Second-Order Greeks; Gamma

How Gamma Affects Delta

Options Trader
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Most options traders have no difficulty in understanding how the first-order Greeks (Theta, Delta, Vega, and the far-less-important Rho) work. When one specific parameter (calendar date, stock price, implied volatility, or interest rate) changes, one of the Greeks provides a very good estimate of how that change affects the value of any option.

These Greeks represent something very important in the options world. Anyone who owns an option position should be concerned with knowing the risk associated with owning that position. And the Greeks come to the rescue because they are used to measure risk. Clarification: The Greeks tell us just how much money we can anticipate earning (or losing) when the price of the underlying asset changes. The estimate will seldom be correct to the nearest one penny, but the estimate is sufficiently accurate that the trader should never be surprised when a big chunk of money is earned or lost. 

If you take the time to use your broker's risk management tools (of course, you can use your own) to draw a picture (i.e., plot stock price vs. P/L on a graph), you will never be surprised by an unexpectedly large loss. That allows you to build a position where the risk of loss resides within your comfort zone. That is accomplished by owning a position with an appropriate position size.

The charts provide a clear picture of how much money may be lost or earned if one week passes, or if the stock moves higher by 5%, or if implied volatility jumps higher by 10%, etc. All option traders must understand these very basic ideas behind options:

  • Delta: Call owners usually* earn profits when the underlying stock rises because calls have positive Delta;
  • Put owners usually* earn profits when the underlying stock falls because puts have negative Delta.

* Not 'always' because another factor may be large enough to offset Delta.

  • Theta: All options come with negative Theta, and lose value as the days pass.
  • Vega: All options come with positive Vega. Thus, options gain value when implied volatility increases. NOTE: It is the implied volatility (IV) that directly affects the option value, but when overall market volatility increases, so does IV. 
First- and Second-order Greeks
First-order Greeks measure how the value of an option changes when one of the parameters affecting the option price changes.
Second-order Greeks measure how the value of a first order Greek changes when one of the parameters affecting the option price changes.

Example: First-Order Greek

When the stock price increases, Delta measures the expected change in the option price.

  • When Delta is 35, call options gain ~35% as much value as the stock (i.e., 35-cents per point).
  • When Delta is -35, put options lose ~35% as much as the stock price change.

When the stock price decreases, Delta still measures the expected change in the option price. 

  • When Delta is 20, call options lose ~20% as much as the stock price change. 
  • When Delta is-30, put options gain ~30% as much as the stock price decrease.

When you own an option (i.e, when your position has positive Gamma), you will discover a certain price range when that Delta increases very quickly as the stock price moves higher. That phenomenon is referred to as "exploding Delta" and produces to significant profits. That range tends to be near 25 to 40 delta.

However, for every option buyer, there is a seller and those exploding deltas is one of the reasons that selling un-hedged (i.e., a position with no protection) options is very risky.

Example: Second-Order Greek

When the stock price increases, Gamma measures the expected change in Delta. In other words, Gamma measures the sensitivity of Delta to a change in the stock price. 

  • When Gamma is 3 and Delta is 26, call options gain 3-Delta (to 29) when the stock price moves one-point higher. 
  • When Gamma is 3 and Delta is -26, put options lose 3-Delta (to 23) when the stock price moves one-point higher.

When the stock price decreases, Gamma measures the expected change in Delta.

  • When Gamma is 5 and Delta is 65, call options lose 5-Delta (to 60) when the stock price moves one-point lower. 
  • When Gamma is 5 and Delta is -65, put options gain 5-Delta (to -70) when the stock price moves one-point lower.

Other than Gamma, other second-order Greeks are seldom used by retail options traders.

Under different circumstances, we observed that a 2-point stock price change did not affect the call option as expected.  That occurred because Delta changed. It was 51 at the original stock price, but after the move, the delta was different. The best estimate for the effect of delta comes from using the average delta -- the midpoint between the starting (i.e., Delta at the original stock price) and ending Delta (Delta at the final stock price).

Gamma Summary

All options have positive gamma.  

When you own an option, add its Gamma to the total position Gamma.

When you sell an option, subtract its Gamma from the position Gamma.

Gamma is greatest when the strike price is close to the stock price [i.e., the option is at (or near) 50-Delta] and declines as the option moves away from the strike price and becomes further in the money (ITM) or further out of the money (OTM). 

By measuring position risk, and then reducing risk (when necessary), you are practicing active risk management.