# Hedging. Part 3. Futures Contract. Futures

The futures price F(t0, t*, T) is given by

F(t0, t*, T) = E[t0, t*][p(t*, T)] = ∫DAe-∫t*Tdx f(t*, x) = F(t0, t*, T) exp {ΩF (t0, t*, T) —– (1)

where F(t0, t*, T) represents the forward price of the same contract F(t0, t*, T) = P(t0, T)/P(t0, t*) and

ΩF (t0, t*, T) = – ∫tot* dt ∫tt* dx σ(t, x) ∫t*T dx’ D(x, x’; t, TFR) σ (t, x’) —– (2)

For μ = 0, equation 2 collapses to

ΩF (t0, t*, T) = – ∫tot* dt ∫tt* dx σ(t, x) ∫t*T dx’ σ (t, x’) —– (3)

which is equal to the one-factor HJM model. For the one factor HJM model with exponential volatility, equation 3 becomes

ΩF (t0, t*, T) = – σ2/2λ3 (1 – e-λ(T – t*))(1 – e-λ(t* – t0))2

Observe that the propagator modifies the product of the volatility functions with μ serving as an additional model parameter. Prices for call options, put options, caps, and floors proceed along similar lines with an identical modification of the volatility functions. Let us take an example and compute the appropriate hedge parameters for futures contract for a period of one year. The proposition expresses the futures price F(t0, t*, T) in terms of the forward price

P(t, T)/P(t, t*) = e-∫t*T dx f (t, x)

and the deterministic quantity ΩF (t0, t*, T) found in equation 2. the dynamics of the futures price dF (t, t*, T) is given by

dF (t, t*, T)/F(t, t*, T) = dΩF (t, t*, T) – ∫t*T dxdf(t, x) —– (4)

(dF (t, t*, T) – E[dF (t, t*, T)])/F(t, t*, T) = -dt ∫t*T dx σ (t, x) A(t, x) —– (5)

squaring both sides to the instantaneous variance of the futures price

Var [dF (t, t*, T)] = dt F2(t, t*, T) ∫t*T dx ∫t*T dx’ σ (t, x) D(x, x’) σ (t, x’) —– (6)

This updates the definition in terms of the futures contract.

Definition: Futures Contract: Let Fi denote the futures price F(t, t*, T) of a contract expiring at time t* on a zero-coupon bond maturing at time Ti. The hedged portfolio in terms of the futures contract is given by

∏(t) = P + ∑i=1NΔiFi

where Frepresents observed market prices. Defining, for notational simplicity,

Li = PFit*Ti dx ∫tT dx’σ (t, x) D(x, x’; t, TFR) σ (t, x’)

Mij = FiFj ∫t*Ti dx ∫tTj dx’σ (t, x) D(x, x’; t, TFR) σ (t, x’)

The hedge parameters and the residual variance when futures contracts are used as underlying instruments have identical expressions to the theorem and the corollary.

Corollary: Hedge Parameters and Residual Variance using Futures Hedge parameters for a futures contract that expires at time t on a zero coupon bond that matures at time Ti equals

Δi = – ∑j=1NLjMij-1

while the variance of the hedged portfolio equals

Var = P2 ∫tT dx ∫tT dx’ σ (t, x) σ (t, x’) D(x, x’; t, TFR) – ∑i=1Nj=1NLjMij-1

for Li and Mij in definition.