Futures Markets (2): Contracts on Interest Rates

Abstract

This chapter provides an in-depth analysis of the two principal types of futures contracts on interest rates. The notional contracts on short-, medium-, or long-term securities presented in Sect. 15.1 are defined relative to an “abstract” reference (a “notional” security ) without physical existence and lead, at the seller’s will, to the delivery of one among different eligible physical securities similar to the notional security. The Short-Term Interest Rate contracts (STIRS), examined in Sect. 15.2, are referenced on a short-term underlying rate (3 months or less) although the contract maturity may be much longer, they are cash settled exclusively, and are based either on a forward-looking interest rate, usually a 3-month LIBOR, or on a backward-looking average of overnight rates.

Appendices

Appendices

These appendices provide two improvements in the techniques for valuing futures contracts. The first concerns the option to change the delivered DS held by the seller of a notional contract. The second allows taking into account the margin call system.

1 Valuation of the Delivery Option

To obtain a closed formula for the delivery option included in a contract on a notional security, we make the following simplifying assumption which is justified by the uniformity of the basket of DS :

Assumption

The forward yield curve is flat for all DS . We denote by ${R}_T^f(t)$ the unique forward rate prevailing on date t ≥ 0 for the maturity T.

Applying the result from Sect. 15.1.4.2, the two candidates for CTD on the expiration date T are the bond with the longest duration using the clean price (indexed by the letter l) and the bond with the shortest duration (indexed by the letter s). The choice of one or the other will depend on the actual bond yield on date T, denoted by R(T).

We showed in Sect. 15.1.4.1, that the futures price on date T is

$$ {F}_T=\min \left(\frac{C_l(T)}{f_l},\frac{C_s(T)}{f_s}\right). $$

Consider, for example, the case where the most probable CTD is the DS with longest duration l (i.e. ${R}_T^f(t)>k$). The value at the expiry of the futures contract can be decomposed into two parts, involving an option of exchange between the two bonds s and l:

$$ {F}_T=\frac{C_l(T)}{f_l}-\max \left(\frac{C_l(T)}{f_l}-\frac{C_s(T)}{f_s};0\right). $$

(15.33)

Written this way, it is clear that the exchange option, with payoff $\max \left(\frac{C_l(T)}{f_l}-\frac{C_s(T)}{f_s};0\right)$, reduces the expiry price of the futures contract as computed in Sect. 15.1.4.1. This exchange option can be valued using Margrabe’s formula (see Chap. 11, Sect. 11.2.7).

Suppose, for example, that the annual standard deviation of the bond rate is 1%. If the option is in the money (${R}_T^f(t)=k$), applying the formula gives a futures price of 99.80%, which can be seen as the theoretical price of the futures without the delivery option (100% of the nominal amount) reduced by the time value of the in-the-money option: 0.20%. This effect is mitigated (the delivery option value is low) when the market yield goes noticeably above or below k since then the likelihood of a change in CTD is very low.

2 Relationship Between Forward and Futures Prices

We examine here the impact of the margin call system on the futures price which, in particular, makes it different from the forward price.

• 1. We first recall two important results (see Appendix 1 to Chap. 9):

  • (a) The futures price, in contrast to the forward price, is a martingale under the risk-neutral probability Q associated with the standard (riskless asset) numeraire ${\beta}_t=\exp \left({\int}_0^tr(s) ds\right)$.

  • (b) The price ${F}_0^T$ on date 0 of the futures contract with underlying S equals the forward price ${\varPhi}_0^T$ corrected by a covariance term (see Eq. (9.A-3) in Appendix 1 to Chap. 9):

$$ {F}_0^T={\varPhi}_0^T\exp \left[\operatorname{cov}\left({\int}_0^Tr(t) dt,\ln {S}_T\right)\right]. $$

(15.34)

The role of covariance is explained as follows: if there is a negative covariance between the short-term rate and the return on the underlying, then an increase in S_t (thus in ${F}_t^T$) goes along with a decrease in the short-term rate. For the buyer of the futures, the positive margin received is invested at a low interest rate. If, on the contrary, S_t (thus ${F}_t^T$) falls, the buyer must finance the negative margin at a higher interest rate. Therefore, the system of margin calls works against the buyer’s interests, implying a price of the futures smaller than that of the forward. Symmetrically, a positive covariance would play in favor of the buyer, implying a futures price higher than that of the forward. Since rates and prices are negatively correlated, the futures prices of financial assets are usually smaller than the corresponding forward prices (while futures rates are higher). Relation (15.34) is general and applies to contracts on both short-term and long-term securities.

• 2. Let us apply Eq. (15.34) to contracts on interest rates beginning with a contract on a short-term rate.

We start with the approximate result in Proposition 10 according to which the price of a short-term futures is equal to the forward price with expiration date T of a zero-coupon with maturity T + K (usually K = 0.25). This result does not take into account the impact of margin calls that affects futures. Noticing that the zero-coupon B_T + K(t) of maturity T + K (duration K on date T) plays the role of S_t in result (b) above, we deduce the short-term (fictitious) futures price defined by Eq. (15.27) with K instead of 0.25²⁰:

The short-term (fictitious) futures price ${\Psi}_0^T$ is linked to the short-term forward price ${\Phi}_0^T$by:

$$ {\Psi}_0^T={\Phi}_0^T\exp \left[-\operatorname{cov}\left({\int}_0^Tr(t) dt,K.{z}_K(T)\right)\right], $$

(15.35)

where z_K(T) is the continuous zero-coupon spot rate of duration K prevailing on expiration date T.

