Forward Pricing for a Defaultable Zero-Coupon Bond

Summary

This note derives a pre-default forward delivery price for a zero-coupon bond that can default before delivery. Under a market-value recovery setup, it expresses the bond’s pre-default value using the risk-free short rate, default hazard, and recovery fraction. It then imposes zero value at inception for the forward payoff, conditional on the forward buyer or seller surviving to delivery, and discounts using the risk-free money-market account.

The resulting forward price is a ratio of conditional expectations: the numerator covers survival and bond value through the bond’s maturity, including recovery effects after delivery, while the denominator reflects survival and discounting up to delivery. The answer notes that the forward may be higher than a standard comparison when that comparison uses the combined rate of interest and hazard. The result depends on the specified recovery convention and model assumptions; it is not a universal adjustment independent of rates, hazard dynamics, recovery, or the comparison contract’s default terms.

Key ideas

• The bond value before default depends on the risk-free rate, hazard rate, and recovery convention.
• A fair forward price is found by setting the survival-weighted, discounted forward payoff to zero.
• Default risk affects both the delivery-date survival factor and the value of the bond after delivery.
• The forward price is represented as a ratio of conditional expectations under the stated setup.
• Comparisons with a default-free forward depend on how the standard case treats hazard and recovery.

Tags

# forward contract on a defaultable zero-coupon bond


# forward contract on a defaultable zero-coupon bond












I'am trying to calculate the price of a forward on a defaultable zero-coupon bond. It is also true that the price will be given by Price a forward contract on a zero-coupon bond ? I guess the defautability of the bond shoud result in higher price for the forward than in the standard case.

Thanks.

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/58040

Based on notations in this question, assuming the market value recovery mechanism, the pre-default value at time $T_1$ of a zero-coupon bond with maturity $T_2$, where $T_1 < T_2$, is given by \begin{align*} P(T_1, T_2) = E\Big(e^{-\int_{T_1}^{T_2}(r_s +(1-R)h_s)ds}\,\big|\, \mathscr{F}_{T_1}\Big). \end{align*} Let $B_t=e^{\int_0^t r_s ds}$ be the credit risk free money-market account value at time $t$. The pre-default forward price $K$ determined at time $t$, for $0\le t \le T_1$, is a value such that \begin{align*} 0 &= E\Big(\pmb{1}_{\tau>T_1}\frac{B_t}{B_{T_1}}(K-P(T_1, T_2)) \,|\,\mathscr{G}_t\Big)\\ &=\pmb{1}_{\tau>t}E\left(\Big(K e^{-\int_t^{T_1}(r_s+h_s) ds} - e^{-\int_t^{T_1}(r_s+h_s) ds-\int_{T_1}^{T_2}(r_s +(1-R)h_s)ds} \Big) \,|\,\mathscr{F}_t\right)\\ &=\pmb{1}_{\tau>t}E\left(\Big(K e^{-\int_t^{T_1}(r_s+h_s) ds} - e^{-\int_t^{T_2}(r_s+h_s) ds+\int_{T_1}^{T_2}Rh_sds} \Big) \,|\,\mathscr{F}_t\right). \end{align*} That is, \begin{align*} K = \frac{E\Big(e^{-\int_t^{T_2}(r_s+h_s) ds+\int_{T_1}^{T_2}Rh_sds} \,|\,\mathscr{F}_t\Big)}{E\Big(e^{-\int_t^{T_1}(r_s+h_s) ds} \,|\,\mathscr{F}_t\Big)}. \end{align*}

Your observation appears correct if you assume that the interest rate is defined by $r_t+h_t$ in the standard case.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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