Greeks-Based Risk (Delta Hedging)

Greeks-Based Risk: Delta Hedging

Delta (Δ) measures how much an option's price changes per $1 move in the underlying asset. Delta hedging neutralizes this exposure by holding an offsetting stock position.

Hedge Quantity

To delta-hedge a position of option_quantity options each with delta Δ:

hedge_quantity = -delta × option_quantity

• Long call (Δ > 0) → short shares to hedge
• Long put (Δ < 0) → long shares to hedge

Hedged P&L

When the underlying moves from S0 to S1:

pnl = option_pnl + hedge_quantity × (S1 - S0)

In a perfect delta hedge, the two components offset each other, leaving a small residual from gamma (second-order effects).

Example

Long 100 call options with Δ = 0.6:
hedge_quantity = −0.6 × 100 = −60 shares (short 60 shares)

If S moves from $100 to$105 and options lose $3: pnl = −3 + (−60) × (105 − 100) = −3 − 300 = **−$303**

This is the gamma P&L — the hedge is not perfect over large moves.