It's Greek to Me - Delta and Gamma

In my previous article, Options - It's Greek to Me, you learned what option Greeks are all about. In this article I will cover delta and gamma. These two measures are probably the most important to understand because they give you a quick measure of risk relative to underlying asset price, a measure to determine hedge ratios, and a snapshot of the probability an option will expire in-the-money (ITM).

If stock price goes up \$1 dollar, how much will a call option premium change?  This is the question delta answers. The delta is the amount an option's price moves relative to the underlying asset price change. Call options have a positive delta (option and asset move in same direction) while put options have a negative delta. For example, if delta of a call option is 0.25 then the option will go up 0.25 for each dollar up move in the asset. If delta of a put option is -0.25 then the option will go up 0.25 for each dollar down move in the asset.

Delta is also dependent on how close the asset price is to the option strike (exercise) price. A far out-of-the-money (OTM) option will have a delta close to zero. An at-the-money (ATM) option typically has a delta around 0.50 and a far ITM option will have a delta close to 1.0.  This change in delta as asset price changes is called gamma.

Gamma is the change in delta relative to a change in the underlying asset. Unlike delta, gamma is highest for ATM options and lowest for deep ITM and OTM options.

The underlying asset always has a delta to 1.0 (100%) and gamma of zero. This is useful in determining equivalent positions and hedge positions. If you buy 200 shares of a stock then you are long 2.0 deltas. To hedge this position by buying puts that are ATM with delta of -0.50,you need to buy 4 puts to completely hedge the long stock (4 X -0.50 = -2.0). The sum of the deltas of the stock position and the option position would then be zero, i.e. not risk at time the positions were established.

Assume we have a stock valued at \$25.00 with 100 days to expiration of its options, volatility is 30%, and interest rate is 6%. We will look at option prices, delta and gamma for OTM, ATM and ITM options.

Strike   Call  Delta  Gamma   Put    Delta  Gamma
20.00   5.422  0.946  0.028  0.096  -0.054  0.028  (Call ITM, Put OTM)
25.00   1.764  0.573  0.100  1.356  -0.427  0.100  (Call ATM, Put ATM)
30.00   0.318  0.164  0.063  5.000  -0.793  0.063  (Call OTM, Put ITM)

You can see that OTM options have low delta and gamma, ATM options delta are close to 0.50 and have the highest gamma, while ITM options have high delta and low gamma. The gamma is telling you the risk associated with a change in underlying asset price. If the ATM option delta is 0.50 with gamma of 0.10 then the delta of the call will go to 0.60 (delta+gamma) if asset price goes up 1-point and the put delta will go to -0.40 with the same 1-point up move of the asset.

You may be asking what this means about risk. In the above scenario, if you bought an ATM call and put with delta of 0.50 you would have a delta neutral straddle. That is, the sum of the call and put deltas is zero. The equivalent stock position is to own zero shares. However, when the underlying asset moved up 1-point and the deltas changed to 0.60 and -0.40 then the sum becomes 0.20 and the equivalent stock position is now long 20 shares. Being long 20 shares is higher risk than owning zero shares.  This also makes it clear that the only way to have a gain is to take the risk.

Another piece of information delta gives you is the probability of the option expiring in the money. Again, look at the above situation. When the deltas were 0.50 for ATM options, you had a 50/50 chance the options would have a value greater than zero at expiration. When the underlying asset when up 1-point, the probability the call option would expire with value increased to 60% (delta = 0.60) whereas the probability the put option would expire with value decreased to 40%.

In summary, if your position delta is positive you want the underlying asset to increase in price. If your position delta is negative you want the underlying asset to decrease in price. If your position gamma is positive you want the underlying asset to move quickly, regardless of direction. If your position gamma is negative you want the underlying asset to move slowly, regardless of direction.

Getting to know and understand delta and gamma places you at a significant advantage over options traders who ignore the Greeks.

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