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 {Ω_F (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

Ω_F (t₀, 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 (t₀, 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 (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 Ω_F (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) = 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 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 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^NΔ_iF_i

where F_i represents observed market prices. Defining, for notational simplicity,

L_i = PF_i ∫_t*^T_i dx ∫_t^T dx’σ (t, x) D(x, x’; t, T_FR) σ (t, x’)

M_ij = F_iF_j ∫_t*^T_i dx ∫_t^T_j 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^NL_jM_ij^-1

while the variance of the hedged portfolio equals

Var = P² ∫_t^T dx ∫_t^T dx’ σ (t, x) σ (t, x’) D(x, x’; t, T_FR) – ∑_i=1^N∑_j=1^NL_jM_ij^-1

for L_i and M_ij in definition.

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Hedging. Part 2. The Best Strategy to Hedge a Bond is to Short a Bond of the Same Maturity.

Definition:

L_i = PP_i ∫_t^T dx ∫_t^T_i dx’σ(t, x) σ(t, x’) D(x, x′; t, T_FR)

M_ij = P_iPj ∫_t^T_i dx ∫_t^T_j dx’σ(t, x) σ(t, x’) D(x, x′; t, T_FR)

This definition allows the residual variance in

P²∫_t^Tdx∫_t^T dx’ σ(t, x) σ(t, x’) D(x, x′; t, T_FR) +2P ∑_i=1^NΔ_iP_i∫_t^Tdx ∫_t^T_i dx’ + ∑_i=1^N∑_j=1^NΔ_iΔ_jP_iP_j∫_t^T_i∫_t^T_jdx’ σ(t, x) σ(t, x’) D(x, x′; t, T_FR)

to be succinctly expressed as

P² ∫_t^T dx ∫_t^T dx’σ(t, x) σ(t, x’) D(x, x′; t, T_FR) + 2 ∑_i=1NΔ_iL_i + ∑_i=1^N∑_j=1^NΔ_iΔ_j M_ij —– (1)

Theorem: Hedge parameter for bond in the field theory model equals

Δ_i = -∑_j=1^N L_j M_ij^-1

and represents the optimal amounts of P(t, T_i) to include in the hedge portfolio when hedging P(t,T). This theorem is proved by differentiating equation 1 with respect to Δi and subsequently solving for Δi. The corollary is proved by substituting the result of theorem into equation 1.

Corollary: Residual variance, the variance of the hedged portfolio equals

V =   P² ∫_t^T dx ∫_t^T dx’σ(t, x) σ(t, x’) D(x, x′; t, T_FR) – ∑_i=1^N∑_j=1^NL_iM_ij^-1L_j

which declines monotonically as N increases. the residual variance in corollary enables the effectiveness of the hedge portfolio to be evaluated. Therefore corollary is the basis for studying the impact of including different bonds in the hedged portfolio. for N = 1, the hedge parameter in the theorem reduces to

Δ₁ = -P/P₁ (∫_t^T dx ∫_t^T1 dx’σ(t, x) σ(t, x’) D(x, x′; t, T_FR))/(∫_t^T1 dx ∫_t^T1 dx’σ(t, x) σ(t, x’) D(x, x′; t, T_FR)) —– (2)

To obtain the HJM limit, constrain the propagator to equal one. The hedge parameter in equation 2 then reduces to

Δ₁ = -P/P₁ (∫_t^T dx ∫_t^T1 dx’σ(t, x) σ(t, x’) )/(∫_t^T1 dx σ(t, x))² = -P/P₁ (∫_t^T dx σ(t, x)/∫_t^T1 dx σ(t, x)) —– (3)

Under the summation of exponential volatility, equation 3 becomes

Δ₁ = -P/P₁ (1 – e^-λ(T-t))/(1 – e^-λ(T₁-t)) —– (4)

Equation 4 coincides with the ratio of the hedged parameters. When T₁ = T, the hedge parameter equals minus one. Economically, this fact states that the best strategy to hedge a bond is to short a bond of the same maturity. This trivial approach reduces the residual variance in equation 1 to zero as ∆₁ = −1 and P = P₁ implies L₁ = M₁₁.