This article presents convenient reduced-form models of the valuation of contingent claims subject to default risk, focusing on applications to the term structure of interest rates for corporate or sovereign bonds. Examples include the valuation of a credit-spread option. This article presents a new approach to modeling term structures of bonds and other contingent claims that are subject to default risk. As in previous “reduced-form” models, we treat default as an unpredictable event governed by a hazard-rate process. 1 Our approach is distinguished by the parameterization of losses at default in terms of the fractional reduction in market value that occurs at default. Specifically, we fix some contingent claim that, in the event of no default, pays X at time T . We take as given an arbitrage-free setting in which all securities are priced in terms of some short-rate process r and equivalent martingale measure Q [see Harrison and Kreps (1979) and Harrison and Pliska (1981)]. Under this “risk-neutral” probability measure, we letht denote the hazard rate for default at time t and let Lt denote the expected fractional loss in market value if default were to occur at time t , conditional

Modeling Term Structures of Defaultable Bonds

In the setting of ‘‘affine’’ jump-diffusion state processes, this paper provides an analytical treatment of a class of transforms, including various Laplace and Fourier transforms as special cases, that allow an analytical treatment of a range of valuation and econometric problems. Example applications include fixed-income pricing models, with a role for intensity-based models of default, as well as a wide range of option-pricing applications. An illustrative example examines the implications of stochastic volatility and jumps for option valuation. This example highlights the impact on option ‘smirks’ of the joint distribution of jumps in volatility and jumps in the underlying asset price, through both jump amplitude as well as jump timing.

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Transform analysis and asset pricing for affine jump-diffusions

1 Introduction 2 The Binomial Model 3 A More General One Period Model 4 Stochastic Integrals 5 Differential Equations 6 Portfolio Dynamics 7 Arbitrage Pricing 8 Completeness and Hedging 9 Parity Relations and Delta Hedging 10 The Martingale Approach to Arbitrage Theory (For advanced readers) 11 The Mathematics of the Martingale Approach (For advanced readers) 12 Black-Scholes from a Martingale Point of View (For advanced readers) 13 Multidimensional Models: Classical Approach 14 Multidimensional Approach: Martingale Approach (For advanced readers) 15 Incomplete Markets 16 Dividends 17 Currency Derivatives 18 Barrier Options 19 Stochastic Optimal Control 20 Bonds and Interest Rates 21 Short Rate Models 22 Martingale Models for the Short Rate 23 Forward Rate Models 24 Change of Numeraire (For advanced readers) 25 LIBOR and Swap Market Models 26 Forwards and Futures Appendix A Measure and Integration (For advanced readers) Appendix B Probability Theory (For advanced readers) Appendix C Martingales and Stopping Times (For advanced readers) References Index

http://www.gbv.de/dms/hebis-mainz/toc/071897542.pdf

Arbitrage Theory in Continuous Time

Recursive methods offer a powerful approach in dynamic macroeconomics. This book contains both an introduction to recursive tools, including standard applications such as asset pricing, and advanced material, including analyses of reputational mechanisms and contract design. The tools are presented with enough technical sophistication to get the reader started working on practical problems. When numerical simulations are called for, the book provides suggestions for how to proceed, as well as references for further reading. The applications cover many substantive issues in macroeconomics, such as equilibrium asset prices, market incompleteness, wealth distribution, fiscal-monetary theories of inflation, government debt, optimal labour and capital taxation, time consistency and credible government policies, optimal social insurance, economic growth and labour market dynamics.

Recursive Macroeconomic Theory

The standard class of affine models produces poor forecasts of future Treasury yields. Better forecasts are generated by assuming that yields follow random walks. The failure of these models is driven by one of their key features: Compensation for risk is a multiple of the variance of the risk. Thus risk compensation cannot vary independently of interest rate volatility. I also describe a broader class of models. These “essentially affine” models retain the tractability of standard models, but allow compensation for interest rate risk to vary independently of interest rate volatility. This additional f lexibility proves useful in forecasting future yields. CAN WE USE F INANCE THEORY to tell us something about the empirical behavior of Treasury yields that we do not already know? In particular, can we sharpen our ability to predict future yields? A long-established fact about Treasury yields is that the current term structure contains information about future term structures. For example, long-maturity bond yields tend to fall over time when the slope of the yield curve is steeper than usual. These predictive relations are based exclusively on the time-series behavior of yields. To rule out arbitrage, the cross-sectional and time-series characteristics of the term structure are linked in an internally consistent way. In principle, imposing these restrictions should allow us to exploit more of the information in the current term structure, and thus improve forecasts. But in practice, existing no-arbitrage models impose other restrictions for the sake of tractability; thus their value as forecasting tools is a priori unclear. I examine the forecasting ability of the affine class of term structure models. By “affine,” I refer to models where zero-coupon bond yields, their physical dynamics, and their equivalent martingale dynamics are all affine functions of an underlying state vector. A variety of nonaffine models have been developed, but the tractability and apparent richness of the affine class has led the finance profession to focus most of its attention on such models. Although forecasting future yields is important in its own right, a model that is consistent with finance theory and produces accurate forecasts can make a deeper contribution to finance. It should allow us to address a key

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Term Premia and Interest Rate Forecasts in Affine Models

This paper presents a consistent and arbitrage-free multifactor model of the term structure of interest rates in which yields at selected fixed maturities follow a parametric muitivariate Markov diffusion process with “stochastic volatility.” the yield of any zero-coupon bond is taken to be a maturity-dependent affine combination of the selected “basis” set of yields. We provide necessary and sufficient conditions on the stochastic model for this affine representation. We include numerical techniques for solving the model, as well as numerical techniques for calculating the prices of term-structure derivative prices. the case of jump diffusions is also considered.

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A yield-factor model of interest rates

This is a survey of multi-factor structure models, concentrating on models in which the term structure has a finite-dimensional (Markov diffusion) state-space representation. The special ‘affine’ case is shown to be tractable.

Multi-factor term structure models

https://www.darrellduffie.com/uploads/pubs/DuffieKan2002.pdf

Rui Kan

Papers

A yield-factor model of interest rates

Multi-factor term structure models

Universal state prices and asymmetric information