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.     

THE GENERAL STRUCTURE OF OPTIMAL INVESTMENT AND CONSUMPTION WITH SMALL TRANSACTION COSTS

We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and time‐varying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors' preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leading‐order corrections due to small transaction costs are obtained.     

The Potential Approach to the Term Structure of Interest Rates and Foreign Exchange Rates

It is possible to specify a model for interest rates in various ways, by giving the dynamics of the spot rate or of the forward rates, for example. A less well–developed approach is to specify the law of the state–price density process directly. In abstract, the state–price density process is a positive supermartingale, and the theory of Markov processes provides a rich framework for the generation of examples of such things. We show how this can be done, and provide simple examples (some familiar, some new) where prices of derivatives can be computed very easily. One benefit of the potential approach is that it becomes very easy to model the yield curve in many countries at once, together with the exchange rates between them.     

THEORY AND CALIBRATION OF SWAP MARKET MODELS

This paper introduces a general framework for market models, named Market Model Approach, through the concept of admissible sets of forward swap rates spanning a given tenor structure. We relate this concept to results in graph theory by showing that a set is admissible if and only if the associated graph is a tree. This connection enables us to enumerate all admissible models for a given tenor structure. Three main classes are identified within this framework and correspond to the co-terminal, co-initial, and co-sliding model. We prove that the LIBOR market model is the only admissible model of a co-sliding type. By focusing on the co-terminal model in a lognormal setting, we develop and compare several approximating analytical formulae for caplets, while swaptions can be priced by a simple Black-type formula. A novel calibration technique is introduced to allow simultaneous calibration to caplet and swaption prices. Empirical calibration of the co-terminal model is shown to be faster, more robust, and more efficient than the same procedure applied to the LIBOR market model. We then argue that the co-terminal approach is the simplest and most convenient market model for pricing and hedging a large variety of exotic interest-rate derivatives.     

Interest Rate,Currency and Equity Derivatives Valuation Using the Potential Approach

Based on the potential approach to interest rate modelling, we introduce a simple tractable model for the unified valuation of interest rate, currency and equity derivatives. Our model is able to accommodate the initial term structure of zero‐coupon bond prices, generate positive and bounded interest rates, and handle cross products such as differential swaps, quanto options and equity swaps. As our model is specified under the actual probability measure, it can be directly used for portfolio risk management and the computation of value at risk. Furthermore, our model yields simple analytical formulas that are easy to calibrate and implement.     

A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.     

Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps

We investigate analytical solvability of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula for the conditional moment generating function of the log-asset price and providing the condition under which this new formula is explicit. The results lay a foundation for a range of valuation, calibration, and econometric problems. We then combine our theoretical results, the Hilbert transform method, various interpolation techniques, with the dimension reduction technique to propose unified simulation schemes for solvable models with affine SV and Lévy jumps. In contrast to traditional exact simulation methods, our approach is applicable to a broad class of models, maintains good accuracy, and enables efficient pricing of discretely monitored path-dependent derivatives. We analyze various sources of errors arising from the simulation approach and present error bounds. Finally, extensive numerical results demonstrate that our method is highly accurate, efficient, simple to implement, and widely applicable.     

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.     

ARBITRAGE‐FREE MULTIFACTOR TERM STRUCTURE MODELS: A THEORY BASED ON STOCHASTIC CONTROL

We present an alternative approach to the pricing of bonds and bond derivatives in a multivariate factor model for the term structure of interest rates that is based on the solution of an optimal stochastic control problem. It can also be seen as an alternative to the classical approach of computing forward prices by forward measures and as such can be extended to other situations where traditionally a change of measure is involved based on a change of numeraire. We finally provide explicit formulas for the computation of bond options in a bivariate linear‐quadratic factor model.     

ON VALUING STOCHASTIC PERPETUITIES USING NEW LONG HORIZON STOCK PRICE MODELS DISTINGUISHING BOOMS,BUSTS, AND BALANCED MARKETS

For longer horizons, assuming no dividend distributions, models for discounted stock prices in balanced markets are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new concept of a balanced market proposed here permits a distinction between such markets and unduly optimistic or pessimistic ones. A tractable example is developed and termed the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. Evidence is provided for long dated equity linked claims paying coupon for time spent by equity above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given the popularity of such claims, the resulting mispricing could possibly take some corrections. Furthermore for these new discounted models, implied volatility curves do not flatten out at the larger maturities.     

Introductory honors economics on the Web

Economics 304KH and 304LH at The University of Texas at Austin are introductory honors economics courses. Our goal was to provide students with more advanced material than available in introductory textbooks, such as a discussion of microeconomics based on one‐dimensional calculus and a survey of macroeconomics based on economic models. The supplemental material consists of Web course notes, on‐line quizzes, simulation and data modules and group interactive modules. This paper focuses on the development of Web economic course materials. We discuss the difficulties of programming mathematical instructional material in HTML and the coming solution in XML, JavaScript, and Java. The effectiveness of the on‐line course materials are also evaluated based on student responses. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Contributions of Professors Fischer Black,Robert Merton and Myron Scholes to the Financial Services Industry

This paper is written as a tribute to Professors Robert Merton and Myron Scholes, winners of the 1997 Nobel Prize in economics, as well as to their collaborator, the late Professor Fischer Black. We first provide a brief and very selective review of their seminal work in contingent claims pricing. We then provide an overview of some of the recent research on stock price dynamics as it relates to contingent claim pricing. The continuing intensity of this research, some 25 years after the publication of the original Black–Scholes paper, must surely be regarded as the ultimate tribute to their work. We discuss jump‐diffusion and stochastic volatility models, subordinated models, fractal models and generalized binomial tree models for stock price dynamics and option pricing. We also address questions as to whether derivatives trading poses a systemic risk in the context of models in which stock price movements are endogenized, and give our views on the ‘LTCM crisis’ and liquidity risk.     

