Option Greeks: Delta

Delta is the change in option premium expected from a small change in the stock price.

It is an important by-product of the Black-Scholes model.

There are three common uses of delta.

  • Measure of option sensitivity
  • Hedge ratio
  • Likelihood of becoming in-the-money

Measure of Option Sensitivity

For a call option, the delta is represented as:

For a put option, the delta is represented as:

Delta indicates how many shares of stock are required to replicate the returns of the option. For example, assume that a call option has a delta of 0.75. If the stock price rises by $1.00, the call option will increase by about 75 cents.

Delta is always positive (between 0 and 1) for long calls and short puts and negative (between 0 and -1) for long puts and short calls. For a European option, the absolute values of the put and call deltas will sum to 1. In the Black-Scholes model, the call delta is exactly equal to N(d1).

  • The delta of an at-the-money option declines linearly over time and approaches 0.50 at expiration

  • The delta of an out-of-the-money option approaches zero as time passes

  • The delta of an in-the-money option approaches 1.0 as time passes

Hedge Ratio

Delta is also used a measure of the hedge ratio.

Assume a short option position has a delta of 0.50. If an investor owns 100 shares of the stock, he can create a perfect hedge by writing 2 calls. This is because the underlying stock always has a delta of one.

Similarly, if a portfolio of options in a stock  is +2.50, the trader would be able to hedge the portfolio by selling short 250 shares of the underlying. This is called delta-hedging.

Likelihood of Becoming In-the-Money

Some traders also use delta is a measure of the probability that an option will be in the money at expiration.

For example, if an out-of the-money option has a delta 0.20, it indicates that the option has a 20% chance of being in the money at expiry, that is the stock price will be above the option striking price at expiration.

Lesson Resources

Download the options pricing and geeks spreadsheet.