Risk-Neutral Pricing in a Two-State Credit Risk Model

Summary

The document frames a request to derive risk-neutral pricing for a zero-coupon bond in a continuous-time, two-state Markov credit model. The states are no default and default. At maturity, the bond pays its full face amount if the issuer has survived, or a reduced recovery amount if default occurred before maturity. The question asks how to justify pricing the bond as a discounted expected payoff under a risk-neutral measure when the measure has not been specified through a Radon-Nikodym derivative.

It contrasts this setup with the familiar Black-Scholes construction, where a change of measure makes the discounted share price a martingale, and asks what condition motivates the measure change in the credit model. No answer, derivation, transition intensities, or specification of the pricing kernel is supplied, so the document is a problem statement rather than a completed method. Any actual derivation would need the model dynamics and market assumptions that determine risk-neutral default probabilities and discounting.

Key ideas

• The model has two states: survival and default.
• The bond pays its full amount upon survival and a reduced amount after default by maturity.
• The question seeks a risk-neutral measure and a derivation for pricing the discounted expected payoff.
• The document does not provide the model parameters or an answer to the derivation request.

Tags

# Deriving the risk-neutral pricing formula for the 2-state credit risk model


# Deriving the risk-neutral pricing formula for the 2-state credit risk model












I am reading Interest rate models by Cairns—specifically the chapter on credit risk. Cairns introduces first the simple 2-state continuous time Markov model for credit risk—with the two states being "No Default" and "Default".

He then uses this model to price a zero-coupon bond maturing at time T. If the company does not default before maturity, a payoff of 1 is received at time T; however, if the company defaults before maturity, then a reduced payoff of $\delta$ is received at time T.

Cairns states that price of the bond at current time t is given by the discounted value of the expected payoff under the risk neutral measure—but no proof is given and the risk neutral measure has not been specified through its Radon-Nikodym derivative.

I would like to know the proof. Many thanks!

Edit: I am well aware of the concept of risk-neutral pricing in the Black-Scholes framework, but the setting here is different: we are dealing with a 2-state Markov model. In the BS framework, we introduced $\mathbb{Q}$ to make the discounted share price a martingale—and to achieve this objective we used the CMG theorem with $\gamma$ as $\frac{\mu - r}{\sigma}$. What do we want to achieve by changing the measure in the 2-state model, and how do we go about changing it?

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