Treasury Bond Futures - New York University
Debt Instruments and Markets
Professor Carpenter
Treasury Bond Futures
Concepts and Buzzwords
Basic Futures Contract
?Underlying asset,
Futures vs. Forward
marking-to-market,
Delivery Options
convergence to cash, conversion factor,
cheapest-to-deliver,
wildcard option,
timing option, end-of-
Reading
month option, implied repo rate, net basis
Veronesi, Chapters 6 and 11
Tuckman, Chapter 14
Treasury Bond Futures
1
Debt Instruments and Markets
Professor Carpenter
Basic Futures Contract
In a basic futures contract without delivery options, the buyer agrees to take delivery of an underlying asset from the seller at a specified expiration date T.
Associated with the contract is the futures price, G(t), which varies in equilibrium with time and market conditions.
On the expiration date, the buyer pays the seller G(T) for the underlying asset.
Marking to Market and Contract Value
Each day prior to the expiration date, the long and short positions are marked to market:
?The buyer gets G(t) - G(t - 1 day).
?The seller gets -(G(t) - G(t - 1 day)).
It costs nothing to get into or out of a futures contract, ignoring transaction costs.
Therefore, in equilibrium, the futures price on any day is set to make the present value of all contract cash flows equal to zero.
Treasury Bond Futures
2
Debt Instruments and Markets
Professor Carpenter
Marking to Market...
Consider buying the contract at any time t and selling it after just one day.
It essentially costs nothing to buy and sell the contract, so the payoff from this strategy is just the profit or loss from the marking to market: G(t+1 day)-G(t).
G(t + 1 day) is random.
G(t) is set today to make the market value of the next day's random payoff G(t+1 day)-G(t) equal to zero.
Marking to Market...
The market value of the random mark-to-market, G(t + 1 day)-G(t), is the cost of replicating that payoff.
We can represent that cost in the usual way as its discounted expected value under the risk-neutral probability distribution.
To make this market value zero, today's futures price must be the expected value of tomorrow's futures price under the risk-neutral probability distribution:
Et{tdt+1 day [G(t + 1 day)-G(t)]}=0
=> G(t) = Et{G(t + 1 day)}.
Treasury Bond Futures
3
Debt Instruments and Markets
Professor Carpenter
Convergence to Cash
Consider entering the futures contract the instant before it expires. The long position would instantly pay the futures price and receive the underlying asset. The payoff would be V(T)-G(T), where V(T) is the spot price of the underlying on the expiration date. In the absence of arbitrage, since it costs nothing to enter into either side of the contract, the (known) payoff must be zero: G(T)=V(T).
Determining the Futures Price Ignoring Delivery Options
Consider a "basic" futures contract on a bond. To determine the current futures price, G(0),
?we start at the expiration date of the futures, when the futures price is equal to the spot price of the underlying bond, ?then work backwards each mark-to-market date to determine the futures price that makes the next marking to market payoff worth zero.
Treasury Bond Futures
4
Debt Instruments and Markets
Professor Carpenter
Example
Consider a futures on a 6%-coupon bond maturing at time 2.
The futures expires at time 1.
The futures contract is marked to market every 6 months.
Class Problem: Time 1 Price of the Underlying 6% Bond Maturing at Time 2
Time 0
Time 0.5
Time 1 Vuu(1)=?
Vud(1)=? Vdd(1)=?
Treasury Bond Futures
5
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