The Hull-White model, developed by John Hull and Alan White, is a foundational framework in credit risk modeling, particularly for valuing credit derivatives such as Credit Default Swaps (CDS). The model represents the hazard rate \( \lambda(t) \)—the instantaneous probability of default—as a stochastic process. This flexibility allows it to capture the changing dynamics of credit risk in response to economic and market conditions.
Survival and Default Probabilities in the Hull-White Model
In credit risk, the survival probability is the likelihood that no default occurs by a given time \( t \). This is mathematically expressed as:
\[ \text{Survival Probability at time } t = P(T > t) = \exp\left(-\int_0^t \lambda(s) \, ds\right), \]
where \( \lambda(s) \) is the hazard rate at time \( s \). The **cumulative probability of default** by time \( t \) is then:
\[ \text{Default Probability at time } t = 1 - \exp\left(-\int_0^t \lambda(s) \, ds\right). \]
When the hazard rate \( \lambda(t) \) is constant, these expressions simplify significantly, which is a common assumption for analytical tractability.
Risk-Neutral Valuation and Applications
The Hull-White model employs the risk-neutral measure for pricing, where payoffs are discounted at the risk-free rate \( r \). This ensures consistency with the no-arbitrage principle. The model is particularly useful for pricing credit derivatives like CDS, which are contingent on default events. The valuation of a CDS involves:
- Calculating the expected loss due to default, adjusted for the recovery rate.
- Discounting the expected loss and premium payments to their present value using the risk-free rate.
Numerical Example: CDS Valuation Using the Hull-White Model
Consider a CDS with the following parameters:
- Hazard Rate (\( \lambda(t) \)): 3% per year (constant).
- Risk-Free Rate (\( r \)): 2% per year (constant).
- Recovery Rate: 40%.
- Maturity: 5 years.
- Notional: $1,000,000.
Here’s how we calculate the probability of default, expected loss, and the CDS valuation:
| Step | Calculation | Result |
|---|---|---|
| 1. Survival Probability at Year 5 | \[ P(T > 5) = \exp(-0.03 \cdot 5) \] | 0.861 |
| 2. Default Probability at Year 5 | \[ 1 - P(T > 5) = 1 - 0.861 \] | 0.139 |
| 3. Loss Given Default (LGD) | \[ \text{LGD} = (1 - \text{Recovery Rate}) \cdot \text{Notional} \] | $600,000 |
| 4. Present Value of Expected Loss | \[ \text{PV(Expected Loss)} = \text{LGD} \cdot \text{Default Probability} \cdot \exp(-r \cdot T) \] | $75,622 |
| 5. Present Value of Premium Payments | \[ \text{PV(Premiums)} = \sum_{t=1}^5 \text{Premium} \cdot \exp(-r \cdot t) \] | $94,214 |
The expected premium payments exceed the expected loss, making this CDS valuable for the protection seller. If the premium payments were lower, the protection buyer might benefit more.
Advantages of the Hull-White Model
The Hull-White model offers several advantages in credit risk modeling:
- Flexibility: The stochastic nature of the hazard rate allows for dynamic modeling of credit risk.
- Market Calibration: The model can be calibrated to market data such as CDS spreads and bond prices for accuracy.
- Analytical Solutions: For certain assumptions, the model provides closed-form solutions, simplifying calculations.
This makes the Hull-White model a robust tool for pricing credit derivatives and managing credit risk in portfolios.
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