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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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  • $\begingroup$ Are you asking, given some other (which?) definition of risk-neutral probability of default, to prove that this one is equivalent? $\endgroup$ – Dimitri Vulis Sep 12 at 20:42
  • $\begingroup$ The author first establishes the model under the real-world measure: he denotes the transition intensity from "No default" to "Default" by $\lambda(t)$, under $\mathbb{P}$. Then he claims that in order to find the bond price, we need to discount the expectation of the payoff, with this expectation taken under $\mathbb{Q}$. That's it. He doesn't provide the proof for why this can be done. $\endgroup$ – Dhruv Gupta Sep 12 at 20:52
  • $\begingroup$ Note: 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? $\endgroup$ – Dhruv Gupta Sep 12 at 21:02

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