α THE BOOK SHORT GAMMA
β NET RISK (BOOK + HEDGE)
Net IR01 / 1bp$0
Net IR Gamma / 1bp²$0
Net Vega / 1bp σ_n$0
Total premium paid$0
Net P&L at ±50bp$0
Net P&L at ±100bp$0
YOUR HEDGEPAYER · 5Y SWAP
IR01 contribution$0
IR Gamma contribution$0
Vega contribution$0
Premium$0
P&L UNDER PARALLEL RATE SHIFT
BOOK ALONEBOOK + HEDGE
UNHEDGED
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All three puzzles solved.
CONVEXITY HEDGING · THE FULL ARC
LEVEL 1 · LINEAR
Match notional, flip direction
Linear positions take linear hedges. Net
IR01 with an opposite swap of equal notional and tenor. The simplest hedge in fixed income.LEVEL 2 · CONVEXITY
Buying gamma costs vega
All long swaptions are gamma-positive — but they all pay vega. ATM gamma scales as
1/√T, vega as √T. Short-dated swaptions give ~20× more gamma per dollar of vega than long-dated.LEVEL 3 · NO FREE HEDGE
You trade one risk for another
Hedging IR Gamma with swaptions cannot zero vega — the instruments themselves carry vega. Optimal answer minimizes the trade-off, never eliminates it.
The arc you walked is the foundation of every rates flow desk. Linear hedging → buying convexity → managing the trade-offs. Most traders learn this implicitly, by blowing up positions. You just got the structured version.
REPLAY
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PROMETHEUS
AI HEDGE ADVISOR