On the Existence of Finite-Dimensional Realizations for Nonlinear Forward Rate Models

We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite-dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite-dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.     

HEATH–JARROW–MORTON INTEREST RATE DYNAMICS AND APPROXIMATELY CONSISTENT FORWARD RATE CURVES

We study a finite-dimensional approach to the Heath–Jarrow–Morton model for interest rate and introduce a notion of approximate consistency for a family of functions in a deterministic and stochastic framework. This amounts to asking the decrease of the minimum distance in least squares sense. We start from a general linearly parameterized set of functions and extend the theory to a nonlinear Nelson–Siegel family. Necessary and sufficient condition to have approximately consistency are given as well as a criterion of stability for the approximation.     

THE TERM STRUCTURE OF INTEREST RATES AS A GAUSSIAN RANDOM FIELD

A simple model of the term structure of interest rates is introduced in which the family of instantaneous forward rates evolves as a continuous Gaussian random field. A necessary and sufficient condition for the associated family of discounted zero-coupon bond prices to be martingales is given, permitting the consistent pricing of interest rate contingent claims. Examples of the pricing of interest-rate caps and the situation when the Gaussian random field may be viewed as a deterministic time change of the standard Brownian sheet are discussed.     

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.     

A Note on the Nelson–Siegel Family

We study a problem posed in Bj"ork and Christensen (1999): Does there exist any nontrivial interest rate model that is consistent with the Nelson–Siegel family? They show that within the Heath–Jarrow–Morton framework with deterministic volatility structure the answer is no. In this paper we give a generalized version of this result including stochastic volatility structure. For that purpose we introduce the class of consistent state space processes, which have the property to provide an arbitrage-free interest rate model when representing the parameters of the Nelson–Siegel family. We characterize the consistent state space Itô processes in terms of their drift and diffusion coefficients. By solving an inverse problem we find their explicit form. It turns out that there exists no nontrivial interest rate model driven by a consistent state space Itô process.     

SEPARABLE TERM STRUCTURES AND THE MAXIMAL DEGREE PROBLEM

This paper discusses separablc term structure diffusion models in an arbitrage-free environment. Using general consistency results we exploit the interplay between the diffusion coefficients and the functions determining the forward curve. We introduce the particular class of polynomial term structure models. We formulate the appropriate conditions under which the diffusion for a quadratic term structure model is necessarily an Ornstein-Uhlenbeck type process. Finally, we explore the maximal degree problem and show that basically any consistent polynomial term structure model is of degree two or less.     

SUPERHEDGING IN ILLIQUID MARKETS

We study superhedging of securities that give random payments possibly at multiple dates. Such securities are common in practice where, due to illiquidity, wealth cannot be transferred quite freely in time. We generalize some classical characterizations of superhedging to markets where trading costs may depend nonlinearly on traded amounts and portfolios may be subject to constraints. In addition to classical frictionless markets and markets with transaction costs or bid‐ask spreads, our model covers markets with nonlinear illiquidity effects for large instantaneous trades. The characterizations are given in terms of stochastic term structures which generalize term structures of interest rates beyond fixed income markets as well as martingale densities beyond stochastic markets with a cash account. The characterizations are valid under a topological condition and a minimal consistency condition, both of which are implied by the no arbitrage condition in the case of classical perfectly liquid market models. We give alternative sufficient conditions that apply to market models with general convex cost functions and portfolio constraints.     

Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rates and Reserves

We consider the problem of a Central Bank that wants the exchange rate to be as close as possible to a given target, and in order to do that uses both the interest rate level and interventions in the foreign exchange market. We model this as a mixed classical‐impulse stochastic control problem, and provide for the first time a solution to that kind of problem. We give examples of solutions that allow us to perform an interesting economic analysis of the optimal strategy of the Central Bank.     

ADMISSIBILITY OF GENERIC MARKET MODELS OF FORWARD SWAP RATES

Our main goal is to re‐examine and extend certain results from the papers by Galluccio et al. and Pietersz and van Regenmortel. We establish several results providing alternate necessary and sufficient conditions for admissibility of a family of forward swaps, that is, the property that it is supported by a (positive) family of bonds associated with the underlying tenor structure. We also derive the generic expression for the joint dynamics of a family of forward swap rates under a single probability measure and we show that these dynamics are uniquely determined by a selection of volatility processes with respect to the set of driving martingales.     

Interest Rate Option Pricing With Poisson-Gaussian Forward Rate Curve Processes

We study a continuous trading bond model where the associated forward rate curve follows a multidimensional Poisson-Gaussian process. the bond market is complete, and the unique arbitrage-free interest rate call option price is explicitly derived.     

