Independent Component Analysis (ICA) to model the term structure of multiple yield curves. The idea is that we first employ ICA to decompose the multivariate time series, then we suggest two ICA methods for dimension reduction and pattern recognition of the term structure. We also compare the results by using an alternative method, Principal Component Analysis (PCA). The empirical studies suggest that the proposed ICA approaches outperform PCA methods in modeling the term structure. This model can be used in financial time series analysis as well as related financial applications.

Previous work on multifactor term structure models has proposed that the short rate process is a function of some unobserved diffusion process. We consider a model in which the short rate process is a function of a Markov chain which represents the "state of the world". This enables us to obtain explicit expressions for the prices of zero-coupon bonds and other securities. Discretizing our model allows the use of signal processing techniques from Hidden Markov Models. This means we can estimate not only the unobserved Markov chain but also the parameters of the model, so the model is self-calibrating. The estimation procedure is tested on a selection of U.S. Treasury bills and bonds.

A number of techniques have been proposed for estimating the term structure, yet solid theoretical foundations and a comparative assessment of the results produced by these techniques are not available. In the present paper we prove, within a well defined mathematical setting, how the existence of discount factors is possible only if the condition of absence of static arbitrage is fulfilled. Besides that we report the results of an extensive set of tests whose scope is to show how the most widely used approaches in this field behave on both real and synthetic data.

A number of numerical methods based on a piecewise polynomial approximation have been proposed for the estimation of the term structure of interest rates. Some drawbacks have been pointed out, such as a possible non monotonic estimated discount function and a highly fluctuating spot and forward rates. In order to overcome these kind of problems, we study the feasibility of an adaptive regression spline technique which use a monotone basis together with two alternative knot location procedures: a deterministic greedy algorithm and its randomized version in a simulated annealing framework. The features of the proposed method are tested on a set of data.

We will develop an efficient method for estimating a smooth nonnegative forward rate sequence using the market price of riskless bonds. This method is an improvement of the classical Carleton–Cooper's method based on standard least square method, which often generates a non-smooth forward rate sequence and hence is not used in practice. The method to be proposed in this paper is intended to resolve this difficulty. We will impose a smoothness condition while maintaining the fitting error within an acceptable level. The resulting optimization problem is shown to be convex in the region of interest. Therefore, we can calculate a globally optimal solution very fast by standard nonlinear programming algorithms. We will demonstrate that this method generates a smooth forward rate sequence at the expense of a very small increase of fitting error.

This paper develops a general equilibrium model of the term structure of interest rates in the presence of the systematic risk of regime shifts. The model elucidates the economic nature of the regime-shift risk premium and introduces a new source of time-variation in bond returns. A closed-form solution for the term structure of interest rates is obtained under an affine model using log-linear approximation. The model is estimated by Efficient Method of Moments. The regime-switching risk is found to be statistically significant and mostly affect the long-end of the yield curve.

The innovative information-based framework for credit risk modeling, proposed recently by Brody, Hughston, and Macrina, is extended to a more general and practically important setup of random interest rates. We first introduce the market model, and we derive an explicit expression for defaultable bond price. Next, the dynamics of the information process and dynamics of defaultable bond are found for both deterministic and random interest rates. Finally, the valuation and hedging of derivative securities are briefly examined. In particular, the valuation formula for a European option on a defaultable bond is established.

In this study, we forecast the term structure of EURIBOR swap rates by means of rolling vector autoregressive (VAR) models. In advance, a principal component analysis (PCA) is adopted to reduce the dimensionality of the term structure. To statistically assess the forecasting performance for particular rates and the level, slope and curvature of the swap term structure, we rely on the Henrikkson–Merton statistic. Economic performance is investigated by means of cash flows implied by alternative trading strategies. Finally, a data-driven, adaptive model selection strategy to "predict the best forecasting model" out of a set of 100 alternative PCA/VAR implementations is shown to outperform forecasting schemes that rely on global homogeneity of the term structure.

We study the term structure of the implied volatility in the presence of a symmetric smile. Exploiting the result by Tehranchi (2009) that a symmetric smile generated by a continuous martingale necessarily comes from a mixture of normal distributions, we derive representation formulae for the at-the-money (ATM) implied volatility level and curvature in a general symmetric model. As a result, the ATM curve is directly related to the Laplace transform of the realized variance. The representation formulae for the implied volatility and its curvature take semi-closed form as soon as this Laplace transform is known explicitly. To deal with the rest of the volatility surface, we build a time dependent SVI-type (Gatheral, 2004) model which matches the ATM and extreme moneyness structure. As an instance of a symmetric model, we consider uncorrelated Heston: in this framework, the SVI approximation displays considerable performances in a wide range of maturities and strikes. All these results can be applied to skewed smiles by considering a displaced model. Finally, a noteworthy fact is that all along the paper we avoid dealing with any complex-valued function.

In this paper, we present a multi-factor continuous-time autoregressive moving-average (CARMA) model for the short and forward interest rates. This model is able to present an adequate statistical description of the short and forward rate dynamics. We show that this is a tractable term structure model and provides closed-form solutions to bond prices, yields, bond option prices, and the term structure of forward rate volatility. We demonstrate the capabilities of our model by calibrating it to a panel of spot rates and the empirical volatility of forward rates simultaneously, making the model consistent with both the spot rate dynamics and forward rate volatility structure.

In this paper, we show how to construct dynamic forward rate models in terms of exogenously specified eigenfunctions (or factor loadings). We also show how to link forward rate models with different number of driving Brownian motions to each other in a way consistent with the implied eigenfunctions. Finally, we discuss how to best parameterize the models in the sense of maximizing the number of free parameters for a given set of eigenfunctions.

