# Price a forward contract on a zero-coupon bond

I'm trying to calculate the price of a forward contract on a zero-coupon bond (ZCB). The forward contract matures at $t_1$ and the ZCB matures at $t_2$. So is the price of the forwards contract just the ratio of (price ZCB that matures at $t_2$) / (price ZCB that matures at $t_1$) ?

• Yes. Think about how you would replicate the forward contract by trading in the two zero bonds. – LocalVolatility Feb 7 '17 at 15:02

## 2 Answers

Another way to obtain this result is, as I mentioned in the comment, to think about how you would replicate the forward contract. It has the following cash-flow structure:

 type          | t                  | t1                      | t2
----------------------------------------------------------------------------
forward       | 0                  | +P(t1, t2) - K          | 0


Here, I also use $P(t_1, t_2)$ to denote the time $t_1$ price of the zero coupon bond with maturity in $t_2$. $K$ is the fair delivery price of the forward contract.

You replicate this contract by taking a long position in the zero-coupon with maturity $t_2$ and financing the purchase by selling the zero coupon bond with maturity in $t_1$ for a notional that yields a current cash-inflow of $P(t, t_2)$. You get

 type          | t                  | t1                      | t2
-----------------------------------------------------------------------------
long ZCB t2   | -P(t, t2)          | +P(t1, t2)              | 0
short ZCB t1  | +P(t, t2)          | -P(t, t2) / P(t, t1)    |
-----------------------------------------------------------------------------
total         | 0                  | +P(t1, t2)              | 0
|                    | -P(t, t2) / P(t, t1)    |


The portfolio has the same cash-flows as the forward in both $t$ and $t_2$. It has the same random cash-flow in $t_1$ ($+P(t_1, t_2)$) and thus the non-random cash-flows at this time also have to agree, i.e. $K = P(t, t_2) / P(t, t_1)$.

• Shouldn't it rather be $1$ at $t_2$ for the forward as well as for the long ZCB $t_2$ ? – ujsgeyrr1f0d0d0r0h1h0j0j_juj Nov 5 '18 at 16:27
• No - that's all fine. Don't understand your question regarding the forward but the position in the zero coupon bond is unwound upon the maturity of the forward contract - so there are no more cash-flows in $t_2$. – LocalVolatility Nov 8 '18 at 7:38
• I am saying that long ZCB t2 gives you +1 at t2, that you don't take in account. And that should modify what is happeing at t2 for the forward as well, to be still able to apply your non arbitrage argument. – ujsgeyrr1f0d0d0r0h1h0j0j_juj Nov 11 '18 at 19:15
• As I wrote before, the position in the zero bond is unwound in $t_1$. Thus, even though the zero bond has a cash-flow in $t_2$ of one, it is not relevant to this argument. Everything is correct here. – LocalVolatility Nov 11 '18 at 21:37

Let $E^{t_1}$ be the expectation operator under the $t_1$-forwad probability measure $Q^{t_1}$, which takes the bond price process $\{P(t, t_1), \, 0\le t \le t_1\}$ as the numeraire. Then, the price of the forward contract, at time $t$, where $0\le t \le t_1$, is given by \begin{align*} E^{t_1}\big(P(t_1, t_2)\mid \mathcal{F}_t\big) &= E^{t_1}\left(\frac{P(t_1, t_2)}{P(t_1, t_1)}\mid \mathcal{F}_t\right)\\ &=\frac{P(t, t_2)}{P(t, t_1)}, \end{align*} as $\left\{\frac{P(t, t_2)}{P(t, t_1)}, 0\le t \le t_1\right\}$ is a martingale under the $t_1$-forwad probability measure $Q^{t_1}$. Here, $\mathcal{F}_t$ is the information set at time $t$.

• Thank you for your replies. I'm not sure if I was calculating the (price ZCB that matures at t2t2) correctly. When the I calculated (price ZCB that matures at t2t2) / (price ZCB that matures at t1t1) I got a ratio of like 1.5. What seemed to work was multiplying the strike times the forwards contract divided by the price that matures at t1. So like K*F(t2)/P(t1) I'm sorry if my notation is terrible. – user3476463 Feb 21 '17 at 5:10
• The price is given by the answer. – Gordon Feb 21 '17 at 12:18
• @Gordon I have question that, if it is a ZCB, so the yield of forward bond price FBP is exactly the same as forward rate? If it's not a ZCB, they are not always same? Suppose we have the relation $$B(t_1,t_2) = G(y_{t_1}),$$ $y_{t_1}$ is the yield of $B(t_1,t_2)$ at $t_1,$ then we have $\textrm{FBP} _0 = G(y_0).$ Is it right? – A.Oreo Aug 15 '17 at 4:17
• @A.Oreo: I will say yes to both. – Gordon Aug 15 '17 at 12:57