Deriving the Default Payment Price from Hazard and Discount Rates

Summary

The document seeks to derive the present value of a fixed payment made at default before a horizon. It begins with an expectation of the discounted payoff conditional on default by the horizon, then gives the default-time density in terms of the hazard rate and survival probability. The target expression integrates the product of the hazard rate, survival, and discounting over possible default times, with the short rate and hazard rate appearing together in the exponential.

The central step is to integrate the discounted payment over the distribution of default times and move between an expectation of an indicator payoff and an integral over time. This is a standard survival-analysis and credit-pricing setup, where the hazard rate weights the likelihood of default at each instant. The document is a derivation question and does not provide the missing calculus or a completed argument using conditional expectations. Its notation also leaves the dependence structure of rates and default intensity implicit, so applying the formula requires assumptions about the underlying probability model.

Key ideas

• The value of a payment at default is an expectation of discounting up to the random default time.
• The hazard rate multiplied by survival probability gives the density of default at a given time.
• Integrating over possible default times leads to a time integral weighted by both hazard and discount factors.
• The derivation depends on assumptions about conditional expectations and the joint model for rates and default.

Tags

# Follow-up Fixed Payment at Default - Pricing


# Follow-up Fixed Payment at Default - Pricing












This question is a follow-up of this question Fixed Payment at Default - Pricing for more clarity.

Starting from the following expression of payoff:

$$ D(0,T) = E(\exp(-\int_{0}^{\tau} r(t)dt) \cdot \mathbb{1}_{\{\tau \leq T\}}) $$

and knowing that $$ \text{Probability}(T \leq \tau \leq T+dT) = \lambda(T) \cdot \exp(-\int_{0}^{T} \lambda(t)dt) \cdot dT $$ we aim to arrive to:

$$ D(0,T) = E(\int_{0}^{T} \lambda(t) \cdot \exp(-\int_{0}^{t} (r(s)+\lambda(s))ds)dt) $$

The idea as stated is to integrate the payoff over all possible times of default, thus form 0 to T.

We thus have: $$ D(0,T) = E(\int_{0}^{T} \exp(-\int_{0}^{t} r(s)ds) \cdot \mathbb{1}_{\{t \leq T\}}dt \mid \mathcal{F}_0) $$ But I am struggling from this step to develop the calculus. Would the law of iterated expectation be needed?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.