Pricing a Bond Forward with a Risk-Neutral Cash Account

Summary

The document explains how to calculate the time-zero delivery price of a bond forward in a discrete-time, risk-neutral fixed-income model. Its pricing relation takes the expected discounted value of the underlying bond at the forward date and divides it by the expected value of the reciprocal cash account at that date. The latter term is the time-zero price of a unit-face zero-coupon bond maturing on the forward date.

This addresses a question about a binomial short-rate model, where the computed forward price appeared to equal a bond price. The response identifies the relevant normalization: the forward value must account for the value of cash over the period, represented by the maturity-matched zero-coupon bond price. The excerpt does not show the model tree or work through the numerical calculation, so it clarifies the formula and denominator but does not verify the particular spreadsheet result.

Key ideas

• A bond forward delivery price uses a risk-neutral expectation of the bond’s future ex-coupon value.
• The expected reciprocal cash account provides the denominator in the forward pricing relation.
• That denominator equals the current price of a unit-face zero-coupon bond maturing at the forward date.
• The document gives the pricing relation but does not demonstrate the specific binomial calculation.

Tags

# Zero Coupon Bond Forward Price


# Zero Coupon Bond Forward Price












I'm currently working on the Coursera Financial Engineering and Risk Management course. In one of the questions I was asked to build a binomial pricing model for fixed-income securities. Specifically a 10-period model with 5% initial short rate, u=1.1, d=0.9, q=1-q=0.5.

One of the questions asked for the bond forward price with maturity at t=4. And the forward price I got was exactly the same as the bond price. Is that correct for zero coupon bonds?

Here's my spreadsheet: https://drive.google.com/file/d/0B2YmiWbV2_98UHBmWEFZZS0xRzA/view?usp=sharing

## Answer by user3747260 (score 1, accepted)

https://quant.stackexchange.com/a/21602

Look at slide 3 of Mod 4: Fixed Income Derivatives: Bond Forwards, that's the relevant equation.

$$ G_0 = \frac{E_0^\mathbb{Q}[Z^j_t/B_t]}{E_0^\mathbb{Q}[1/B_t]} $$

Where $G_0$ denotes the price of the forward at $t=0$, $E_0^\mathbb{Q}$ is the risk-neutral price at $t=0$, $Z^j_t$ denotes the ex-coupon price of the bond at time $t$ and state $j$, and $B_t$ is the value of the cash account at time $t$.

You need to divide by the risk-neutral price of the reciprocal of the cash account at t=4, which is equal to the t=0 price of a zero-coupon bond with face value of 1 that matures at t=4

*edited for a more complete answer

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