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

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Definition:

Li = PPi ∫tT dx ∫tTi dx’σ(t, x) σ(t, x’) D(x, x′; t, TFR)

Mij = PiPj ∫tTi dx ∫tTj dx’σ(t, x) σ(t, x’) D(x, x′; t, TFR)

This definition allows the residual variance in

P2tTdx∫tT dx’ σ(t, x) σ(t, x’) D(x, x′; t, TFR) +2P ∑i=1NΔiPitTdx ∫tTdx’ + ∑i=1Nj=1NΔiΔjPiPjtTitTjdx’ σ(t, x) σ(t, x’) D(x, x′; t, TFR)

to be succinctly expressed as

P2 ∫tT dx ∫tT dx’σ(t, x) σ(t, x’) D(x, x′; t, TFR) + 2 ∑i=1iLi + ∑i=1Nj=1NΔiΔj Mij —– (1)

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

Δi = -∑j=1N Lj Mij-1

and represents the optimal amounts of P(t, Ti) 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 ∫tT dx ∫tT dx’σ(t, x) σ(t, x’) D(x, x′; t, TFR) – ∑i=1Nj=1NLiMij-1Lj

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 (∫tT dx ∫tT1 dx’σ(t, x) σ(t, x’) D(x, x′; t, TFR))/(∫tT1 dx ∫tT1 dx’σ(t, x) σ(t, x’) D(x, x′; t, TFR)) —– (2)

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