Futures

The futures price F(t_{0}, t_{*}, T) is given by

F(t_{0}, t_{*}, T) = E[t_{0}, t_{*}][p(t_{*}, T)] = ∫DAe^{-∫t*Tdx f(t*, x)} = F(t_{0}, t_{*}, T) exp {Ω_{F} (t_{0}, t_{*}, T) —– (1)

where F(t_{0}, t_{*}, T) represents the forward price of the same contract F(t_{0}, t_{*}, T) = P(t_{0}, T)/P(t_{0}, t_{*}) and

Ω_{F} (t_{0}, t_{*}, T) = – ∫_{to}^{t*} dt ∫_{t}^{t*} dx σ(t, x) ∫_{t*}^{T} dx’ D(x, x’; t, T_{FR}) σ (t, x’) —– (2)

For μ = 0, equation 2 collapses to

Ω_{F} (t_{0}, t_{*}, T) = – ∫_{to}^{t*} dt ∫_{t}^{t*} 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

Ω_{F} (t_{0}, t_{*}, T) = – σ^{2}/2λ^{3} (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(t_{0}, 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 Ω_{F} (t_{0}, 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) = dΩ_{F} (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 F^{2}(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 F_{i} denote the futures price F(t, t_{*}, T) of a contract expiring at time t_{*} on a zero-coupon bond maturing at time T_{i}. The hedged portfolio in terms of the futures contract is given by

∏(t) = P + ∑_{i=1}^{N}Δ_{i}F_{i}

where F_{i }represents observed market prices. Defining, for notational simplicity,

L_{i} = PF_{i} ∫_{t*}^{Ti} dx ∫_{t}^{T} dx’σ (t, x) D(x, x’; t, T_{FR}) σ (t, x’)

M_{ij} = F_{i}F_{j} ∫_{t*}^{Ti} dx ∫_{t}^{Tj} dx’σ (t, x) D(x, x’; t, T_{FR}) 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 T_{i} equals

Δ_{i} = – ∑_{j=1}^{N}L_{j}M_{ij}^{-1}

while the variance of the hedged portfolio equals

Var = P^{2} ∫_{t}^{T} dx ∫_{t}^{T} dx’ σ (t, x) σ (t, x’) D(x, x’; t, T_{FR}) – ∑_{i=1}^{N}∑_{j=1}^{N}L_{j}M_{ij}^{-1}

for L_{i} and M_{ij} in definition.