Futures and Forwards

Abstract

A forward or a futures contract fixes today the terms of a transaction to be carried out on a future date. In organized exchanges, futures contracts are standardized in their amounts, maturities, and the quality of the underlying asset. They are subject to daily margin calls and cleared by official clearing houses. As a result, they are practically void of counterparty risk. By contrast, forward contracts are traded over the counter, tailor-made, and flexible as the parties design the terms of the contract at their own convenience, but are in general less liquid and fraught by counterparty risk. After a general presentation of futures and forwards, we examine the relationship between the forward/futures price and the spot price (cash and carry). We then study hedging with these instruments and finally present the main contracts according to their underlying assets.

Appendix

1.1 The Relationship Between Forward and Futures Prices

The prices of forward and futures contracts written on the same underlying and with the same maturity T do, in general, differ slightly. However:

Proposition 1

Under AAO and with deterministic interest rates, the futures price equals the forward price.

Proof

Let us use the notation:

• S_t: the spot price on t of an asset x .

• F_t: the price at t of a futures contract with maturity T written on x (F_T = S_T) .

• Φ_t: the price at t of a forward contract on x with the same maturity (Φ_T = S_T).

• r_T–t(t): the continuous rate prevailing at the instant t for transactions with duration T–t and assumed to be deterministic.

• Consider the period (0, T), just T days long, the j^th day being bracketed by the instants (j–1, j) for j = 1,…, T.

First, let us consider strategy A consisting in holding during day j (from j–1 to j) n(j–1) futures contracts with maturity T, with

$$ n\left(j-1\right)=e{{{}^{-\left(T-j\right)r}}_{T-j}}^{(j)}. $$

This strategy provides each day j a margin = n(j–1)[F_j – F_j–1] which one loans or borrows between the dates j and T. The sum of these margins capitalized at T can be written

$\sum \limits_{j=1}^T{e}^{\left(T-j\right){r}_{T-j}(j)}n\left(j-1\right)\left[{F}_j-{F}_{j-1}\right]=\sum \limits_{j=1}^T\left[{F}_j-{F}_{j-1}\right]={F}_T-{F}_0={S}_T-{F}_0.$

Remark

• Strategy A only involves a single cash flow (random, taking place at T and equal to S_T – F₀) .

• Strategy A is only possible if rates are deterministic since in the contrary case n(j–1) cannot be determined at time j–1 as a function of the rate r_T–j(j) which will only become known at time j.

Now, let us consider strategy B consisting of buying on date 0 a forward contract at the price Φ₀ and held to maturity; B only leads to a single (random) cash flow on date T, which writes:

$$ {\Phi}_T-{\Phi}_0={S}_T-{\Phi}_0. $$

If Φ₀ > F₀ the strategy “A–B” consisting in adopting strategy A and selling a forward contract would generate a single, certain, cash flow on date T equal to

S_T – F₀ – [S_T – Φ₀] = Φ₀ – F₀ > 0.

This would therefore constitute an arbitrage. Strategy “B–A” would be an arbitrage if F₀ > Φ₀. Consequently, at any instant denoted by 0, under AAO the equality F₀ = Φ₀ must hold.

Proposition 2

When interest rates are stochastic, the futures price exceeds the forward price if the interest rates are positively correlated with the underlying asset price, and falls below in the reverse case.

This result has an intuitive explanation. If rate and price are positively correlated, for a price hike the positive margin received by the holder of the futures contract will be invested at a rate presumably increasing. In the reverse case the negative margin will be borrowed at a falling rate. In this way, for a buyer of futures, the positive effect of an appreciation is amplified and the negative effect of a depreciation is damped: the existence of margin calls benefits the buyer, which explains why the futures price is (slightly) higher than the forward price if the interest rate and the underlying asset price are positively correlated. The opposite is true if rate and price are negatively correlated.

*This intuitive result can be proved with a continuous-time stochastic model. Consider the risk-neutral probability Q (different from the true probability P) under which the futures price F_t is a martingale.³⁰ Since F_T = S_T, we obtain

$$ {F}_t={E_t}^Q\ \left[{S}_T\right] $$

(9.A-1)

where E_t^Q is the conditional expectation under Q conditioned on the information available at t.

Applying Itô’s Lemma to the forward price, that is to the ratio ${\Phi}_t\left(S,B\right)=\frac{S_t}{B_T(t)}$, where B_T(t) denotes the price of the maturity-T zero-coupon bond at t, we have

$$ \frac{d{\Phi}_t}{\Phi_t}=\frac{d{S}_t}{S_t}-\frac{d{B}_T(t)}{B_T(t)}-\mathit{\operatorname{cov}}\left(\frac{d{S}_t}{S_t},\frac{d{B}_T(t)}{B_T(t)}\right)+\mathit{\operatorname{var}}\left(\frac{d{B}_T(t)}{B_T(t)}\right) $$

$$ \frac{d{S}_t}{S_t}-\frac{d{B}_T(t)}{B_T(t)}-\mathit{\operatorname{cov}}\left(\frac{d{\Phi}_t}{\Phi_t},\frac{d{B}_T(t)}{B_T(t)}\right). $$

And since, under Q, the derivatives of δS/S and δB/B are both equal to the risk-free rate r(t), the derivative of $\frac{d{\Phi}_t}{\Phi_t}$equals – cov ($\frac{d{\Phi}_t}{\Phi_t},\frac{d{B}_T(t)}{B_T(t)}\Big)$) which is denoted by –σ_ΦB(t)δt. Using the notation σ_Φ for the volatility of the forward price and W for a Brownian process under Q, we can then write

$$ \frac{d{\Phi}_t}{\Phi_t}=-{\sigma}_{\Phi B}(t) dt+{\sigma}_{\Phi}(t) dW, $$

which implies that ${\Phi}_t{e}^{\underset{0}{\int^t}{\sigma}_{\Phi B}(u) du}$ is a martingale with respect to Q; from this follows:

${\Phi}_t{e}^{\underset{0}{\int^t}{\sigma}_{\Phi B}(u) du}={E}_tQ\left[{\Phi}_T{e}^{\underset{0}{\int^T}{\sigma}_{\Phi B}(u) du}\right]$, or else

$$ {\Phi}_t={E}_t^Q\left[{S}_T{e}^{\underset{t}{\int^T}{\sigma}_{\Phi B}(u) du}\right]. $$

(9.A-2)

Assuming that ${\sigma}_{\Phi B}(t) dt\equiv \operatorname{cov}\kern-0.025em \left(\frac{d{\Phi}_t}{\Phi_t},\frac{d{B}_T(t)}{B_T(t)}\right)$ is deterministic, (9.A-2) simplifies to:

${\Phi}_t={e}^{\underset{t}{\int^T}{\sigma}_{\Phi B}(u) du}{E}_t^Q\left[{S}_T\right]$, which, by (9.A-1), implies the following relation between forward and futures prices:

$$ {\Phi}_t={e}^{\underset{t}{\int^T}{\sigma}_{\Phi B}(u) du}{F}_t. $$

(9.A-3)

(9.A-3) shows that the futures price exceeds the forward price if and only if ${\int}_t^T{\sigma}_{\Phi B}(u) du$ < 0, i.e., if and only if the covariance between the relative variations of the forward price and the zero-coupon price, which will hold on average from now until the end of the contract, is negative, therefore if and only if the covariance between the variations of the forward price and those of the zero-coupon rate is positive.