# How does one price the market value and estimate the fair value of a bond futures roll?

Consider the current situation: we are entering December, meaning that the December futures are being rolled into the March futures (i.e. traders are selling their holdings of December futures contracts before they expire so that they don't have to take delivery, and correspondingly buying the March futures contracts).

There are two things I am wondering about:

1. Pricing the market value of a futures roll.

2. Estimating the fair value of a futures roll.

Pricing futures roll

I'm guessing this is either the spread between the bond futures (Dec and March) or the spread between their fair values, which is computed as the cash (market) value of the CTD bond plus the cost of carry.

Estimating FV of futures roll

I've been given a hint about what one might estimate as the fair value of the futures roll. I think it involves the net basis of each of the CTD bonds. This makes sense, as the net basis is the gross basis accounting for the implied repo rate, which is the cost of borrowing now for delivery in the future.

If you could help formulate these definitions and calculations for me then it would be much appreciated. In order to do examples in the calculations (if you like), then just use random values for price, yield, net basis, repo rate etc.

Thanks.

The market price of the roll (aka calendar spread) is defined as $$(\text{front contract price} - \text{back contract price}) \times 32,$$ where the $${}\times32$$ part converts the price into "32nds," the standard quoting convention for Treasury futures calendar spreads.

Estimating the fair value of the roll, in principle, is straightforward. We'd compute the fair value of the two bond futures contracts and take the difference. We can then compare the market quoted spread against this fair value to assess richness/cheapness. The challenge, of course, is to estimate the fair value of bond futures themselves, which can be very involved. There are several related questions that could be of help; e.g., Pricing Treasury Futures.

In practice, relative value is a small component of calendar spread trading, particularly since a large source for mispricing historically, the delivery option, is pretty much zero in the current environment. So IMO, it's better to focus on trader positioning, duration, and other major drivers. You may find this question relevant to your quest.

P.S. You mentioned a hint to use net basis to gauge fair value. This could work in the current environment. Recall that net basis is the market's pricing for the delivery option. If you're 100% sure that the delivery option has no value (which is frequently the case today), then the fair value of net basis is zero. A positive (or negative) net basis suggests that the bond futures contract is cheap (or rich). However, there are many technical reasons as to why net basis might not be converge to zero.

• I would add to Helin's comments that one technical reason why the net basis might not converge is due to the conversion factor which is published by the CME. These conversion factors depend on the coupon and the remaining maturing from the last delivery date. Sometimes depending on the tenor, the remaining maturity is rounded up/down. I think for TY, it is rounded to the nearest 3-months. – VanillaCall Dec 2 '18 at 16:50

There are two equations that help me understand this:

1) Gross Basis = Spot CTD Price - Conversion Factor * Futures Price If the Gross basis is positive, this means that it is a positive carry. In other words, buying the underlying CTD and delivering it against selling the futures results in a gain

2) Net Basis = Forward CTD Price - Conversion Factor * Futures Price This basically adjusts the spot CTD to the forward delivery date so it has been carry adjusted (removed). If the net basis is positive, then the Forward CTD is rich compared to the Futures. If the net basis is negative, the Forward CTD is cheap to the Futures. Generally, a positive net basis means there is optionality allowing the seller of the Futures contract to deliver different securities. This happens when securities are close CTD candidates and small shifts between them can result in one being the CTD. This optionality is priced into the futures contract as a lower price because the investor who is long the contract need to be compensated for this optionality risk.

There are two important things to consider when assessing the value of a futures roll. These are the difference in market price of the front and back month contracts:

$$\Delta f=f_{\text{back}}-f_{\text{front}},$$

and the difference between the net basis on the deliverable bonds of the futures contracts:

$$\Delta b=b_{\text{front}}-b_{\text{back}}.$$

The first is obvious - we will be selling one and buying the other, so the spread between the prices must come directly into the value of the roll as $$\Delta f$$. The net basis the value of holding the bond into delivery, adjusted for the effects of implied repo.

This means that we lose some net basis (i.e. that of the front month contract, removing our inverse of delivery optionality) and gain some more net basis (i.e. that of the back month contract, because we reintroduce that inverse of delivery optionality).

Thus, the fair value can be written like this:

$$FV = \Delta f - \Delta b=(f_{\text{back}}-f_{\text{front}})-(b_{\text{front}}-b_{\text{back}})$$

• How do you have optionality on the the back contract? When you roll into the back contract, you're short optionality since you're long the futures contract. So selling the front and buying the back should give you net short optionality. – VanillaCall Feb 20 '19 at 11:52
• Hi sorry, wrong way around! I had in my head "buying basis" instead of "buying futures", I've edited it now. – quanty Feb 20 '19 at 18:26