# Pricing Treasury futures

I've recently learned that at the delivery of Treasury futures the short side can decide which of the $n$ Treasury bonds (with relevant maturities) to deliver. If the short side chooses to deliver the $i$-th bond, the the long side has to pay $c_i\cdot F$ for delivery, where $c_i$ is the conversion factor for the $i$-th bond, which is the value everybody knows. If the bond price at delivery is $B_i$ then the short side delivers the cheapest-to-deliver (CDT) $\hat i$-th bond where $$\tag{1} \hat i = \operatorname{argmin}\limits_{1\leq i\leq n}\left(B_i - c_i\cdot F\right).$$ As a result, CDT bonds depend on the $F$ whereas $F$ itself shall be some kind of expected value of the delivered bond, that is $$\tag{2} F = \frac{\Bbb E B_{\hat i}}{\Bbb E c_\hat i}.$$ From these arguments it seems that to find $F$ one has to solve a fixpoint problem $(1)-(2)$, and it does not seem to be trivial even to prove existance there. Am I missing something?

In case I don't miss anything, I think the problem can be simplified as follows. To avoid all the discounting/margin account issues, let's talk about the forward contract the expires on one of the stocks $S_i$ where $1\leq i\leq n$. To each of these stocks we assign some constant adjustment weighting $c_i$, and at the maturity of the contract the short side has to choose $i$ and deliver the cash difference $S_i - c_i\cdot F$, where $F$ is the forward price we agree upon now, and $S_i$ is the stock price at the delivery. I assume that neither of stocks pays dividends, and interest rates are $0$. Of course, since it's a cash delivery, the short side will deliver $S_{\hat i}$ $$\hat i = \operatorname{argmin}\limits_{1\leq i\leq n}\left(S_i - c_i\cdot F\right).$$ Under these conditions it may be ok to assume that the forward price satisfies $$\Bbb E[S_\hat i - c_\hat i\cdot F] = 0$$ which of course translates into $(2)$.

• Thanks for this question! I have been confused by this fact already for a while. I also think that then the Treasury futures should be some kind of simple forward (with future margin calculation) with the bond as underlying. This would be an alternative formulation of your (2). – Ric May 28 '15 at 15:48
• @Richard: thanks for your interest! Regarding your last point, I've added some explanation. – Ulysses May 29 '15 at 7:10
• But the conversion factor is deterministic. The expectation is w.r.t the martingale measure. Then $E[B_i]$ is the futures price of the bond. Then the whole story is: choose conversion factors to make the basket comaprable, find the cheapest to deliver (ctd) out of that basket, if you have it calculate the usual futures/forward price. But still in order to find the ctd we need the future price ... – Ric May 29 '15 at 7:15
• @Richard: conversion factor is not deterministic in the sense that $\hat i$ is random – Ulysses May 29 '15 at 7:16
• Ok, now I understand what you mean ...your question and your thoughts are mathematically rigorous. I just wonder how this is done in practice? They don't solve fixed point problems. I guess they just trade futures $F$ and if this $F$ deviates too much from its fair values (given by the forward of the CTD) then it is corrected (due to arbitrage reasons). – Ric May 29 '15 at 7:19

Treasury futures are actually really complicated... There are complete books dedicated to this topic (e.g., The Treasury Bond Basis) and really good sell-side research papers ("Understanding Treasury Bond Futures" by Salomon Brothers) that I highly recommend.

You're actually very much on the right track, but I'll try to paint a somewhat complete picture.

Let's start by assuming that the delivery basket has one one bond, then ignoring margining (the usual forward-future difference), then the future's fair price is simply the bond's forward price (for settlement on the delivery date).

Let's now assume that the delivery basket is composed of multiple bonds. This introduces the so-called "quality option." As you mentioned, it's the short that determines which bond to deliver. Effectively, the long has sold an option to the short.

• Let's first assume that today IS the delivery date. Then the short will simply look at all the bonds in the delivery basket, calculate their invoice prices ($f \cdot CF_i + AI_i$, where $f$ is the futures price, $CF$ is the conversion factor, and $AI$ is the accrued interest), and choose the bond with the lowest invoice price. To preclude arbitrage opportunities, the price of the future must be $\min_i P_i / CF_i$, where $P_i$ is each bond's clean price. The bond with the lowest invoice price is delivered and is called the cheapest-to-deliver (CTD).

• If today is a date before the delivery date and if the CTD is known deterministically (as is frequently the case in recent years), then the fair futures price is simply $\min_i F_i / CF_i$, where $F_i$ is the forward price of each bond.

• It gets interesting when the CTD is NOT known at the time of pricing; i.e., there's the probability of a "CTD switch." Suddenly, the quality option becomes valuable! Because it's an option that the short has bought from the long, the quality option reduces the futures price from the min forward price: $$f = \min_i\left( \frac{F_i}{CF_i} \right) - \text{delivery option value}.$$

The question is of course how to come up with the delivery option value (DOV). In practice, it's actually easier to calculate the futures price and back out the DOV, rather than the other way around.

Conceptually, let's assume that you have an interest rate model at hand. Using this model, you can create a large number of scenarios of bond yields as of the delivery date. For each scenario, you can look at who the CTD is and set the futures price to be the converted forward price of that bond, in that scenario. The futures fair price is then the probability-weighted average of all the converted forward prices across all the different scenarios.

As a very simple example, let's say there are two bonds up for delivery and yields can either go up or down with equal probabilities. When rates go up, bond A is the CTD and the fair futures price in that case should be $F^A_\text{up} / CF_A$. When rates go down, bond B is the CTD and the fair price is $F^B_\text{dn} / CF_B$. Then the futures price must be the average of these two numbers.

What gets more interesting is that

1. The delivery period is not one day but a full month (for US Treasury futures). This creates the so-called timing option. Practically speaking, the timing option is worthless – if carry is positive, you deliver as late as possible. But you can easily incorporate this option into your model (it's like pricing an American option that allows continuous exercises).

2. Futures stop trading about a week before the last delivery date. During this week, futures price is fixed, but the cash market is still trading and the CTD can switch yet again! This creates the end-of-month option and can be valuable sometimes. This requires some very fancy modeling that's better discussed elsewhere...