Price of bond future, given a specific interest rate?

Question

I'm interested in calculating what a theoretical price of the ZB or UB(Ultra Bond) futures would be priced at, given an interest rate of 1%. Or 0%

If the 30Y interest rate is around 1%, what will the ZB or UB be priced at?

If the 30Y interest rate is around 0%, what will the ZB, or UB be priced at

Answer 1

Treasury bond futures are surprisingly complicated - this is an attempt at a short explanation, it will obviously gloss over some details, but hopefully gives you a flavour of how they are priced.

The most important fact is that the underlying is not a single bond, but a basket of bonds. For example, the US Treasury Bond Futures contract spec says that you can deliver

U.S. Treasury bonds that have remaining term to maturity of at least 15 years and less than 25 years from the first day of the futures delivery month.

This is generally around 40 different bonds (assuming four new issues per year for a ten year period). Obviously, when it comes to delivery, the short will want to deliver whichever bond costs them the least - this bond is known as the cheapest-to-deliver, and one way to understand the bond futures contract is as a forward on the cheapest-to-deliver bond (where the relevant risk-free rate is the term repo rate to delivery, since bonds can generally be financed more cheaply than stocks).

The ~40 bonds deliverable into the futures contract all have different coupons. In the absence of other factors, the cheapest bond to deliver would simply be the one with the the lowest coupon - so there wouldn't be much point having a basket of deliverables at all. To get around this, each bond with price $p_i$ has an associated conversion factor $c_i$, which is roughly the price of the bond, per $1 of face value, if its yield was 6%.

The contract specification says that the amount paid by the long on delivery of the bonds is

The delivery invoice amount equals the futures settlement price times a conversion factor, plus accrued interest.

i.e. if the futures finally settle at $f$, the amount paid is $f\times c_*$ where $c_*$ is the conversion factor for the cheapest to deliver bond. In mathematical terms,

$$ f = \min_i (p_i/c_i) $$

If this was all there was to it, the futures price would closely track the forward price of the cheapest-to-deliver. However, there are several other factors that need to be taken into account -

1. The cheapest-to-deliver bond might change over the life of the contract, representing an even better deal for the short (who can now deliver a cheaper bond). The option to change the bond you are going to deliver is the switch option or quality option.
2. The short can deliver the bond on any day of the delivery month. Generally there is one day which is most effective to deliver on (typically the last day of the month, unless the carrying cost of the position is very negative) but it may change. This option is known as the timing option.
3. The bond futures contract settles about a week before the last delivery day, at which point the settlement price (which is one component of the invoice price) is fixed. However, the cheapest-to-deliver bond can still change, to the advantage of the short. This is known as the end-of-month option.
4. The close of trading in the futures market is about 2 hours before the close of trading in the cash bond market. Generally the short will have to buy/sell some additional bonds, over and above the ones they already hold as a futures hedge, in order to make delivery. If there is a sufficiently large move in the cash market after the futures have settled, the short may be able to get a good discount on these additional bonds. This is known as the wild-card option.

Notice that all of these options are in the favour of the short. Therefore, to compensate the long for selling all of these options, the futures need to trade at a discount to the forward price of the cheapest-to-deliver. Roughly,

Futures Price = (Forward Price of Cheapest-to Deliver) / (Conversion factor) - Option Value

The one thing that we haven't talked about here is the yield of the 30 year bond! Obviously this is a factor, but the yield is taken into account in the price of the bonds in the deliverable basket. Other factors can be more important in the determining the futures price, like the level and slope of the yield curve (which determines the cheapest-to-deliver), repo rates (which determine the forward values, and whether the carry is positive or negative) and yield volatilities (which influence the price of the various options).