Proof

Applying Eq. (15.34) in the case where S_T = B_T + K(T) gives

$$ {\Psi}_0^T=\kern1em {\varPhi}_0^T\exp \left(\operatorname{cov}\left({\int}_0^Tr(t) dt;\;\ln \Big({B}_{T+K}(T)\right)\right). $$

We obtain the desired result by using the fact that, by definition

$$ {B}_{T+K}(T)={e}^{-K.{Z}_K(T)}. $$

Again, note that the covariance in Eq. (15.35) is that of two processes strongly positively correlated. Indeed, the two terms depend on the evolution of short-term rates between 0 and T: ${\int}_0^Tr(s) ds$ depends on the evolution of the instantaneous rate over the period [0, T], and z_K(T) is the rate of duration K that will prevail at T. Since this covariance is multiplied by (–1), the futures price is smaller than the corresponding forward price.

• 3. An analogous result holds for a contract on a long-term, notional rate and for a contract on a bond price.

Let ${F}_0^{i,T}$ be the price on date 0 of a fictitious futures contract written on DS i and of maturity T. By applying Eq. (15.34), the futures and forward clean prices are related by

$$ {F}_0^{i,T}={\Phi}_0^{i,T}\exp \left(\operatorname{cov}\left({\int}_0^Tr(s) ds;\ln {C}_i(T)\right)\right), $$

where C_i is the clean spot price of DS i.

The relationship between the futures price of a notional contract and its forward price is obtained by applying the last equation to the CTD m.

More precise relations can be obtained by combining this last result with an interest-rate model such as those developed in Chap. 17. Using the notations of that chapter, we denote by F(t, T, T_B) and Φ(t, T, T_B), respectively, the price on date t of a T-futures contract and a T-forward contract written on a bond B(t,T_B) with maturity T_B > T. Using the one-factor Heath–Jarrow–Morton model (Chap. 17, Sect. 17.2.1.4), according to which the instantaneous volatility of the instantaneous forward rate is the constant σ, we derive an explicit formula for the relationship between F(t, T, T_B) and Φ(t, T, T_B) (Eq. 17.24 in Chap. 17):

$$ F\left(t,T,{T}_B\right)=\varPhi \left(t,T,{T}_B\right)\exp \left[-\frac{\sigma^2}{2}\left({T}_B-T\right){\left(T-t\right)}^2\right], $$

(15.36)

which is Eq. (15.12) in the text providing the convexity adjustment between the futures and forward clean prices of a zero-coupon bond due to margin calls .

• 4. Using the same one-factor Heath–Jarrow–Morton model, which is equivalent to the Ho and Lee (1986) model, we can derive a closed-form relation for the convexity adjustment relevant to contracts on short-term rates (note that, by contrast, Eq. (15.35) is general and relative to prices).

Assuming that the instantaneous spot interest rate obeys:

$$ dr=\theta (t) dt+\sigma dW(t), $$

(15.37)

where W(t) is a Wiener process, the price P(t,T) of the T-maturity zero-coupon bond is, in the risk-neutral world, equal to:

$$ P\left(t,T\right)=A\left(t,T\right){e}^{-r\left(T-t\right)}, $$

(15.38)

where A(t, T) is a deterministic function of time.

Define the current (continuously compounded) forward rate f(t, T₁,T₂) for lending/borrowing between T₁ and T₂ as:

$$ f\left(t,{T}_1,{T}_2\right)=\frac{LnP\left(t,{T}_1\right)- LnP\left(t,{T}_2\right)}{T_2-{T}_1}. $$

Note that in the main text relative to STIR contracts, T₁ is denoted by T and T₂ by T+K.

Applying Ito’s lemma to f(t, T₁,T₂) using Eqs. (15.37) and (15.38) yields:

$$ {\displaystyle \begin{array}{l} df\left(t,{T}_1,{T}_2\right)=\frac{1}{T_2-{T}_1}\left[\left({T}_2-{T}_1\right)\sigma dW+\frac{1}{2}{\sigma}^2\left[{\left({T}_2-t\right)}^2-{\left({T}_1-t\right)}^2\right] dt\right]\\ {}\kern6em =\sigma dW+\frac{{\left({T}_2-t\right)}^2-{\left({T}_1-t\right)}^2}{2\left({T}_2-{T}_1\right)}{\sigma}^2 dt\end{array}} $$

(15.39)

The risk-neutral conditional expectation of df(.) is the second, deterministic, term on the r.h.s. of Eq. (15.39). By integrating this term between 0 and T₁, we obtain the risk-neutral expected change in the forward rate between the current date (0) and date T₁:

$$ {E}_t\left[f\left({T}_1\right)-f(t)\right]=\int_0^{T_1}\frac{{\left({T}_2-t\right)}^2-{\left({T}_1-t\right)}^2}{2\left({T}_2-{T}_1\right)}{\sigma}^2 dt=\frac{1}{2}{\sigma}^2{T}_1{T}_2. $$

Now, we know that under the risk-neutral probability, the futures rate is a martingale (see the Appendix to Chap. 9) so that the conditional expectation of its change between any two dates is 0. Therefore, the convexity adjustment between the forward and futures rates is conform to that given in Eq. (15.28) in the main text, i.e., 0.5σ²T₁T₂.