Section 26.4

Hedging and Approximations

Delta-hedging positions, delta-gamma price approximations, charm and the decay of delta, volatility smiles.

Toolkit

Delta-hedging eliminates first-order price risk, leaving exposure to higher-order Greeks and to the dynamics of the Greeks themselves. A hedged portfolio’s P&L over a small move is governed by the Taylor expansion of the option price in $S$ and $t$ :

dC \approx \Delta\,dS + \tfrac12\Gamma\,dS^2 + \Theta\,dt + \text{(vega, etc.)}.

Key facts:

$\Delta=\partial C/\partial S$ , $\Gamma=\partial^2C/\partial S^2$ ; for ATM options $\Delta\approx 0.5$ (call) or $-0.5$ (put).
Charm $=\partial\Delta/\partial t$ measures delta decay: as expiry nears, delta migrates toward $1$ (ITM) or $0$ (OTM), requiring rebalancing of the hedge even with $S$ fixed.
Under “sticky delta”, implied vol $\sigma(\Delta)$ depends only on moneyness via delta; a strike change $dK=-dS$ shifts delta by $-\Gamma\,dK$ , letting one locate the new point on the smile.

Problems