An option's price can be influenced by a number of factors that can either help or hurt traders depending on the type of options positions they have established. Successful traders understand the factors that influence options pricing, which includes the "Greeks"—a set of risk measures that indicate how exposed an option is to time-value decay, implied volatility and pricing changes in an underlying security. This article explains the significance of four Greek risk measures—delta, theta, vega, and gamma—and explains their importance. We review option characteristics that will help you better understand the Greeks.

Influences on an Option's Price

Table 1 lists the major influences on both a call and put option's price. The plus or minus sign indicates an option's price direction resulting from a change in one of the listed variables. For example, when there is a rise in implied volatility, there is an increase in the price of an option as long as other variables remain static.

Options Increase in Volatility Decrease in Volatility Increase in Time to Expiration Decrease in Time to Expiration Increase in the Underlying Decrease in the Underlying
Calls + - + - + -
Puts + - + - - +

Table 1: Major influences on an option's price

Bear in mind that results will differ depending on whether a trader is long or short. If a trader is long on a call option, a rise in implied volatility will be favorable because higher volatility is typically priced into the option premium. On the other hand, if a trader has established a short call option position, a rise in implied volatility will have an inverse (or negative effect). The writer of a naked option, whether a put or a call, would not benefit from a rise in volatility because writers want the price of the option to decline.

Tables 2 and 3 present the same variables but in terms of long and short call options (Table 2) and long and short put options (Table 3). Note that a decrease in implied volatility, reduced time to expiration, and a fall in the price of the underlying will benefit the short call holder. At the same time, an increase in volatility, a greater time remaining on the option, and a rise in the underlying will benefit the long call holder. A short put holder benefits from a decrease in implied volatility, a reduced time remaining until expiration, and a rise in the price of the underlying while a long put holder benefits from an increase in implied volatility, a greater time remaining until expiration, and a decrease in the price of the underlying. 

Call

Options


 


 


Increase in Volatility Decrease in Volatility Increase in Time to Expiration Decrease in Time to Expiration Increase in the Underlying Decrease in the Underlying
Long + - + - + -
Short - + - + - +

Table 2: Major influences on a short and long call option's price

Put

Options


 


 


Increase in Volatility Decrease in Volatility Increase in Time to Expiration Decrease in Time to Expiration Increase in the Underlying Decrease in the Underlying
Long + - + - - +
Short - + - + + -

Table 3: Major influences on a short and long put option's price

Interest rates play a negligible role in a position during the life of most option trades, so we exclude this price variable from the discussion. However, it is worth noting that higher interest rates make call options more expensive and put options less expensive all other things being equal.

This summary of pricing influences provides context for an examination of the risk categories used to gauge the relative impact of these variables. The Greeks allow traders to project changes in an option's price.

The Greeks

Table 4 describes four primary risk measures—the Greeks—that a trader should consider before opening an option position. 

Delta

Delta is a measure of the change in an option's price (premium of an option) resulting from a change in the underlying security. The value of delta ranges from -100 to 0 for puts and 0 to 100 for calls (multiplied by 100 to shift the decimal). Puts generate negative delta because they have a negative relationship with the underlying; that is, put premiums fall when the underlying rises and vice versa.

Conversely, call options have a positive relationship with the price of the underlying: if the underlying rises, so does the call premium provided there are no changes in other variables such as implied volatility or time remaining until expiration. If the price of the underlying falls, the call premium will also decline provided all other things remain constant.

An at-the-money option has a delta value of approximately 50 (0.5 without the decimal shift), which means the premium will rise or fall by half a point with a one-point move up or down in the underlying. For example, if an at-the-money wheat call option has a delta of 0.5 and wheat rises by 10 cents, the premium on the option will increase by approximately 5 cents (0.5 x 10 = 5) or $250 (each cent in a premium is worth $50).