"利率通道"调控模式对我国利率市场化改革的启示

金融自由化理论的发展对各国金融改革提供了理论支持,我国正在进行的利率市场化改革,迫切需要构建适合实际和发展需要的利率调控模式,所以必须了解“利率通道”调控模式的运行机制,运用实证数据对“利率通道”调控模式进行模拟分析,从而建立以“利率通道”调控为主、公开市场操作调控为辅的利率调控模式作为我国利率市场化改革的选择。     

MOMENT EXPLOSIONS AND STATIONARY DISTRIBUTIONS IN AFFINE DIFFUSION MODELS

Many of the most widely used models in finance fall within the affine family of diffusion processes. The affine family combines modeling flexibility with substantial tractability, particularly through transform analysis; these models are used both for econometric modeling and for pricing and hedging of derivative securities. We analyze the tail behavior, the range of finite exponential moments, and the convergence to stationarity in affine models, focusing on the class of canonical models defined by Dai and Singleton (2000) . We show that these models have limiting stationary distributions and characterize these limits. We show that the tails of both the transient and stationary distributions of these models are necessarily exponential or Gaussian; in the non-Gaussian case, we characterize the tail decay rate for any linear combination of factors. We also give necessary and sufficient conditions for a linear combination of factors to be Gaussian. Our results follow from an investigation into the stability properties of the systems of ordinary differential equations associated with affine diffusions.     

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.     

EQUILIBRIUM STATE PRICES IN A STOCHASTIC VOLATILITY MODEL1

In a stochastic volatility model, the no-free-lunch assumption does not induce a unique arbitrage price because of market incompleteness. In this paper, we consider a contingent claim on the primitive asset, traded in zero net supply. Given a system of Arrow-Debreu state prices, we provide necessary and sufficient conditions for consistency with an intertemporal additive equilibrium model that we fully characterize. We show that the risk premia corresponding to the minimal martingale of Föllmer and Schweizer (1991) are consistent with logarithmic preferences, while the Hull and White model (1987) (volatility risk premium independent of the asset price) is consistent with a class of utility functions including constant relative risk aversion (CRRA) ones.     

Consistent recalibration of yield curve models

The analytical tractability of affine (short rate) models, such as the Vasi?ek and the Cox–Ingersoll–Ross (CIR) models, has made them a popular choice for modeling the dynamics of interest rates. However, in order to properly account for the dynamics of real data, these models must exhibit time‐dependent or even stochastic parameters. This breaks their tractability, and modeling and simulating become an arduous task. We introduce a new class of Heath–Jarrow–Morton (HJM) models that both fit the dynamics of real market data and remain tractable. We call these models consistent recalibration (CRC) models. CRC models appear as limits of concatenations of forward rate increments, each belonging to a Hull–White extended affine factor model with possibly different parameters. That is, we construct HJM models from “tangent” affine models. We develop a theory for continuous path versions of such models and discuss their numerical implementations within the Vasi?ek and CIR frameworks.     

关于我国利率市场化若干问题的研究

实行利率市场化有利于增强我国银行对外资银行的竞争力,是我国金融体制改革的必然选择。研究利率市场化对我国宏观投资效益、存贷款先后次序安排及汇率的效应分析,有利于改进资金使用的宏观效益。我国是发展中国家,金融市场发育尚不完善,应选择渐近式实施利率市场化进程为宜。     

The Term Structure of Simple Forward Rates with Jump Risk

This paper characterizes the arbitrage-free dynamics of interest rates, in the presence of both jumps and diffusion, when the term structure is modeled through simple forward rates (i.e., through discretely compounded forward rates evolving continuously in time) or forward swap rates. Whereas instantaneous continuously compounded rates form the basis of most traditional interest rate models, simply compounded rates and their parameters are more directly observable in practice and are the basis of recent research on "market models." We consider very general types of jump processes, modeled through marked point processes, allowing randomness in jump sizes and dependence between jump sizes, jump times, and interest rates. We make explicit how jump and diffusion risk premia enter into the dynamics of simple forward rates. We also formulate reasonably tractable subclasses of models and provide pricing formulas for some derivative securities, including interest rate caps and options on swaps. Through these formulas, we illustrate the effect of jumps on implied volatilities in interest rate derivatives.     

TERM STRUCTURES OF IMPLIED VOLATILITIES: ABSENCE OF ARBITRAGE AND EXISTENCE RESULTS

This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h . We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h . For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h , namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions.     

SOCIAL DISCOUNTING AND THE LONG RATE OF INTEREST

The well‐known theorem of Dybvig, Ingersoll, and Ross shows that the long zero‐coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long‐term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long “simple” interest rate (or Libor rate) is finite, then this rate (unlike the zero‐coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis and certain generalizations thereof, that illustrate explicitly the good asymptotic behavior of the resulting discount bond systems. The conditions necessary for the existence of such “hyperbolic” and “generalized hyperbolic” long rates are those of so‐called social discounting, which allow for long‐term cash flows to be treated as broadly “just as important” as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage‐free valuation framework for the cost‐benefit analysis and risk management of long‐term social projects, such as those associated with sustainable energy, resource conservation, and climate change.     

Short Rate Dynamics and Regime Shifts*

We characterize the dynamics of the US short‐term interest rate using a Markov regime‐switching model. Using a test developed by Garcia, we show that there are two regimes in the data: In one regime, the short rate behaves like a random walk with low volatility; in another regime, it exhibits strong mean reversion and high volatility. In our model, the sensitivity of interest rate volatility to the level of interest rate is much lower than what is commonly found in the literature. We also show that the findings of nonlinear drift in Aït‐Sahalia and Stanton, using nonparametric methods, are consistent with our regime‐switching model.