# Delta and Gamma

In options trading, investors use risk measures known as the Greeks (that is theta, vega, delta, and gamma) to evaluate investment options and gauge a potential investment’s exposure to factors such as time, price moves in the underlying, and volatility of the underlying.

## The Definition of Delta

The Options Industry Council (OIC) is a great resource for investor education. On its about page, the OIC states “The Options Industry Council was created as an industry cooperative to provide education about the benefits and risks of exchange-listed equity options.”

The OIC defines delta as “the theoretical estimate of how much an option’s premium may change given a $1 move in the underlying. For an option with a Delta of .50, an investor can expect about a $.50 move in that option’s premium given a $1 move, up or down, in the underlying.”

Delta should be understood as a theoretical model and not a scientific measurement. Other factors like implied volatility and time to expiration can also impact an option’s premium. As other factors influence an option’s premium, an investor cannot rely on delta alone to predict how a change in price of an underlying security will impact the premium – one must also consider the other factors (i.e. implied volatility and time to expiration.

## How is Delta Measured?

For puts, delta is measured from -1.00 to 0; for calls, delta is measured from 0 to 1.00. For example, a call option with a .50 delta means that for a $1 increase in the underlying stock, the premium may increase by $0.50. Similarly, a put option with a -.50 delta indicates that if the underlying stock increased by $1, the put option premium would decrease by $0.50.

## How does Delta Work?

Let’s look at in-the-money and out-of-the-money calls and puts to see how delta works (as a theoretical model) to predict how a change in the underlying stock price will affect an option’s premium.

### Out of the Money Calls

As review, a call is out of the money when the strike price is higher than the stock price. Delta values for out of the money calls range from .50 to 0, and as calls get further out of the money (i.e. as strike price gets higher and higher above the stock price), delta approaches 0. In other words if the stock price is 100 the delta of a 120 call might be .20, while the delta of a 130 call might be .10. The delta of a 150 call might be .01, which means a $1 increase in the stock price would result in very little if any positive impact on the premium of the 150 call.

Premium of OTM call option | Delta of OTM Call option | Change in Stock Price | Change in Premium (delta * change in underlying) | New Premium |

$4.50 | 0.49 | $1.00 | $0.49 | $4.99 |

$4.50 | 0.49 | $0.50 | $0.25 | $4.75 |

$4.50 | 0.05 | $1.00 | $0.05 | $4.55 |

$4.50 | 0.05 | $0.25 | $0.01 | $4.51 |

$4.50 | 0.05 | $(1.00) | $(0.05) | $4.45 |

### In the Money Calls

As review, a call is in the money when the strike price is lower than the stock price. Delta values for in the money calls range from .50 to 1, and as calls get further into the money (i.e. as the strike price gets lower and lower below the stock price), delta approaches 1. In other words if the stock price is 100 the delta of an 80 call might be .80, while the delta of a 70 call might be .90. As delta gets closer to 1, an investor could theoretically expect the premium price for an option to move generally on a 1:1 basis with changes in stock price.

Premium of ITM call option | Delta of ITM Call option | Change in Stock Price | Change in Premium (delta * change in underlying) | New Premium |

$4.50 | 0.86 | $1.00 | $0.86 | $5.36 |

$4.50 | 0.86 | $0.50 | $0.43 | $4.93 |

$4.50 | 0.75 | $1.00 | $0.75 | $5.25 |

$4.50 | 0.75 | $0.25 | $0.19 | $4.69 |

$4.50 | 0.75 | $(1.00) | $(0.75) | $3.75 |

### Out of the Money Puts

As review, a put is out of the money when strike price is lower than the stock price. Delta values for out of the money puts range from -0.50 to 0, and as puts get further out of the money (i.e. as the strike price gets lower and lower below the stock price), delta approaches 0. In other words, if the stock price is 100 the delta of an 80 put might be-.40, while the delta of a 60 put might be -0.10. The delta of a 50 put might be -.01, which means a $1 increase in the stock price would result in very little if any negative impact on the premium of the 50 put.

Premium of ITM Put option | Delta of ITM Put option | Change in Stock Price | Change in Premium (delta * change in underlying) | New Premium |

$4.50 | -0.49 | $1.00 | $(0.49) | $4.01 |

$4.50 | -0.49 | $0.50 | $(0.25) | $4.26 |

$4.50 | -0.02 | $1.00 | $(0.02) | $4.48 |

$4.50 | -0.02 | $0.25 | $(0.01) | $4.50 |

$4.50 | -0.02 | $(1.00) | $0.02 | $4.52 |

### In the Money Puts

As review, a put is in the money when the strike price is higher than the stock price. Delta values for in the money puts range from -.50 to -1.00, and as puts get further into the money (i.e. as the strike price gets higher and higher above the stock price), delta approaches -1.00. In other words, if the stock price is 100 the delta of a 110 put might be -.60, while the delta of a 130 put might be -.80. The delta of a 150 put might be -1.00, which means an investor could theoretically expect the premium price for an option to move in a -1:1 basis with changes in stock price.

Premium of ITM Put option | Delta of ITM Put option | Change in Stock Price | Change in Premium (delta * change in underlying) | New Premium |

$4.50 | -0.51 | $1.00 | $(0.51) | $3.99 |

$4.50 | -0.51 | $0.50 | $(0.26) | $4.25 |

$4.50 | -0.98 | $1.00 | $(0.98) | $3.52 |

$4.50 | -0.98 | $0.25 | $(0.25) | $4.26 |

$4.50 | -0.98 | $(1.00) | $0.98 | $5.48 |

## Gamma

Whereas delta provides an investor with a theoretical amount by which an option’s premium will change given a $1 change in the underlying stock, gamma provides an investor with a theoretical amount by which an option’s delta will change given a $1 change in the underlying stock. Simply put, delta measures the theoretical change in premium given a $1 stock move, gamma measures the theoretical change in delta given a $1 stock move.

Gamma is measured from 0.00 to 1.00.

Although gamma seems like a topic that’s worth knowing about, at this stage I don’t see the benefit of doing a deep dive on gamma – it is well laid out by the Options Industry Council here.