Limiting Connection between Discrete and Continuous Time Forward Interest Rate Curve Models

We study discrete time Heath–Jarrow–Morton (HJM) type of interest rate curve models, where the forward interest rates – in contrast to the classical HJM models – are driven by a random field. Our main aim is to investigate the relationship between the discrete time forward interest rate curve model and its continuous time counterpart. We derive a general result on the convergence of discrete time models and we give special focus on the nearly unit root spatial autoregression model.     

A multiple-curve HJM model of interbank risk

In the aftermath of the 2007?C2009 financial crisis, a variety of spreads have developed between quantities that had been essentially the same until then, notably LIBOR?COIS spreads, LIBOR?COIS swap spreads, and basis swap spreads. By the end of 2011, with the sovereign credit crisis, these spreads were again significant. In this paper we study the valuation of LIBOR interest rate derivatives in a multiple-curve setup, which accounts for the spreads between a risk-free discount curve and LIBOR curves. Towards this end we resort to a defaultable HJM methodology, in which these spreads are explained by an implied default intensity of the LIBOR contributing banks, possibly in conjunction with an additional liquidity factor. Markovian short rate specifications are given in the form of an extended CIR and a Lévy Hull?CWhite model for a risk-free short rate and a LIBOR short spread. The use of Lévy drivers leads to the more parsimonious specification. Numerical values of the FRA spreads and the basis swap spreads computed with the latter largely cover the ranges of values observed even at the peak of the 2007?C2009 crisis.     

Convexity Bias in the Pricing of Eurodollar Swaps

The traditional use of LIBOR futures prices to obtain surrogates for the Eurodollar forward rates is proved to yield a systematic bias in the pricing of Eurodollar swaps when one assumes that the yield curve is well described by the Heath-Jarrow-Morton model. The resulting theoretical inequality is consistent with the empirical observations of Burghardt and Hoskins (1995), and it provide a theoretical basis for price anomalies that are suggested by more recent empirical data.     

On Gaussian HJM framework for Eurodollar Futures

One of the standard tools for the theoretical analysis of fixed income securities and their associated derivatives is the term structure model of Heath, Jarrow and Morton. In this paper the question, what specific HJM model is consistent with the observed price of an Eurodollar Futures contract? is discussed. Eurodollar Futures, apart from being the most heavily traded futures are connected to London Inter Bank Offered Rate (LIBOR) and to domestic monetary conditions. The answer to the above question will help in pricing any new derivative on Eurodollar Futures or the one that is not heavily traded. A simple tool to measure the adequacy of different HJM structures that may be used to model Eurodollar Futures price process is suggested. Moreover, the question of estimation of parameters of these models by different methods—method of realized volatility, method of maximum likelihood (ML) and a two‐stage method that combines both the realized volatility and ML—is addressed. Although it sounds like a typical statistical procedure, one must be careful in applying standard statistical techniques that are not suitable under arbitrage theory, in particular, ML method. Copyright © 2010 John Wiley & Sons, Ltd.     

Small-noise limit of the quasi-Gaussian log-normal HJM model

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 greatly simplifies their numerical implementation. We present a qualitative study of the solutions of the quasi-Gaussian log-normal HJM model. Using a small-noise deterministic limit we show that the short rate may explode to infinity in finite time. This implies the explosion of the Eurodollar futures prices in this model. We derive explicit explosion criteria under mild assumptions on the shape of the yield curve.     

Energy futures prices: term structure models with Kalman filter estimation

We present a class of multi-factor stochastic models for energy futures prices, similar to the interest rate futures models recently formulated by Heath. We do not postulate directly the risk-neutral processes followed by futures prices, but define energy futures prices in terms of a spot price, not directly observable, driven by several stochastic factors. Our formulation leads to an expression for futures prices which is well suited to the application of Kalman filtering techniques together with maximum likelihood estimation methods. Based on these techniques, we perform an empirical study of a one- and a two-factor model for futures prices for natural gas.     

Strong consistency of maximum likelihood estimators for a discrete-time random field HJM-type interest rate model

We consider a discrete time Heath–Jarrow–Morton-type forward interest rate model, where the interest rate curves are driven by a geometric spatial autoregression field. Strong consistency of maximum likelihood estimators is proved for stable and unstable no-arbitrage models containing a simple stochastic discounting factor. This research was supported by the Hungarian Scientific Research Fund under Grant No. OTKA–T048544/2005.     

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.     

A mean reverting process for pricing treasury bills and futures contracts

An analytically tractable, discrete-time single-factor model is developed for valuing treasury bills and futures contracts. It uses a multiplicative binomial foward process that creates neither negative nor implausibly large positive interest factors, and which can incorporate different possible degrees of mean reversion. The paper derives explicit formulae for bill prices, futures prices, their conditional variances and risk premia in a setting that relates the evolution of the term structure more closely to both model and data than do other similar works. In contrast to other term-structure constrained models, this paper emphasizes that in a one-factor model the martingale probabilities cannot be treated independently of the perturbation functions. The paper's empirical methods also differ from the customary approaches. Instead of comparing differences between model-predicted and observed prices, the paper applies ARCH methodology to test model-predicted ratios of conditional variances to risk premia. Our tests find influences exogenous to the model, but these factors do not seem capable of being explained with two-factor models using only interest rates.     

