FI 4000
Synthetic Treasury Bill Forward Contracts
Using Put and Call Options on Treasury Bills
Throughout this note, a period is a fixed amount of time (like three months) and we assume that there are Treasury bills with maturity in as many periods as the exposition suggests. The price today (spot price) of a Treasury bill maturing in t periods is denoted , expressed as a percent of par value. The corresponding spot rate is denoted by , so by definition of spot rate,
= .
We also assume that European call and put options on Treasury bills of all expiration dates and exercise prices are available.
Suppose you want buy forward in t periods a Treasury bill that has one period to maturity. One can do this synthetically by trading in options on Treasury bills. Consider options on the Treasury bill with t + 1 periods to maturity. Purchase a call with t periods to expiration and exercise price equal to the future value in t periods of the spot price of the Treasury bill. Similarly sell a put with the same expiration and exercise price. Denote the price of the call and put by and , respectively. By put/call parity ((17.2) page 555 in (BKM)),
- = -
= -
= - = 0.(1)
Notice that we now have a position in options, long a call, short a put, that costs 0, just as in a forward contract. Let denote the value in t periods of a Treasury bill with one period to maturity. In t periods, the value of this options position will be
At Maturity of Options
Long call0-
Short put0
Net- -
which is the same as a long forward position in a Treasury bill with one period to maturity, where is the forward price.
But we know what the forward price as a percent of par must be if there are no arbitrage possibilities. From formula (**) of the handout notes “Zero Coupon Bond Forward Contract (Are Forward Rates Good For Anything?),” it is = , where is the one period forward rate t periods hence. Look again at what was chosen to be, the future value in t periods of the spot priceof the underlying Treasury bill, i. e.,
= = = = == ,(2)
since = . Equations (1) and (2) verify that we have indeed created a long forward position in a one period Treasury bill using put and call options on Treasury bills. Such a position is referred to as a synthetic forward position.
To create a synthetic short forward position, just go long the put option and short the calloption, both with exercise price given by equation (2).