The HullWhite 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 HullWhite 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 HullWhite 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 riskneutral valuation, implying that expected values of payoffs are calculated using the riskfree rate. Finally, the parameters of the HullWhite 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 HullWhite 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.

RiskFree Rate: Constant at 2% per year.

Recovery Rate: 40%.

CDS Maturity: 5 years.

CDS Notional: $1,000,000.
Calculations:
1.
Survival Probability by Year 5: Calculated as e^(0.03 * 5), which results in approximately 0.861.
2.
Cumulative Probability of Default by Year 5: 1 minus the survival probability, giving approximately 0.139.
3.
CDS Premium Payments: Assuming a CDS premium rate of 2% annually, the annual premium payment is $20,000.
4.
Expected Loss: Loss Given Default (LGD) is calculated as (1  Recovery Rate) * Notional, resulting in $600,000.
5.
CDS Valuation: The present value of the expected loss is calculated using the formula LGD * Probability of Default * e^(RiskFree Rate * T), which amounts to about $75,622.
The
present value of premium payments, calculated by summing the discounted annual payments at the riskfree rate over 5 years, is about $94,214.
HullWhiteModel
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FinancialModeling
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CDSValuation
RiskManagement
EconomicModeling
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StochasticProcesses
DefaultProbability
RiskNeutralValuation
FinanceTheory
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DerivativesPricing
QuantitativeFinance
FinancialEngineering
MarketDataAnalysis
EconomicUncertainty
FinancialMathematics
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