**5.5 Valuation of T-Note and T-Bond Futures**

Assume that the daily resettlement implied from marking to market has no pricing implications (e.g., interest rates are constant and so the theoretical futures price equals the forward price), but that the settlement process of the future provides the short side with a delivery option. We will consider only one type of delivery option, the quality option, although the valuation principle extends in the same way to all types of delivery options. Finally, assume that there are no coupon payments between the present time and the time of the future’s maturity.

The Quality Delivery Option

The quality option arises because there is no unique underlying security for the T*-*bond (T*-*note) futures contract. Instead, the seller has the right to choose which underlying T*-*bond (or T*-*note) will be used to satisfy settlement, from a set of acceptable securities.

The arbitrage-free futures price under these conditions is constructed from the following two alternatives (Hemler 1990):

I) Buy the ith bond (with face value equal to $100,000) from the set of acceptable underlying bonds for settling the Treasury bond futures contract and, at the same time, sell one futures contract.

ii) Buy a quality option that pays either 0 or the maximum incremental value from switching to bond j from bond I at the time of settlement. In addition, invest the present value of the invoiced amount from the short to the long, computed using bond I (with face value equal to $100,000) and discounted at the spot rate of interest for the period of time between the present and the time of settlement.

Recall that the invoiced amount for a T*-*bond or T*-*note future is:

where *DQP *is the decimal value of the settlement price, *CF* is the conversion factor, and *AI**DD * is the accrued interest at the time of the delivery date.

At the time of settlement, the value of alternative ii) is the terminal option value (which either finishes in the money or out of the money) plus the invoiced amount using bond i. The value of alternative i) equals the same because the amount you receive over the invoiced amount based on bond i exactly equals the terminal value of the option.

Similarly, at any time prior to settlement, the two alternatives yield identical payoffs and so the present value of each alternative is as follows:

where the superscript is relative to bond i and,

*q**t(n) *= Value at time t of the conversion option defined relative to bond i,

*DD* = Delivery day for settlement of the futures contract,

*DQP *= Decimal value of the future’s price,

*S**t* = Spot Asked price for the deliverable bond (or note), i, at time *t*,

*Ai**t*, *AI**DD* = Accrued interest at the present time, t, and at the time of the delivery date, DD,

*CF #9;*
*I *= Conversion factor for bond I.

*trDD* = Spot rate of interest defined over the remaining life of the futures contract

Solving for the futures price, *DQP*, yields:

This arbitrage-free bond futures (or forward) value is the theoretical forward value reduced by the value of the quality option. Other options can be defined and added in the same way and, if coupon payments are made during the remaining life of the future, then alternative ii) is modified by adding zero-coupon bonds that have maturities which match the timing and the magnitude of the coupon payments.

Finally, you can observe that, because it is the short side that acquires these options, arbitrage-free pricing implies that this side does not acquire these options for nothing. Instead, the value of the option is reflected in the *reduced price* of a future that the short side receives.