The Hull-White Model in Simple Terms

The Hull-White model, developed by John Hull and Alan White, is a cornerstone framework for modeling short-term interest rates. It is widely used in finance for valuing interest rate derivatives, bonds, and other instruments sensitive to interest rate changes. The model’s flexibility also allows its adaptation for credit risk applications, such as valuing Credit Default Swaps (CDS).

Core Equation of the Hull-White Model

The Hull-White model assumes that the short-term interest rate \( r(t) \) evolves as a stochastic process given by:

\[ dr(t) = \left[\theta(t) - a r(t)\right] dt + \sigma dW(t) \]

Where:

• \( r(t) \): Short-term interest rate at time \( t \).
• \( \theta(t) \): Time-dependent drift term ensuring the model fits the observed yield curve.
• \( a \): Mean reversion speed; controls how quickly \( r(t) \) reverts to its long-term mean.
• \( \sigma \): Volatility of the interest rate changes.
• \( dW(t) \): Wiener process representing the random shock.

The term \( \theta(t) \) is calibrated to ensure that the model matches the initial term structure of interest rates. The inclusion of the mean reversion term \( -a r(t) \) ensures that \( r(t) \) does not drift too far from its long-term average.

Adaptation for Credit Risk

In credit risk applications, the Hull-White model is modified to model the hazard rate \( \lambda(t) \)—the instantaneous probability of default at time \( t \). This hazard rate represents the intensity of default risk at a given moment. The survival probability, which is the probability that no default occurs by time \( t \), is derived from \( \lambda(t) \) and is expressed as:

\[ P(T > t) = \exp\left(-\int_0^t \lambda(s) \, ds\right) \]

The corresponding default probability is the complement of the survival probability:

\[ P(T \leq t) = 1 - P(T > t) \]

When \( \lambda(t) \) is assumed constant, these expressions simplify significantly, making the model easier to compute. However, the stochastic nature of \( \lambda(t) \) in the Hull-White framework allows it to dynamically adjust to changing market conditions.

Numerical Example

Consider a Credit Default Swap (CDS) where the hazard rate \( \lambda(t) \) is constant at 3% per year. The maturity of the CDS is 5 years, and we want to calculate the survival probability and default probability at the end of 5 years.

Using the survival probability formula:

\[ P(T > 5) = \exp(-\lambda \cdot t) \]

Substituting \( \lambda = 0.03 \) and \( t = 5 \):

\[ P(T > 5) = \exp(-0.03 \cdot 5) = \exp(-0.15) \approx 0.861 \]

The default probability is then:

\[ P(T \leq 5) = 1 - P(T > 5) = 1 - 0.861 = 0.139 \]

This means there is an 86.1% chance the counterparty will survive the 5-year period and a 13.9% chance it will default.

Advantages of the Hull-White Model

The Hull-White model offers several advantages:

• Flexibility: The time-dependent drift term \( \theta(t) \) ensures that the model can fit the observed term structure of interest rates.
• Mean Reversion: The inclusion of mean reversion ensures stability and prevents unrealistic behavior of \( r(t) \) or \( \lambda(t) \).
• Market Calibration: It can be calibrated to observed market data, such as yield curves and credit spreads, ensuring accuracy.
• Analytical Solutions: For specific assumptions, the model provides closed-form solutions, simplifying calculations.

This makes the Hull-White model a robust and versatile tool for financial modeling in both interest rate and credit risk domains.