Vega Theta Delta Gamma
Measures Impact of a Change in Volatility Measures Impact of a Change in Time Remaining Measures Impact of a Change in the Price of Underlying Measures the Rate of Change of Delta

Table 4: The main Greeks

As the option gets further in the money, delta approaches 100 on a call and -100 on a put with the extremes eliciting a one-for-one relationship between changes in the option price and changes in the price of the underlying. In effect, at delta values of -100 and 100, the option behaves like the underlying in terms of price changes. This behavior occurs with little or no time value as most of the value of the option is intrinsic. We will come back to the concept of time value when we discuss theta.

Three things to keep in mind with delta:

1. Delta tends to increase closer to expiration for near or at-the-money options.

2. Delta is further evaluated by gamma, which is a measure of delta's rate of change.

3. Delta can also change in reaction to implied volatility changes.

Gamma

Gamma, also known as the first derivative of delta, measures delta's rate of change. Table 5 shows how much delta changes following a one-point move in the price of the underlying. When call options are deep out of the money, they generally have a small delta because changes in the underlying generate tiny changes in pricing. However, the delta becomes larger as the call option gets closer to the money.

Table 5

In Table 5, delta is rising as we read the figures from left to right, and it is shown with values for gamma at different levels of the underlying. The column showing profit/loss (P/L) of -200 represents the at-the-money strike of 930, and each column represents a one-point change in the underlying. At-the-money gamma is -0.79, which means that for every one-point move of the underlying, delta will increase by exactly 0.79 (for both delta and gamma the decimal has been shifted two digits by multiplying by 100).

If you move right to the next column (which represents a one-point move higher to 931 from 930), you can see that delta is -53.13 (an increase of .79 from -52.34). Delta rises as this short call option gets into the money, and the negative sign means that the position is losing because it is a short position (in other words, the position delta is negative). Therefore, with a negative delta of -51.34, the position will lose 0.51 (rounded) points in premium with the next one-point rise in the underlying.

There are some additional points to keep in mind about gamma:

1. Gamma is smallest for deep out-of-the-money and deep in-the-money options.

2. Gamma is highest when the option gets near the money

3. Gamma is positive for long options and negative for short options.

Theta

Theta is not used much by traders, but it is an important conceptual dimension. Theta measures the rate of decline of time-premium resulting from the passage of time. In other words, an option premium that is not intrinsic value will decline at an increasing rate as expiration nears. Table 6 shows theta values at different time intervals for an S&P 500 Dec at-the-money call option. The strike price is 930. As you can see, theta increases as expiration gets closer (T+25 is expiration). At T+19, which is six days before expiration, theta has reached 93.3, which in this case tells us that the option is now losing $93.30 per day, up from $45.40 per day at T+0 when the trader hypothetically opened the position.

- T+0 T+6 T+13 T+19
Theta 45.4 51.85 65.2 93.3

Table 6: Theta values for short S&P Dec 930 call option

Some additional points about theta to consider when trading:

1. Theta can be high for out-of-the-money options if they carry a lot of implied volatility.

2. Theta is typically highest for at-the-money options

3. Theta will increase sharply in the last few weeks before expiration and can severely undermine a long option holder's position, especially if implied volatility declines at the same time.

Vega

Vega, the fourth and final risk measure, quantifies risk exposure to implied volatility changes. Vega tells us approximately how much an option price will increase or decrease given an increase or decrease in the level of implied volatility. Option sellers benefit from a fall in implied volatility, but it is just the reverse for option buyers. In Table 5, the short call has a negative vega, which indicates the position will gain if implied volatility falls (hence the inverse relationship indicated by the negative sign). The value of vega indicates by how much the position will gain. Using at-the-money vega, which is -96.94, a short call position will gain $96.94.for each 1% drop in implied volatility. Conversely, the position would lose $96.94 in reaction to a 1% rise in implied volatility.

Additional points to keep in mind regarding vega:

1. Vega can increase or decrease without price changes of the underlying (due to changes in implied volatility).

2. Vega can increase in reaction to quick moves in the underlying. 

3. Vega falls as the option gets closer to expiration.

The Bottom Line

The price of an option is influenced by changes in the underlying, the time to expiration, implied volatility, and the way that traders measure the impact of those variables.