Stochastic models for bidding strategies on oligopoly electricity market

In this paper we consider the forward/futures contracts and Asian-type call options for power delivery as important components of the bidding strategies of the players’ profits on the electricity market. We show how these derivatives can affect their profit. We use linear asymmetric supply function equilibrium (SFE) and Cournot models to develop firms’ optimal bidding strategies by including forward/futures contracts and Asian-type options. We extend the methodology proposed by Niu et al. (IEEE Trans Power Syst 20(4):1859–1867, 2005), where only forward contracts for power delivery were considered in the SFE model.     

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. This research has been supported by the Hungarian Scientific Research Fund under Grants No. OTKA–F046061/2004 and OTKA–T048544/2005.     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

On the on-line rent-or-buy problem in probabilistic environments

Fujiwara and Iwama [In: The 13th Annual International Symposium on Algorithms and Computation, pp. 476–488 (2002)] first integrated probability distribution into the classical competitive analysis to study the rental problem. They assumed that the future inputs are drawn from an exponential distribution, and obtained the optimal competitive strategy and the competitive ratio by the derivative method. In this paper, we introduce the interest rate and tax rate into the continuous model of Fujiwra and Iwama [In: The 13th Annual International Symposium on Algorithms and Computation, pp. 476–488 (2002)]. Moreover, we use the forward difference method in different probabilistic environments to consider discrete leasing models both with and without the interest rate. We not only give the optimal competitive strategies and their competitive ratios in theory, but also give numerical results. We find that with the introduction of the interest rate and tax rate, the uncertainty involved in the process of decision making will diminish and the optimal purchasing date will be put off.     

Warped products of Hadamard spaces

In the Riemannian case, our approach to warped products illuminates curvature formulas that previously seemed formal and somewhat mysterious. Moreover, the geometric approach allows us to study warped products in a much more general class of spaces. For complete metric spaces, it is known that nonpositive curvature in the Alexandrov sense is preserved by gluing on isometric closed convex subsets and by Gromov–Hausdorff limits with strictly positive convexity radius; we show it is also preserved by warped products with convex warping functions. Received: 9 January 1998/ Revised version: 12 March 1998     

On a class of optimization problems emerging when hedging with short term futures contracts

This paper generalizes earlier work by G. Larcher and the author about hedging with short-term futures contracts, a problem which was considered in connection with the debacle of the German company Metallgesellschaft. While the original problem corresponded to the simplest possible model for the price process, i.e. Brownian motion, we give here solutions to more general models, i.e. a mean reverting model (Ornstein–Uhlenbeck process) and geometric Brownian motion. Furthermore we allow for interest rates greater than 0.     

Integrated simulation and optimization models for tracking international fixed income indices

Portfolio managers in the international fixed income markets must address jointly the interest rate risk in each market and the exchange rate volatility across markets. This paper develops integrated simulation and optimization models that address these issues in a common framework. Monte Carlo simulation procedures generate jointly scenarios of interest and exchange rates and, thereby, scenarios of holding period returns of the available securities. The portfolio manager’s risk tolerance is incorporated either through a utility function or a (modified) mean absolute deviation function. The optimization models prescribe asset allocation weights among the different markets and also resolve bond-picking decisions. Therefore several interrelated decisions are cast in a common framework. Two models – an expected utility maximization and a mean absolute deviation minimization – are implemented and tested empirically in tracking a composite index of the international bond markets. Backtesting over the period January 1997 to July 1998 illustrate the efficacy of the optimization models in dealing with uncertainty and tracking effectively the volatile index. Of particular interest is the empirical demostration that the integrative models generate portfolios that dominate the portfolios obtained using classical disintegrated approaches. Received: November 24, 1998 / Accepted: October 1, 2000?Published online December 15, 2000     

Lognormality of rates and term structure models

A term structure model with lognormal type volatility structure is proposed. The Heath, Jarrow and Morton (HJM) framework, coupled with the theory of stochastic evolution equations in infinite dimensions, is used to show that the resulting instantaneous rates are well defined (they do not explode) and remain positive, contrary to those derived in [2]. They are also bounded from below and above by lognormal processes. The model can be used to price and hedge caps, swaptions and other interest rate and currency derivatives including the Eurodollar futures contract, which requires integrability of one over zero coupon bond. This extends results obtained by Sandmann and Sondermann in [22] and [23] for Markovian lognormal short rates to (non-Markovian) lognormal forward rates. We show also existence of invariant measures for the proposed term structure dynamics     

Pricing caps with HJM models: The benefits of humped volatility

In this paper we compare different multifactor HJM models with humped volatility structures, to each other and to models with strictly decreasing volatility. All the models are estimated on Euribor and swap rates panel data maximizing the quasi-likelihood function obtained from the Kalman filter. We develop the analysis in two steps: first we study the in-sample properties of the estimated models, then we test the pricing performance on caps. We find the humped volatility specification to greatly improve the model estimation and to provide sufficiently accurate cap prices, although the models has been calibrated on interest rates data and not on cap prices. Moreover, we find the two-factor humped volatility model to outperform the three-factor models in pricing caps.     

关于货币市场结构转换利率期限结构的实证研究

基本的利率期限结构模型均未能将结构转换效应考虑进来,因此为了探讨结构转换架构下利率期限结构模型的特性,本文在中国货币市场利率数据的基础上对基本利率期限结构模型和结构转换利率期限结构模型进行了比较研究,结果发现中国货币市场利率动态中存在明显的结构转换效应,且在结构转换效应中其本身也存在着不稳定性,这充分反映了中国货币市场在发展过程中的不成熟特征.