Jarrow–Turnbull Model

Overview

The Jarrow–Turnbull model is an intensity-based (reduced-form) credit-risk model used to price credit-sensitive securities and estimate default probabilities. Developed by Robert Jarrow and Stuart Turnbull in the 1990s, it integrates stochastic interest-rate dynamics with a default hazard rate to value corporate debt and credit derivatives under a risk-neutral framework.

How it works

Default is modeled as an unpredictable (surprising) event governed by a stochastic hazard (intensity) process rather than as the predictable outcome of a firm’s observable asset path.
Interest rates are modeled dynamically (often with multi-factor term-structure models). The model jointly accounts for the term structure of interest rates and credit risk when discounting future cash flows.
Pricing uses risk-neutral probabilities: survival probabilities are derived from the hazard rate, and expected discounted cash flows account for possible default and any recovery assumptions.
Calibration is typically done to market data such as credit spreads, bond prices, and CDS quotes so the model reflects prevailing market-implied default risk.

Reduced-form vs. structural models

Structural models (e.g., the Merton framework) derive default from the firm’s asset value crossing a default barrier. They require modeling the firm’s assets and liabilities and typically predict default at predictable times (such as maturity).
Reduced-form models like Jarrow–Turnbull treat default as an exogenous, random time with a market-driven intensity. They do not require direct modeling of the firm’s asset value and are often preferred for pricing and hedging because they can be calibrated directly to market instruments.

Practical considerations and limitations

The Jarrow–Turnbull model is widely used alongside structural approaches; practitioners often combine insights from both types.
Strengths: flexible calibration to market prices, explicit treatment of interest-rate and credit interactions, and suitability for pricing liquid credit instruments.
Limitations: the hazard-rate specification abstracts from firm fundamentals and may miss idiosyncratic economic drivers of default; model outputs depend on quality and availability of market data and on chosen recovery assumptions.

Key takeaways

Jarrow–Turnbull is a foundational reduced-form, intensity-based credit-risk model.
It prices defaultable securities by combining stochastic interest-rate models with a default intensity.
It is market-calibrated and commonly used for pricing and hedging credit instruments, often in conjunction with structural models.

Jarrow Turnbull Model