AFM syllabus

In order to manage a portfolio of options, the dealer must know how the value of the options will vary with changes in the various factors affecting their price.

Such assessments of sensitivity are measured by:

Delta

Delta = Change in option price / Change in price of underlying security

Delta is a measure of how much an option premium changes in response to a change in the security price. 

For instance, if a change in share price of 5p results in a change in the option premium of 1p, then the delta has a value of (1p/5p) 0.2.

Therefore, the writer of options needs to hold five times the number of options than shares to achieve a delta hedge. 

A delta value ranges between 0 and +1 for call options, and between 0 and -1 for put options. 

The actual delta value depends on how far it is in-the-money or out- of-the-money.

The absolute value of the delta moves towards 1 (or -1) as the option goes further in-the-money (where the price of the option moves in line as the price of the underlying asset) and shifts towards 0 as the option goes out-of-the-money (where the price of the option is insensitive to changes in the price of an underlying asset)

At-the- money calls have a delta value of 0.5, and at-the-money puts have a delta value of -0.5.

Gamma

Gamma  = Change in the delta value / Change in the price of the underlying security

Gamma measures the amount by which the delta value changes as underlying security prices change.

Theta

Theta measures how much the option premium changes with the passage of time. 

The passage of time affects the price of any derivative instrument because derivatives eventually expire. An option will have a lower value as it approaches maturity. 

Thus:

Theta  = Change in the option price (due to changes in value) / Change in time to expiry