Delta Hedging and Dynamic Rebalancing

Delta tells you how many shares to buy

Delta = roughly how much the option gains for every $1 the stock rises. After selling 10 calls (1,000 shares) with a Delta of 0.5, buying 500 shares offsets small moves, making the position Delta neutral.

Delta changes, so the hedge has to follow

When the stock rises, Delta grows and you need to buy more; when it falls, you trim. The speed at which Delta changes is Gamma. Constantly adjusting the shares is dynamic hedging, and the more violently price jumps, the harder it is for rebalancing to keep up.

After hedging, it's a contest of volatilities

Delta hedging removes directional risk; the remaining P&L mostly depends on which is bigger, actual volatility or the implied volatility (IV) used to price the option. If things are calmer, the seller earns the time value; if wilder, rebalancing always buys high and sells low, and the seller loses.

Key takeaways

• After selling calls, buy Delta × the contract's shares in stock. For small moves the two sides offset; this is called Delta neutral.
• When the stock moves, Delta changes (that change is called Gamma), so you have to keep rebalancing: dynamic hedging.
• The hedged P&L depends mainly on whether realized volatility or the implied volatility the option was priced at is bigger.
• When prices jump sharply, rebalancing can't keep up, and an option seller can lose money even while hedged.

Try it yourself

You can do these exercises in this lesson's interactive simulation:

1. Keep the "Normal" market and "Rebalance daily," then press ▶ to the end: roughly what's the total P&L at expiry? Then look at the "No hedge" row on the right.
2. While it plays, watch "Shares needed vs. actually held": when the stock rises, is the system buying or selling?
3. Switch to "Hedge only at the start" and replay: from which day do the cyan and amber bars clearly stop matching?
4. Switch to the "Wild" market: even with daily rebalancing, why does the seller still lose? What happened on the day of the gap?

Quiz and explanations

1. You sell 10 calls (100 shares each) with Delta = 0.6. How many shares must you buy to be Delta neutral?

Answer: 600 shares.0.6 × 10 × 100 = 600 shares.

2. After the stock rises, these calls' Delta becomes 0.7. To stay neutral you should…

Answer: Buy another 100 shares.You now need 0.7 × 1,000 = 700 shares, 100 more than before.

3. You sell a call priced at 30% volatility and Delta hedge it daily. The stock's realized volatility turns out to be only 15%. The seller most likely…

Answer: Makes money: realized volatility was below the priced volatility.Delta hedging removes directional risk, so the remaining P&L mainly comes from realized minus implied volatility. With a calmer stock, the seller earns the time value.

Terms in this lesson

• Delta: Roughly how much an option's price changes when the underlying rises $1; near 1 deep in the money, near 0 far out of the money.
• Delta neutral (Delta Neutral): A portfolio whose total Delta is 0, so small moves in the underlying barely change its value.
• dynamic hedging (Dynamic Hedging): Continually adjusting the hedge as prices move to keep the portfolio neutral.
• Gamma: How much Delta changes when the underlying moves $1; the higher the Gamma, the more often a Delta hedge needs adjusting.
• implied volatility (IV) (Implied Volatility (IV)): The volatility backed out of an option's market price, reflecting the market's expectation of future swings; the higher the IV, the pricier the option.
• volatility (Volatility): How wildly the underlying's price swings; the bigger the swings, the more expensive options get.