The pricing of foreign currency options under jump‐diffusion processes

In this article, the authors derive explicit formulas for European foreign exchange (FX) call and put option values when the exchange rate dynamics are governed by jump‐diffusion processes. The authors use a simple general equilibrium international asset pricing model with continuous trading and frictionless international capital markets. The domestic and foreign price level are introduced as state variables that contain jumps caused by monetary shocks and catastrophic events such as 9/11 or Hurricane Katrina. The domestic and foreign interest rates are stochastic and endogenously determined in the model and are shown to be critically affected by the jump risk of the foreign exchange. The model shows that the behavior of FX options is affected through the impact of state variables and parameters on the nominal interest rates. The model contrasts with those of M. Garman and S. Kohlhagen (1983) and O. Grabbe (1983), whose models have exogenously determined interest rates. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:669–695, 2007     

BILINEAR TERM STRUCTURE MODEL

The Gaussian Affine Term Structure Model (ATSM) introduced by Duffie and Kan is often used in finance to price derivatives written on interest rates or to compute the reserve to hedge a portfolio of credits (CreditVaR), and in macroeconomic applications to study the links between real activity and financial variables. However, a standard three‐factor ATSM, for instance, implies a deterministic affine relationship between any set of four rates, with different times‐to‐maturity, and these relationships are not observed in practice. In this paper, we introduce a new class of affine term structure models, called Bilinear Term Structure Model (BTSM). This extension breaks down the deterministic relationships between rates in structural factor models by introducing lagged factor values, and the linear dependence by considering quadratic effects of the factors.     

Coordinating electronic commerce activities in MANIFOLD

Modern electronic commerce environments are heavily web-based and involve issues such as distributed execution, multiuser interactive access or interface with and use of middleware platforms. Thus, their components exhibit the properties of communication, cooperation and coordination as in CSCW, groupware or workflow management systems. In this paper we examine the potential of using coordination technology to model electronic commerce activities and we show the benefits of such an approach. Furthermore, we argue that control-oriented, event-driven coordination models (which enjoy some inherent properties such as security) are more suitable for electronic commerce than data-driven ones which are based on accessing an open shared communication medium in almost unrestricted ways. This revised version was published online in August 2006 with corrections to the Cover Date.     

A new use of importance sampling to reduce computational burden in simulation estimation

Simulation estimators (Lerman and Manski 1981; McFadden, Econometrica 57(5):995–1026, 1989; Pakes and Pollard, Econometrica 57:1027–1057, 1989) have been of great use to applied economists and marketers. They are simple and relatively easy to use, even for very complicated empirical models. That said, they can be computationally demanding, since these complicated models often need to be solved numerically, and these models need to be solved many times within an estimation procedure. This paper suggests methods that combine importance sampling techniques with changes-of-variables to address this caveat. These methods can dramatically reduce the number of times a particular model needs to be solved in an estimation procedure, significantly decreasing computational burden. The methods have other advantages as well, e.g. they can smooth otherwise non-smooth objective functions and can allow one to compute derivatives analytically. There are also caveats—if one is not careful, they can magnify simulation error. We illustrate with examples and a small Monte-Carlo study.     

RATING BASED LÉVY LIBOR MODEL

In this paper, we consider modeling of credit risk within the Libor market models. We extend the classical definition of the default‐free forward Libor rate and develop the rating based Libor market model to cover defaultable bonds with credit ratings. As driving processes for the dynamics of the default‐free and the predefault term structure of Libor rates, time‐inhomogeneous Lévy processes are used. Credit migration is modeled by a conditional Markov chain, whose properties are preserved under different forward Libor measures. Conditions for absence of arbitrage in the model are derived and valuation formulae for some common credit derivatives in this setup are presented.     

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.     

The impact of return nonnormality on exchange options

The Margrabe formula is used extensively by theorists and practitioners not only on exchange options, but also on executive compensation schemes, real options, weather and commodity derivatives, etc. However, the crucial assumption of a bivariate normal distribution is not fully satisfied in almost all applications. The impact of nonnormality on exchange options is studied by using a bivariate Gram–Charlier approximation. For near-the-money exchange options, skewness and coskewness induce price corrections that are linear in moneyness, whereas kurtosis and cokurtosis induce quadratic price corrections. The nonnormality helps to explain the implied correlation smile observed in practice. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:845–870, 2008     

Affine multiple yield curve models

We provide a general and tractable framework under which all multiple yield curve modeling approaches based on affine processes, be it short rate, Libor market, or Heath–Jarrow–Morton modeling, can be consolidated. We model a numéraire process and multiplicative spreads between Libor rates and simply compounded overnight indexed swap rates as functions of an underlying affine process. Besides allowing for ordered spreads and an exact fit to the initially observed term structures, this general framework leads to tractable valuation formulas for caplets and swaptions and embeds all existing multicurve affine models. The proposed approach also gives rise to new developments, such as a short rate type model driven by a Wishart process, for which we derive a closed‐form pricing formula for caplets. The empirical performance of two specifications of our framework is illustrated by calibration to market data.