Dynamic hedging: gamma slippage and the cost of convexity
Continuous re-hedging eliminates first-order risk but pays a gamma cost when the underlying moves. The cost is exactly what the option premium pays for.
You’ve delta-hedged. So a small move in the stock leaves you flat? Almost — but delta itself MOVES with the stock (that’s gamma). After the move, your hedge is wrong. You re-balance, take a tiny loss, and repeat. That tiny loss IS the cost of convexity. Long-gamma traders earn on movement; short-gamma traders pay for it. We’ll measure this loss for one trade.
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Worked example
You sold one call ($100$ shares) with $\Delta = 0.5$ and $\Gamma = 0.05$ at stock $\$100$. The stock moves to $\$102$. Estimate the gamma-driven P&L over the move (assuming you continuously delta-hedge).
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Reflection
Why does the gamma-P&L scale with $(\Delta S)^2$ and not $\Delta S$? If you sold a short-dated, near-the-money option, where do gamma and theta concentrate, and what does that imply about the rebalancing schedule you'd run?