Pricing Bond Forwards from Discounted Bond Values
Summary
The document derives the no-arbitrage forward price for a coupon-bearing bond delivered at a future date. It starts from the condition that a newly entered forward has zero value, so the discounted expected delivery value minus the forward price must have expectation zero under the risk-neutral measure. Because the forward price is known at inception, it can be factored out of the conditional expectation.
The derivation then uses the martingale property of discounted bond prices to relate the expected discounted delivery bond value to its current price. This yields a forward price equal to the current coupon-bearing bond price divided by the price of a zero-coupon bond maturing on the delivery date. The explanation assumes risk-neutral valuation, a forward maturing before the underlying bond, and a consistent discounting framework. It does not discuss coupon timing details, market frictions, or alternative settlement conventions.
Key ideas
- A fair forward contract has zero value when initiated.
- Risk-neutral valuation discounts the delivery payoff to the valuation date.
- The forward price is the ratio of the current coupon-bearing bond value to the delivery-date discount bond price.
- The derivation relies on discounted bond prices being martingales under the risk-neutral measure.
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Full text
# formula for pricing bond-futures
# formula for pricing bond-futures
Is anybody able to help me understanding why does $P_t(S)$ appear in the solution to the following problem; deriving the price of bond forward contracts? Thank you Given:
- $r_t$, the instantaneous rate process
- $P_t(T)$, the price of a zero-coupon bond at $t$ and expiring at $T$
- $F_t(S,T)$, the price at $t$ of a forward expiring at $S\leq T$
- $B_S(T)$, the price at $S$ of a coupon-bearing bond expiring at $T>t$
The payoff for going long the forward is $B_S(T)-F_t(S,T)$, and the position is costless at inception such that, in the absence of arbitrage, the forward satisfies $F_t(S,T)=\frac{1}{P_t(S)}\mathbb{E}_t\left(e^{-\int_t^Sr_\tau d\tau} B_S(T)\right)$ where $\mathbb{E}_t$ denotes the expectation
## Answer by Sebastian (score 0, accepted)
https://quant.stackexchange.com/a/68911
I assume the forward contract matures at time $S$ and $0 \leq t < S < T.$ Let $F_t(S,T)$ be the price that makes the forward contract be worth zero at time $t$. Then by the risk-neutral valuation formula we obtain $$0= \mathbb E^{\mathbb Q} \left[(B_S(T) - F_t(S,T))e^{-\int_t^S r(u)du} | \mathcal F_t \right].$$ Since $ F_t(S,T))e^{-\int_t^S r(u)du}$ is $\mathcal F_t$-measurable, $$F_t(S,T)=e^{\int_t^S r(u)du} \mathbb E^{\mathbb Q} \left[B_S(T)e^{-\int_t^S r(u)du} | \mathcal F_t \right].$$ Under the risk-neutral probability measure $\mathbb Q$ the discounted bearing-bond is a martingale, so $$F_t(S,T)=e^{\int_t^S r(u)du} B_t(T)=\frac{1}{P_t(S)}B_t(T).$$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.