Heath-Jarrow-Morton Research Articles

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.

A Wiener Chaos Approach to Hyperbolic SPDEs

We construct generalized weighted Wiener chaos solutions for hyperbolic linear SPDEs driven by a cylindrical Brownian motion. Explicit conditions for the existence, uniqueness, and regularity of generalized (Wiener Chaos) solutions are established in Sobolev spaces. An equivalence relation between the Wiener Chaos solution and the traditional one is established. The Heath–Jarrow–Morton (HJM) forward rate model is used as an example to illustrate the general construction.

A FINITE-DIMENSIONAL HJM MODEL: HOW IMPORTANT IS ARBITRAGE-FREE EVOLUTION?

We consider a two-factor Heath–Jarrow–Morton (HJM) model under the risk neutral measure and show that it may be decoupled into a particular dynamic Nelson–Siegel (NS) model plus a somewhat counter-intuitive adjustment (lying outside the NS family) which keeps it arbitrage-free. We assess the importance of the adjustment for arbitrage-free pricing by comparing the HJM model with a novel NS model which is selected using projection techniques. We analyze forward curves and derivative prices generated by the HJM and projected NS model and consider two real-world case studies. Our analysis shows that the influence of the adjustment term on arbitrage-free evolution is small.

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.

Quantile hedging for equity-linked life insurance contracts in a stochastic interest rate economy

In this paper we examine equity-linked life insurance contracts in a stochastic interest rate economy via quantile hedging whose purpose is to look for the optimal probability of a successful hedge under initial budget constraint. Most of the existing studies have focused on valuing equity-linked life insurance contracts by quantile hedging or in a framework of stochastic interest rates. However, a few have taken into account simultaneously the two techniques, which make valuing equity-linked life insurance contracts more difficult. We model the term structure of interest rates by classical HJM model that imbeds stochastic interest rate economy into one containing an arbitrary number of additional risky assets. By means of the change of measure approach, we give explicit formulas for the fair values of the following four products: deterministic payoff contract, pure equity-linked life contract, equity-linked life contract with guarantee, equity-linked life contract with minimum guarantees and capped benefits. We find that the explicit formulas are mainly composed of normal distribution functions and two-dimensional normal distribution functions. We also investigate sensibility of the survival probability using data of interest rates, stock prices and life table from China.

Simulation of coupon bond European and barrier options in quantum finance

Coupon bond European and barrier options are studied in the framework of quantum finance. The prices of European and barrier options are analyzed by generating sample values of the forward interest rates f ( t , x ) using a two-dimensional Gaussian quantum field A ( t , x ) . The strong correlations of forward interest rates are described by the stiff propagator of the quantum field A ( t , x ) . Using the Cholesky decomposition, A ( t , x ) is expressed in terms of white noise. The simulation results for European coupon bond and barrier options are compared with approximate formulas, which are obtained as power series in the volatility of the forward interest rates. The simulation shows that the simulated price deviates from the approximate value for large volatilities. The numerical algorithm is flexible and can be used for pricing any kind of option. It is shown that the three-factor HJM model can be derived from the quantum finance formulation.

A Poisson-Gaussian model to price European options on the extremum of several risky assets within the HJM framework

This paper generalizes European call options on the extremum of several risky assets in a Poisson-Gaussian model which allows both the risky assets and stochastic interest rates moving randomly with jump risks. The stochastic interest rate is assumed to follow an extended multi-factor HJM model with jumps. The authors provide explicitly the closed-form solutions of these options through the change of numeraire technique and examine the effects of both jump risks and stochastic interest rate on the option price with numerical experiment. The model can be seen as an extension of Stulz (1982), Johnson (1987) and Lindset (2006).

Application of the Kusuoka approximation with a tree-based branching algorithm to the pricing of interest-rate derivatives under the HJM model

AbstractThis paper demonstrates the application of a new higher-order weak approximation, called the Kusuoka approximation, with discrete random variables to non-commutative multi-factor models. Our experiments show that using the Heath–Jarrow–Morton model to price interest-rate derivatives can be practically feasible if the Kusuoka approximation is used along with the tree-based branching algorithm.

Nota do Editor

On 4/14/2010 died in Rio de Janeiro Manuel Octávio Bessada Lion. Bessada was an official of the Central Bank of Brazil, the author of several books (some of them today a reference in various courses in finance and economics), professor of financial mathematics, derivatives, equities and statistics. He was a pioneer in issues of open market and derivatives in Brazil. His performance at the Central Bank, his books and courses played an important role in the formation and consolidation of the financial market. Although not solely dedicated to research, he has contributed several articles in technical journals and academic. The last one, "Pricing Asian Interest Rate Options with a Three-Factor HJM Model," published in vol. 8, no. 1, 2010 of the Journal of Finance. Bessada will be missed by his relatives and friends.

CMS Spread Options and Similar Options in Multi-Factor HJM Framework

Constant maturity swaps (CMS), CMS spreads and similar products are analyzed in multi-factor HJM models. For Gaussian models, which include some Libor Market Models and the G2 model, explicit approximated formula are provided. The approximations are done through two different approaches: an exact solution to an approximated equation and an approximated solution to the exact equation. The first approach has been used previously in the literature for other models, the second is new. The approximations provide prices with error less than 0.02 basis point for the best approach; this is better than the previous literature and negligible in practice. The approach can be used to price standard CMS and CMS spreads and also for similar exotic products.

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.