We consider an arbitrage-free futures price model of Heath–Jarrow–Morton type which is driven by a multidimensional Wiener process and a marked point process. We find necessary and sufficient conditions for this model to produce a log futures curve that changes only through parallel shifts. The same analysis is carried out for the case when the log futures curve changes only through proportional shifts. We prove that there exist nontrivial parallel and proportional shifting log futures curves and we show how to specify the futures price model in order to obtain them. Additionally the shift functions are characterized. Finally, we consider the case of all other single-factor affine models which are neither parallel nor proportional shifting curves. We find necessary and sufficient conditions for the purely Wiener-driven log futures model to admit such other affine shifting curve and we characterize the shift functions.

We introduce here the idea of a long-term swap rate, characterized as the fair rate of an overnight indexed swap (OIS) with infinitely many exchanges. Furthermore, we analyze the relationship between the long-term swap rate, the long-term yield, (F. Biagini, A. Gnoatto & M. Härtel (2018) Affine HJM Framework on $S_d^{+}$ and long-term yield, Applied Mathematics and Optimization77 (3), 405–441, F. Biagini & M. Härtel (2014) Behavior of long-term yields in a lévy term structure, International Journal of Theoretical and Applied Finance17 (3), 1–24, N. El Karoui, A. Frachot & H. Geman (1997) A note on the behavior of long zero coupon rates in a no arbitrage framework. Working Paper. Available at Researchgate: https://www.researchgate.net/publication/5066730), and the long-term simple rate (D. C. Brody & L. P. Hughston (2016) Social discounting and the long rate of interest, Mathematical Finance28 (1), 306–334) as long-term discounting rate. Finally, we investigate the existence of these long-term rates in two-term structure methodologies, the Flesaker–Hughston model and the linear-rational model. A numerical example illustrates how our results can be used to estimate the nonoptional component of a CoCo bond.

In this paper, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipović’s maximal degree theorem for exponential polynomial models, as well as an explicit characterization of the set of feasible parameters in the case when the factor process is bounded. Extensions to time-inhomogeneous and multi-factor polynomial models are also considered.

We provide a full classification of all attainable term structure shapes in the two-factor Vasicek model of interest rates. In particular, we show that the shapes normal, inverse, humped, dipped and hump-dip are always attainable. In certain parameter regimes, up to four additional shapes can be produced. Our results apply to both forward and yield curves and show that the correlation and the difference in mean-reversion speeds of the two factor processes play a key role in determining the scope of attainable shapes. The key mathematical tool is the theory of total positivity, pioneered by Samuel Karlin and others in the 1950s.

In this paper, we analyze the shapes of forward curves and yield curves that can be attained in the two-factor Vasicek model. We show how to partition the state space of the model, such that each partition is associated to a particular shape (normal, inverse, humped, etc.). The partitions and the corresponding shapes are determined by the winding number of a single curve with possible singularities and self-intersections, which can be constructed as the envelope of a family of lines. Building on these results, we classify possible transitions between term structure shapes, give results on attainability of shapes conditional on the level of the short rate, and propose a simple method to determine the relative frequency of different shapes of the forward curve and the yield curve.

Estimation of benchmark yield curve in developing markets is often influenced by liquidity concentration. Based on an affine term structure model, we develop a long run liquidity weighted fitting method to address the trading concentration phenomenon arising from horizon-induced clientele equilibrium as well as information discovery. Specifically, we employ arguments from models of liquidity concentration and benchmark security information. After examining time series behavior of price errors against our fitted model, we find results consistent with both the horizon and information hypotheses. Our evidence indicates that trading liquidity carries information effect in the long run, which cannot be fully captured in the short run. Trading liquidity plays a key role in long run term structure fitting. Markets for liquid benchmark government bond issues collectively form a long term equilibrium. Compared with previous studies, our results provide a robust and realistic characterization of the spot rate term structure and related price forecasting over time, which in turn help portfolio investment of fixed income and long run pricing of financial instruments.

This paper is a study of the term structure of interest rates based on the Heath–Jarrow–Morton (HJM) models with Hull–White volatility function. Under fast mean-reverting stochastic volatility, we obtain an analytic formula for an approximate bond price with estimated error using a Markovian transform method combined with a singular perturbation method. The stochastic volatility correction effect against time-to-maturity is revealed so that it can capture more of the complexities of the interest rate term structure.

A variety of approaches have been proposed to extend classical fixed income portfolio immunization theory to cases where shifts in the term structure are not parallel. Following Reitano (1991a, 1991b, 1992, 1996) and Poitras (2007), this paper uses partial durations and convexities to specify benchmark partial immunization bounds for non-parallel term structure shifts. Theoretical results are obtained by exploiting properties of the multivariate Taylor series expansion of the spot interest rate pricing function. It is demonstrated that the partial immunization bounds can be effectively manipulated by adequate selection of the securities being used to immunize the portfolio. The inclusion of time values permits the results obtained to be related to previous studies by Christensen and Sorensen (1994), Chance and Jordan (1996), Barber and Copper (1997) and Poitras (2005, Ch. 5) on the time value-convexity tradeoff.

We present a dynamic term structure model in which interest rates of all maturities are bounded from below at zero. Positivity and continuity, combined with no arbitrage, result in only one functional form for the term structure with three sources of risk. We cast the model into a state-space form and extract the three sources of systematic risk from both the US Treasury yields and the US dollar swap rates. We analyze the different dynamic behaviors of the two markets during credit crises and liquidity squeezes.