Pricing Forwards on Defaultable Coupon Bonds

Summary

The document sets out the notation for a defaultable coupon bond and compares its forward price with the standard result for a non-defaultable coupon bond. In the standard case, the forward price is the bond value less coupons paid before settlement, divided by the discount bond price to settlement. This gives a relationship between the forward, past coupons, and the current bond price.

The author asks whether a similar relationship can be derived for the risky bond, whose coupons depend on survival and which pays recovery at default. The document offers no derivation or answer, so it does not resolve how default timing, recovery, or credit risk should enter the forward valuation. It is useful as a statement of the pricing problem and the standard-bond benchmark, but readers would need a credit-risk model and additional assumptions to obtain a specific risky-forward formula.

Key ideas

• A defaultable coupon bond pays coupons only if it survives and may pay recovery at default.
• The standard coupon-bond forward price deducts coupons paid before settlement from the current bond value and discounts to the settlement date.
• The document asks whether this identity extends to a defaultable bond but does not provide a solution.
• Any risky-bond forward valuation must account for default timing, recovery, and the chosen credit-risk assumptions.

Tags

# Forward contract on a defaultable coupon bearing bond


# Forward contract on a defaultable coupon bearing bond












Notations :

$P(t,T)$ : the $t$-price of a coupon bearing bond paying coupons $C_i$ at $T_i$ maturing at $T$

$B(t,T)$ : the $t$-price of a non defaultable zero coupon bond paying 1 at $T$

$P_r(t,T)$ : the $t$-price of a (risky bond) defaultable coupon bearing bond paying coupons $C_i$ at $T_i$ if no default, else paying a fixed recovery $R$ at time of default $\tau$ and maturing at $T$

According to Price a forward contract on a zero-coupon bond

The forward price of a forward contract on a standard coupon bearing bond settling at $t_1$ could be expressed as :

$$F(t_1,T)= \mathbb{E}^{t_1}\left ( P(t_1,T) \right | \mathbb{F}_t)= \frac{\sum_{T_i\geq t_1}C_iB(t,T_i)+B(t,T)}{B(t,t_1)}= \frac{P(t,T)-\sum_{T_i\leq t_1}C_iB(t,T_i)}{B(t,t_1)}$$

Hence we could write Forward Bond($t_1$) + past coupons($t_1$) = standard bond($t$). (*)

Likewise, how to establish the forward expression for $ P_r (t_1, T) $? does an expression similar to (*) hold? I find it difficult to account for the defaultability of the risky bond.

Thanks !

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