Pricing Calls on Defaultable Bonds with Stochastic Rates and Hazard Rates

# Price of a Bond-Call option in the defaultable framework


# Price of a Bond-Call option in the defaultable framework












I would like to compute the price for a Call option written on a defaultable bond as underlying. Suppose you have the following dynamic under the risk free measure $\mathcal{Q}$ for the interest rate: $$ d r_t = ( a^r - b^r r_t) \, dt + \sigma^r dW^r_t \\ $$ with initial point $r_0$, and the following dynamic for the hazard rate: $$ d \gamma_t = ( a^\gamma - b^\gamma \gamma_t) \, dt + \sigma^\gamma dW^\gamma_t \\ $$ with initial point $\gamma_0$. I'm gonna call $p_1(T_1, T_2)$ the price at time $T_1$ for a defaultable bond with maturity $T_2$ and interest rate $r_t$. We can also suppose that there exists a risk-free bond $p_0(T_1, T_2)$ (again with interest rate $r_t$).

I will write $\tau$ in order to denote the random variable of the default time so that its density is

$$ \gamma_t e^{-\int_0^t \gamma_s \, ds} $$

The very classical formula for the price of the call option in this case is:

$$ E^{Q} \left[ I_{\tau > T_1} e^{-\int_0^{T_1} r_s \, ds} \left( p_1(T_1, T_2) - K \right)^+ \right] $$

By using a classical theorem (Theorem 9.23; Quantitative Risk Management; McNeil, Frey, Embrechts) we have the following result:

$$ = E^{Q} \left[ e^{\int_0^{T_1} r_s + \gamma_s \, ds } \left( p_1(T_1, T_2) - K \right)^+ \right] I_{\tau > 0} . $$

Here I have some trouble: I would like to do a change of numeraire (the really standard technique) and use the Black-Scholes formula. The problem is the "spred term", i.e. the exponential with $\gamma$. Infact let me do the usual splitting of the Call payoff: $$ = E^{Q} \left[ e^{-\int_0^{T_1} r_s + \gamma_s \, ds } p_1(T_1, T_2) I_{ p_1(T_1, T_2) - K >0} \right] - K E^{Q} \left[ e^{-\int_0^{T_1} r_s + \gamma(s) \, ds } I_{ p_1(T_1, T_2) - K >0} \right] $$

In order to compute the first term I will change the numeraire by using $p_0$:

$$ p_0(0, T_1)E^{Q^{p_0}} \left[ e^{-\int_0^{T_1} \gamma_s \, ds } I_{ p_1(T_1, T_2) - K >0} \right] = p_0(0, T_1)E^{Q^{p_0}} \left[ e^{-\int_0^{T_1} \gamma(s) \, ds } I_{ \frac{p_1(T_1, T_2)}{p_0(T_1, T_1)} - K >0} \right] $$ where in the last passage I've used the fact that $p_0(T_1, T_1) = 1$. At this point I'm little bit lost. Infact in the case of non defaultable bond everything is fine since there is no $\gamma$-term and I can use the Black-Scholes formula.

My question is: what is, more or less, the idea to compute the price in the defaultable framework?

Ciao! AM