Delta (Call & Put)

Delta: The First Greek

Delta (Δ) measures how much the option price changes for a $1 move in the underlying stock price:

$\Delta = \frac{\partial V}{\partial S}$

Black-Scholes Delta

For a European call: $\Delta_{\text{call}} = N(d_1)$

For a European put: $\Delta_{\text{put}} = N(d_1) - 1$

Since $0 \leq N(d_1) \leq 1$:

• Call delta is always between 0 and 1
• Put delta is always between -1 and 0

Interpretation

• Delta ≈ 0.6 means: the call gains ~$0.60 for every$1 the stock rises
• An ATM option has delta ≈ 0.5 (call) or ≈ -0.5 (put)
• A deep ITM call has delta ≈ 1 (moves dollar-for-dollar with the stock)
• A deep OTM call has delta ≈ 0 (barely reacts to stock moves)

Delta as Probability

Delta is also approximately equal to the risk-neutral probability that the option expires in-the-money.

Delta Hedging

If you sell 1 call option (delta = 0.6), you can hedge by buying 0.6 shares of stock. This creates a delta-neutral portfolio that is insensitive to small stock moves.