Hedging: Duration, Regression, and Model-Based Approaches
Fixed Income Trading and Risk Management by Alexander Düring (ISBN 9781119756354)
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Hedging is adding a liquid offset so an existing position stops moving with the market. Chapter 36 walks three rungs of sophistication: same-yield DV01, regression, and full curve model.
Why hedging defines price
Traders think in hedged terms. Buy a bond, pay fixed on swap at x bp I-spread; breakeven sale is x minus swap bid-offer. Swap curve moves drive cash bond quotes. Shared hedge conventions become market correlation.
Residual idiosyncratic risk always remains. Only a flat book is truly neutral. Rates work as a hedge canvas because curve structure is relatively stable versus single-name equity or credit noise.
Duration-neutral hedge
dP = D̄ dy. For two bonds: N₁ D̄₁ + N₂ D̄₂ = 0 ⇒ N₂ = −N₁ D̄₁/D̄₂.
Same with market values and modified duration. Desks think notionals; asset managers often think market value.
Catch: one dy drives both legs. Works for same market, similar maturity. Cross-market or 2Y vs 30Y “duration neutral” is often meaningless.
Regression hedges
Allow dV = N₁ D̄₁ dy₁ + N₂ D̄₂ dy₂ = 0 with dy₂ = β dy₁.
Or skip yields: dP₂ = β dP₁ (price regression). Works for futures and non-yield underlyings (oil hedging a corp bond? still linear locally).
Standard regression assumes noise only on y. Reverse regression β’ ≠ 1/β. Misspecification. PCA (Chapter 33) is the symmetric fix when both legs are noisy.
Yield curve model hedges
Flat curve model: only parallel shift, hedge is duration neutral (sum of weighted DV01s = 0).
Structural models: bond model price P̂ᵢ = φ(i, μ(t), ω). Hedge ∂V/∂μⱼ = 0 for each state variable μⱼ.
k-factor model needs k hedge instruments in general (k+1 names if one leg is client bond fixed). Solve linear system; sensitivities often numerical.
Calibrating for hedging ≠ best fit to today’s prices. Good hedge model minimizes overnight unexplained P&L on bond moves, including carry/rolldown path.
PCA-neutral butterflies
State variables = projections of yields on first two PCA factors. ∂yᵢ/∂μⱼ = factor loadings fⱼᵢ. Butterfly weights from 3×3 system with bullet notional ±100 and zero exposure to factors 1 and 2.
Sign of bullet depends rich vs cheap.
When each hedge tier breaks
Duration neutral fails when the hedge leg references a different curve segment or country. Hedging a 7Y KfW with a 10Y Bund future might look DV01-matched today and fall apart on a Italy-Germany spread move.
Regression fails when correlation is unstable (regime change, policy shock). Rolling windows help but lag.
Model hedges fail when the model is calibrated for fit, not for P&L explanation. A beautiful Nelson-Siegel fit that misses overnight carry can produce “hedged” books with steady bleed.
Düring’s practical hierarchy: use the simplest hedge that matches the risks you actually fear for the horizon you hold. Adding factors because you can does not remove idiosyncratic risk; it trades it for estimation error.
Link back to futures and basis
Chapter 28’s futures hedges are a special case of price regression when yield is undefined on the future. Basis risk is the residual Chapter 32 warns about when bond RV trades use futures for cheap execution. Chapter 36 makes the math explicit: unless β is stable, your “hedged” widener is still a directional bet on basis.
Chapter 36 closes the loop from Chapter 31 curve trades to desk implementation. You pick the hedge instrument by liquidity, then pick the method by how much structure you believe will move together. Duration match is a special case, not the default for global books.