# Option Gamma

This is part of option Greeks tutorials. See also delta, theta, vega, rho.

## What Is Gamma

Gamma is closely related to delta – both measure an option's sensitivity to underlying price, although each in a different way. While delta indicates how much option premium will change if underlying price increases by \$1, gamma measures how much the delta itself will change if underlying price increases by \$1. While delta is the speed of option price change, gamma is the acceleration.

## Example

Consider a \$35 strike call option on a stock that is currently trading at \$35 (the option is at the money). With 20 days to expiration, implied volatility of 30% and interest rate at 2.50%, the option's premium is \$1.00, delta is 0.52 and gamma is 0.16.

The delta indicates that the option's price will rise by \$0.52 (to \$1.52) if the underlying stock increases by \$1 (to \$36). In reality, if the stock goes up to \$36, the option's price gets to \$1.60 (8 cents higher than predicted by the delta). Was the delta incorrect?

One characteristic of delta (and all the other Greeks) is that it is not constant. As underlying price changes, not only the option premium will change, but also the delta. In our example, the delta was 0.52 when the stock was at \$35, but it gradually increased as the stock was going up. With the stock at \$35.50, the delta was already 0.60. With the stock at \$36, the delta got to 0.68. As the underlying price increased by \$1 from \$35 to \$36, the option's delta increased by 0.16 from 0.52 to 0.68. This is the gamma of 0.16.

## Gamma Values

Unlike delta, which is positive for calls and negative for puts, gamma is positive for both call and put options.

Unlike delta, which can't be greater than 1 for a single option, there is no theoretical upper limit on possible gamma values.

Like delta, gamma can reach different values depending on moneyness – the relationship between an option's strike price and current underlying price.

## Gamma and Option Moneyness

Gamma is highest (delta changes fastest) when an option is near or at the money. With underlying price close to the option's strike price, delta is close to the middle of its possible range (near 0.50 for calls or -0.50 for puts) and even a small change in underlying price can cause a significant change in delta.

When an option is far out of the money, its delta is close to zero. Neither the option's premium, nor its delta are very sensitive to changes in underlying price. For instance, with a \$50 strike call option it doesn't make much difference if the underlying is at \$20 or \$25 – it is still far out of the money in either case. As you have already guessed, gamma is close to zero for far out of the money options.

Lastly, when an option is deep in the money, its delta is very close to +1 (for calls) or -1 (for puts), but it also changes very slowly. The delta of a \$50 strike call option would still be close to +1 if the underlying dropped from \$100 to \$90 or increase from \$100 to \$110. Gamma is close to zero for deep in the money options.

When you draw a chart of gamma with underlying price in the X-axis, it often looks like the familiar bell curve: it peaks around the middle (at the money) and approaches zero on both ends (out of the money, in the money).

## Gamma and Time to Expiration

Besides moneyness, gamma is also affected by passing time. As expiration nears, gamma of at-the-money options increases and the bell-curve-shaped chart of gamma becomes more peaked. If we think of gamma as a measure of option's instability, it is no surprise that those options which are at the money and with very little time to expiration are the most instable, with highest gamma.

Conversely, as expiration approaches, both out-of-the-money and in-the-money options lose gamma. Both ends of the bell curve are pushed even closer to zero.

## Gamma and Volatility

Volatility affects gamma quite similarly as time. Higher volatility is like more time to expiration; lower volatility is like less time.

Rising volatility increases out of the money and in the money gamma, while at the money gamma falls.

Decreasing volatility increases at the money gamma, while out of the money and in the money gamma decline.

At this point you should have a good understanding of what gamma is and how it behaves for different options under different circumstances. In the last part of this tutorial we will discuss why and how we can use gamma in practical trading situations. As with other Greeks, it is all about risk management.

The greatest benefit of gamma is that it reveals hidden risk exposures which delta can't identify.

Consider a straddle – a popular option strategy composed of one call option and one put option with same strike price and expiration. When you choose the at-the-money strike, a straddle has total delta close to zero, because the call option's positive delta (close to +0.50) and the put option's negative delta (close to -0.50) cancel one another.

Strategies like straddles are often called delta neutral or non-directional, which means unaffected by underlying price direction. When trading straddles, we don't care which way the underlying security moves; we are more concerned about factors like volatility and passing time. However, the term non-directional can be dangerously misleading.

As explained above, if both the options are at the money, they have high gamma. Like all other Greeks, gamma is additive – to calculate total gamma of a portfolio of multiple options, simply add up all the long options' gammas and subtract all the short options' gammas.

As a result, a straddle may have zero delta, but it certainly has very high gamma. This means that its delta will rapidly increase if the underlying price goes up and rapidly decrease (to negative territory) if underlying price falls. A straddle is only delta-neutral at the money. When underlying price rises above the strike, a straddle quickly becomes bullish (profits from further underlying price increase); when underlying price falls below the strike, it quickly becomes bearish (profits from further decline).

Thanks to positive gamma, your profits tend to accelerate, while your losses tend to slow down if the underlying makes a big move. This is good, but every trade has two sides. The buyer of a straddle has high gamma and accelerating profits, but the seller of a straddle has high negative gamma and accelerating losses.

Traders are often attracted to negative gamma strategies like short straddles. They like the idea of making money from time decay and not having to care about market direction. Being delta neutral may lead to a false sense of security, unless the trader understands gamma. This does not mean that short gamma strategies should be avoided – they can indeed be very profitable. But their risks must not be underestimated. Paying attention to gamma is especially important with larger, more complex positions.

## Gamma Hedging

Like delta hedging is used to eliminate unwanted delta exposure, a similar concept of gamma hedging is used to neutralize gamma.

Gamma can only be hedged with options. Unlike in delta hedging, it is not possible to use the underlying security, as it has zero gamma.

The most effective way to gamma hedge is often to buy at the money, short term options, because they tend to have both highest gamma and highest liquidity. That said, when adding options to a portfolio, one should keep in mind that not only gamma, but also other Greeks (such as delta, vega or theta) may be affected.

Like with delta hedging, a gamma hedged position requires constant monitoring and rebalancing, as both delta and gamma change with changing market circumstances and with passing time.

## How to Calculate Gamma

Mathematically, gamma is the second derivative of option price (or first derivative of delta) with respect to underlying price. Like in the other Greeks tutorials, I have tried to avoid the mathematics and focused mainly on the logic and practical trading. Those interested in the exact formulas can find them in Black-Scholes Greeks Formulas and Option Greeks in Excel.

## Summary

• Gamma measures how much delta will change if underlying price increases by \$1.
• All options have positive gamma. All short option positions have negative gamma.
• Gamma is highest at the money. At the money gamma increases with passing time or decreasing volatility.
• Positive gamma means your profits accelerate in big moves.
• Negative gamma means your losses accelerate and can be very dangerous.

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