Hedging. Part 1.

LongShortHedgeAllocation

Hedging a zero coupon bond denoted P(t,T) using other zero coupon bonds is accomplished by minimizing the residual variance of the hedged portfolio. The hedged portfolio Π(t) is represented as

Π(t) = P (t, T ) + ∑i=1NΔiP(t, Ti)

where ∆i denotes the amount of the ith bond P(t, Ti) included in the hedged portfolio. Notethe bonds P (t, T) and P (t, Ti) are determined by observing their market values at time t. It is the instantaneous change in the portfolio value that is stochastic. Therefore, the volatility of this change is computed to ascertain the efficacy of the hedge portfolio.

For starters, consider the variance of an individual bond in the field theory model. The definition P (t, T) = exp(-∫tT dxf(t, x)) for zero coupon bond prices implies that

dP(t, T)/P(t, T) = f(t, t)dt – ∫tTdxdf(t, x) = (r(t) – ∫tTdxα(t, x) – ∫tTdxσ(t, x)A(t, x))dt

and E[dP(t, T)/P(t, T) = r(t) – ∫tTdxα(t, x)dt since, E[A(t, x)] = 0. Therefore

dP(t, T)/P(t, T) – E[dP(t, T)/P(t, T) = – ∫tTdxσ(t, x)A(t, x))dt —– (1)

Squaring this expression and invoking the result that E[A(t, x)A(t, x′)] = δ(0)D(x, x′; t, TFR) = D(x, x′; t, TFR) /dt results in the instantaneous bond price variance

Var [dP(t, T)] = dt P2(t, T)∫tTdx ∫tT dx’σ(t, x) D(x, x′; t, TFR) σ(t, x’) —– (2)

As an intermediate step, the instantaneous variance of a bond portfolio is considered. For a portfolio of bonds, ∏ = ∑i=1NΔiP(t, Ti), the following results follow directly

d∏(t) – E[d∏(t)] = -dt ∑i=1NΔiP(t, Ti) ∫tTi dxσ(t, x)A(t, x) —– (3)

and

Var [d∏(t)] = dt ∑i=1Nj=1NΔiΔjP(t, Ti)P(t, Tj) ∫tTdx ∫tTj dx σ(t, x) D(x, x′; t, TFR) σ(t, x’) —– (4)

The (residual) variance of the hedged portfolio

Π(t) = P (t, T ) + ∑i=1NΔiP(t, Ti) ∫tTdx ∫tTdx’

may now be computed in a straightforward manner. For notational simplicity, the bonds P(t,Ti) (being used to hedge the original bond) and P(t,T) are denoted Pi and P respectively. Equation (4) implies the hedged portfolio’s variance equals the final result shown below

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) —– (5)

Observe that the residual variance depends on the correlation between forward rates described by the propagator. Ultimately, the effectiveness of the hedge portfolio is an empirical question since perfect hedging is not possible without shorting the original bond. Minimizing the residual variance in equation (5) with respect to the hedge parameters Δi is an application of standard calculus.

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