Apreçamento de Opções Asiáticas de Taxa de Juros através de um Modelo HJM de Três Fatores

O apreçamento de instrumentos derivativos de taxa de juros tem atraído a atenção de muitos pesquisadores. Gerentes de risco, operadores e mais genericamente, todos os participantes do mercado, procuram informações nos derivativos. Neste trabalho, implementamos o modelo HJM para apreçar opções de IDI. O objetivo é demonstrar a importância dos componentes principais da estrutura de taxa de juros no apreçamento e no hedge dos derivativos no mercado brasileiro. Os resultados indicam que o modelo HJM consistentemente subapreça as opções de IDI no mercado de balcão e superapreça as opções de prazo maior negociadas na BM&F. Além disso, observa-se que o erro de apreçamento apresenta uma relação diretamente positiva com o tempo para vencimento e negativa com a proximidade do dinheiro. 
 
 (Artigo disponível em inglês somente)

A Type of HJM Based Affine Model: Theory and Empirical Evidence

In this paper a type of Heath, Jarrow and Morton (1992) (HJM) based affine model is derived theoretically. This type of affine model is obtained by applying Linear Realization Theory to construct Finite Dimensional Realizations (FDRs) of the Gaussian HJM model. The algorithms of constructing Standard Observable Canonical Realization and Jordan Canonical Realization are introduced sequentially. And it is shown that the commonly adopted FDR is actually Jordan Canonical Realization. The empirical results show that a two-factor model of this type provides great fit to the term structure of interest rates data. The resulting state variables have clear economic interpretations. And it is found that the short end of the term structure can be precisely considered as a “medium-run factor” which uniformly shifts the yield curve. This finding has an important implication for bond portfolios management, and also helps us better understand the interactions between macro-economy and term structure dynamics.

Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

We give sufficient conditions for the existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.

The Term Structure of Interest Rates

This paper reviews the term structure of interest rates literature relating to the arbitrage-free pricing and hedging of interest rate derivatives. Term structure theory is emphasized. Topics included are the HJM model, forward and futures contracts, the expectations hypothesis, and the pricing of caps/floors. Directions for future research are discussed.

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.

Arbitrage-Free Modeling of the Term Structure of Interest Rates (in Danish)

Initially we set up the new framework developed by Heath, Jarrow and Morton (1987) and specifies the process for bond-prices, spot-rates and forward-rates assuming a deterministic volatility structure. However, the main contribution of this paper is in determining the bridge between this new modelling framework and the traditional yield-curve modelling approach.In that connection we consider the following one-factor models: Cox, Ingersoll and Ross (1985), Vasicek (1977) and Longstaff (1989). With respect to two-factor models we analyse the following 2 models: Longstaff and Schwartz (1991) and Vasicek and Fong (1991). Lastly, we also consider the general multi-factor gaussian yield-curve model from Langetieg (1980). With respect to these models we derive the embedded volatility structure and relate them to the HJM modelling framework.Next, using the Hull og White (1990b)/(1993) model, it is shown that when introducing a time-dependent parameter in the drift-specification and in the volatility-specification that it is possible to match both the initial yield-curve and the initial volatility structure - this derivation is performed using the HJM framework.Lastly, multiple candidate trinomial models is being derived/analysed for the Hull and White model.

Fractional term structure models: No-arbitrage and consistency

In this work we introduce Heath-Jarrow-Morton (HJM) interest rate models driven by fractional Brownian motions. By using support arguments we prove that the resulting model is arbitrage free under proportional transaction costs in the same spirit of Guasoni [Math. Finance 16 (2006) 569-582]. In particular, we obtain a drift condition which is similar in nature to the classical HJM no-arbitrage drift restriction. The second part of this paper deals with consistency problems related to the fractional HJM dynamics. We give a fairly complete characterization of finite-dimensional invariant manifolds for HJM models with fractional Brownian motion by means of Nagumo-type conditions. As an application, we investigate consistency of Nelson-Siegel family with respect to Ho-Lee and Hull-White models. It turns out that similar to the Brownian case such a family does not go well with the fractional HJM dynamics with deterministic volatility. In fact, there is no nontrivial fractional interest rate model consistent with the Nelson-Siegel family.

Pricing and hedging of quanto range accrual notes under Gaussian HJM with cross‐currency Levy processes

AbstractThis study analyzes the pricing and hedging problems for quanto range accrual notes (RANs) under the Heath‐Jarrow‐Morton (HJM) framework with Levy processes for instantaneous domestic and foreign forward interest rates. We consider the effects of jump risk on both interest rates and exchange rates in the pricing of the notes. We first derive the pricing formula for quanto double interest rate digital options and quanto contingent payoff options; then we apply the method proposed by Turnbull (Journal of Derivatives, 1995, 3, 92–101) to replicate the quanto RAN by a combination of the quanto double interest rate digital options and the quanto contingent payoff options. Using the pricing formulas derived in this study, we obtain the hedging position for each issue of quanto RANs. In addition, by simulation and assuming the jump risk to follow a compound Poisson process, we further analyze the effects of jump risk and exchange rate risk on the coupons receivable in holding a RAN. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:973–998, 2009

On the valuation of interest rate products under multi-factor HJM term-structures

We consider the valuation of interest rate products with effected cash flow under a multifactor Heath–Jarrow–Morton (HJM) model of the term-structure of interest rates by hierarchical approximation. At the higher-level, we apply a stochastic spectral approximation of the forward rates and exhaust an indexed family of regularized Hamilton–Jacobi characterizations of the value function. At the lower-level, we utilize penalization and an extrapolation method-of-lines finite element method. Application to interest rate caps and an American discount bond option are considered in order to demonstrate the applicability of the method.