Introduction

The theoretical futures price for a Treasury bond futures contract plays a crucial role in derivatives trading and risk management. It represents the expected price at which the futures contract should trade in the market. Calculating the theoretical futures price involves considering factors such as the spot price of the underlying bond, the yield-to-maturity, the coupon rate, and the time to maturity. This chapter delves into the methodology for calculating the theoretical futures price and its significance in financial markets.

Calculating the Theoretical Futures Price

The theoretical futures price (F) for a Treasury bond futures contract can be calculated using the following formula:

$$F=\frac{P-A I}{C} \times(1+r)^T$$ Where:

• $P$ is the current price of the underlying Treasury bond.
• $A I$ is the accrued interest on the bond.
• $C$ is the bond’s coupon payment.
• $r$ is the risk-free interest rate.
• $T$ is the time to maturity of the bond in years.

Explanation:

• The first part of the formula $\left(\frac{P-A I}{C}\right)$ calculates the number of coupon payments that have accrued since the last coupon payment date, adjusting for accrued interest.
• The second part $\left((1+r)^T\right)$ accounts for the time value of money by discounting the future cash flows to the present value using the risk-free interest rate.

Example: Suppose we have a Treasury bond with a current price (P) of USD 1,200, accrued interest (AI) of USD 30, a coupon payment (C) of USD 40, a risk-free interest rate (r) of 3%, and a time to maturity (T) of 5 years.

$$F=\frac{1200-30}{40} \times(1+0.03)^5$$ $$F=29.25 \times 1.159274 \approx 33.89$$

In this example, the theoretical futures price for the Treasury bond futures contract is approximately USD 33.89.

Significance

Theoretical futures prices provide valuable insights into the market’s expectations and help traders make informed decisions. If the calculated theoretical price differs significantly from the actual market price, arbitrage opportunities may arise.