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Pricing a Forward on a Zero-Coupon Bond with a Short-Rate Lattice

Article Quant Q&A · Author: jwedmore
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Summary

The document describes how to price a forward contract on a zero-coupon bond using no-arbitrage. The example gives a bond with a stated face value and current price, a forward maturity before the bond’s maturity, and a binomial short-rate lattice. The key relationship is that the current price of the longer-maturity bond equals the price of a zero-coupon bond maturing at the forward date multiplied by the forward delivery price.

This reduces the calculation to finding the intermediate-maturity zero-coupon price from the short-rate lattice, then using the no-arbitrage relationship to obtain the forward price. The document does not show the lattice calculation or provide a numerical forward price, so it offers the pricing principle rather than a complete worked solution. Its setup also does not discuss details such as settlement conventions or any market frictions.

Key ideas

  • No-arbitrage links the longer-maturity bond price to an intermediate bond price and the forward delivery price.
  • The short-rate lattice is used to calculate the zero-coupon bond price at the forward maturity.
  • The document states the pricing relationship but does not carry out the lattice valuation or report a forward price.

Tags

Fixed incomeDerivatives pricingArbitrage
Full text
# Forward Contract Price on Zero Coupon Bond


# Forward Contract Price on Zero Coupon Bond












I'm trying to calculate the forward contract on a zero coupon bond where the forward contract matures at t=4. The zero coupon bond matures at t=10 and has a face value of 100. The price of that bond is 61.62

$n=10-period$ binomial model for the short-rate The lattice parameters are: $r(0,0) = 5\%, u = 1.1, d = 0.9d, q = 0.5, 1−q = 0.5$

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/50803

To calculate the forward price $F$ of a zero coupon bond at t=4, note that arbitrage considerations imply that $$Z(0,10)= Z(0,4) F$$. This essentially means that investing in a 4 year zero coupon bond together with a forward contract to invest from year 4 to year 10 must be the same as investing in a 10 year bond. So you need to first calculate Z(0,4) from your lattice.

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.