The Hull-White model, developed by John Hull and Alan White, is a reduced form model used for valuing credit derivatives like Credit Default Swaps (CDS). In this model, the hazard rate λ(t)
represents the instantaneous probability of default. This hazard rate is often modeled as a stochastic process, reflecting the uncertainty and changing economic conditions over time.

The probability of default is linked to the hazard rate. The survival probability, or the probability that no default
occurs by a given time t, is expressed as exp(-∫0^t λ(s) ds). Consequently, the cumulative probability of default by time t is 1 minus this survival probability.

One key aspect of the Hull-White model is its use of stochastic intensity for the hazard rate. This approach provides
flexibility in modeling the default process, allowing for a more realistic representation of credit risk dynamics.

In terms of application, the Hull-White model is primarily used for pricing credit derivatives. These derivatives'
payoffs depend on the occurrence of a default event, and the model offers a framework for estimating the present value of these contingent payoffs.

The model employs risk-neutral valuation, implying that expected values of payoffs are calculated using the risk-free
rate. Finally, the parameters of the Hull-White model are typically calibrated using market data, which includes historical default rates, bond spreads, and credit derivative prices.

Let’s go through the numerical example using the Hull-White model to calculate the probability of default and price a
Credit Default Swap (CDS):

Assumptions:

- Hazard Rate (λ(t)): Constant at 0.03 (or 3%) per year.
- Risk-Free Rate: Constant at 2% per year.
- Recovery Rate: 40%.
- CDS Maturity: 5 years.
- CDS Notional: $1,000,000.

Calculations:

- Survival Probability by Year 5: Calculated as e^(-0.03 * 5), which results in approximately 0.861.
- Cumulative Probability of Default by Year 5: 1 minus the survival probability, giving approximately 0.139.
- CDS Premium Payments: Assuming a CDS premium rate of 2% annually, the annual premium payment is $20,000.
- Expected Loss: Loss Given Default (LGD) is calculated as (1 - Recovery Rate) * Notional, resulting in $600,000.
- CDS Valuation: The present value of the expected loss is calculated using the formula LGD * Probability of Default * e^(-Risk-Free Rate * T), which amounts to about $75,622.

The present value of premium payments, calculated by summing the discounted annual payments at the risk-free rate over 5 years, is about $94,214.

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