greeks delta and gamma of a stock option

The Greeks: Delta and Gamma of a Stock Option

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The greeks, such as Delta and Gamma, of stock options help us explain the price of the stock option contract. These are theoretical concepts, which are directionally correct, but the option price changes are not guaranteed to follow any exact ‘formula’. In this post we will explore the delta of a stock option and also understand what gamma is.

What is Delta of a Stock Option?

Delta is a theoretical measure of price movement in the option contract per dollar price movement in the price of the underlying asset (let’s assume underlying asset is a stock).

Option Price Calculation based on Delta: What Happens to Option Price when the Stock Price Changes?

Price Change of Call Option

Investors buy a call option when they want to benefit from an increase in the stock price. When the stock price goes up, the call option price also goes up. Hence, Delta of a Call Option is Positive. Delta helps us calculate what the theoretical change in the call option price would be based on price movement in the stock.

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Let’s consider the following scenario:

  • Current Stock Price = $100
  • Call Option Premium (Current Value) = $15
  • Delta of the Call Option = 0.25 or 25%
Stock Price rises to $101

The Call Option premium rises by 0.25 * ($101 – $100) = $0.25, to reach $15.25

Stock Price rises to $105

The Call Option premium rises by 0.25 * ($105 – $100) = $1.25, to reach $16.25

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Stock Price falls to $99

The Call Option premium falls by 0.25 * ($100 – $99) = $0.25, to reach $14.75

Stock Price falls to $95

The Call Option premium falls by 0.25 * ($100 – $95) = $1.25, to reach $13.75


delta of a stock option

Price Change of Put Option

Investors buy a put option when they want to benefit from a decrease in the stock price. When the stock price goes down, the put option price goes up. Hence, Delta of a Put Option is Negative. Delta helps us calculate what the theoretical change in the put option price would be based on price movement in the stock.

Let’s consider the following scenario:

  • Current Stock Price = $100
  • Put Option Premium (Current Value) = $5
  • Delta of the Put Option = -0.10 or -10%
Stock Price falls to $99

The Put Option premium rises by 0.10 * ($100 – $99) = $0.10, to reach $5.10

Stock Price falls to $95

The Put Option premium rises by 0.10 * ($100 – $95) = $0.50, to reach $5.50

Stock Price rises to $101

The Put Option premium falls by 0.10 * ($101 – $100) = $0.10, to reach $4.90

Stock Price rises to $105

The Put Option premium falls by 0.10 * ($105 – $100) = $0.50, to reach $4.50


How does Strike Price Impact the Delta of a Stock Option?

Out of Money Call Options with strike price far from the stock’s current price have low delta – a dollar change in stock price will move the call option price by a lower dollar amount.

Out of Money Call Options with strike price closer to the stock’s current price have higher delta.

In the Money Call Options have higher delta than Out of Money Call Options.

Deep In the Money Call Options have higher delta than ITM Call Options with strike prices closer to the stock’s current price.

delta of a stock option

Generally, the lower the strike price of the Call Option, the higher the Delta.

What is Gamma of a Stock Option?

So, now that we understand what Delta is, let’s see how Gamma relates to Delta of a Stock Option. You might have wondered whether Delta itself changes based on some other factors. In fact, delta changes!

Gamma of a Stock Option actually measures the rate of change in delta. Gamma is always positive and is the highest for At The Money Stock Options (i.e. for options with strike price closest to the stock’s current price). It reduces as the strike price gets further from the current stock price, in either direction.

For analogy, if delta is speed of a car, gamma is the rate of change in speed, or acceleration of the car.

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