In the last two weeks’ articles, which you can read here and here, I’ve been discussing the movements in option prices in response to market forces. I’ve focused on the equal and offsetting effects of option price changes on the two parties to the option contract: the option buyer and the option writer (seller). We’ll extend that discussion today.
Last week we began looking at an example using the SPY October 173 call option. We imagined two traders: Rosie on the long side and Thorne on the short side. At that time these calls were almost exactly at the money, since SPY was at $173.05. The call itself was trading at $1.87.
We noted that the delta of this call was .54. This meant that for the next $1.00 change in SPY, up or down, the price of the 173 call would change by 54 cents, in the same direction. We know it would be in the same direction, because the delta was a positive number. If the delta had been a negative number, then that would have meant that the price of the option would change in the opposite direction from the underlying (SPY). Call options always have positive values for delta, while put options have negative delta values.
Remember that there are two people in each option trade, and the profit or loss of each is equal and opposite to that of the other. Since the call option itself has a positive delta, Rosie, our call buyer, owns that positive delta. She makes money when the underlying goes up. Rosie’s position has positive delta.
Thorne, on the other side of this trade, has sold the call. Since he did not own the call before he sold it, he is short the call. He sold it short in the belief that the price of SPY would drop. (Note that for the sake of simplicity in this example, neither Rosie nor Thorne has any position in SPY itself. Thorne’s short call is naked – not covered, as it would be if he simultaneously owned SPY). If SPY were to drop by $1.00, the 173 call would drop by 54 cents, from $1.87 to $1.33. If that happened, Thorne could buy back (buy to close) the call for $1.33. Since he had sold it for $1.87 originally, he would pocket the $.54 profit.
Since Thorne profits from a drop in the price of the underlying, we say that his position has negative delta. Note that “negative” here in no sense means “bad” or “undesirable.” “Negative” simply means “one of the two possible directions.” We could just as easily call them “left delta” and “right delta” or “east delta” and “west delta.” The point is just that there are two types of delta that have opposite meanings.
Thorne, our option seller, wants a position with negative delta. That is the type of position that profits when the underlying price goes down, and that is what he expects. Negative delta is what he gets when he sells the call. You could say that his position is short 1 call, or that it is long minus-1 call. Since he is short something that has positive delta, his position has negative delta.
Just to sum up the positions of the players with the two types of options:
Calls always have positive delta – their prices move in the same direction as the underlying.
Puts always have negative delta – their prices move in the opposite direction from the underlying.
The positions of call owners have positive delta, while those of call sellers have negative delta.
The positions of put owners have negative delta, while those of put sellers have positive delta.
Moving on from Delta, let’s look at its close relative, Gamma. We noted last week that this option had a value for gamma of .10. This meant that if SPY moved by $1.00, then the delta of that option would change by .10, and that change would be in the same direction as underlying price. In the case of a $1.00 drop, delta would change by .10. Since delta started at +.54, and price dropped, the new delta would be (+.54 – .10) = .44. If instead SPY increased in price by $1.00, the option’s new delta would be (+.54 + .10)= .64.
The definition of gamma is the change in delta for a 1-point increase in the underlying asset price. A put’s delta becomes a smaller negative number when underlying price rises, and a larger negative number when it drops. A call’s delta becomes a larger positive number when underlying price rises and a smaller positive number when price drops. So all options, both puts and calls, have positive gamma values. We algebraically add the gamma to the option’s current delta to get its new delta after a $1 price increase; and algebraically subtract the gamma from the option’s current delta to get its new delta after a $1 price drop.
Option owners’ positions (whether they own puts or calls) have positive gamma, while option sellers’ positions have negative gamma. While delta measures the speed of option price change, gamma measures the acceleration of option price change as the underlying continues to move.
OK, gamma measures the change in delta when the underlying moves. But why should we need it? Why should delta change in the first place?
Recall that another aspect of delta is that it is the approximate probability of the option’s finishing in the money. With our call strike at $173, and SPY also just a little above $173, then the chance of SPY still being above $173 at October expiration was just a little better than 50-50. If SPY went up by any amount the option would be in the money; if it went down by any amount as large as just $.05 it would be out of the money.
That’s why the delta was just a little higher than 50-50, at .54. But if SPY should move up by $1.00, to $174.05, then its probability of being above $173 at any future time will have become higher. From $174, SPY could go up, sideways, or even down by $1.04, and the 173 call would still be in the money. The probability of its ending up in the money would therefore be higher. The Gamma value of .10 tells us how much higher. It says that if SPY can make the first $1.00 move, its probability thereafter of being above $173 at October expiration will have gone from 54% to 64%. At that point Gamma will change too. It is highest when options are at the money, and less when they are away from the money in either direction.
If instead of rising, SPY should drop by $1.00 to $172.05, then its probability of ending up above $173 will have decreased. Again, the gamma value of .10 tells us how much – that probability (which is also delta) will have dropped from 54% to 44%.
So how does this affect Rosie and Thorne? Rosie, being long the option, has positive gamma. Upward movement in SPY helps her at an accelerating rate. She makes $.54 on the first $1.00 move, then another $.64 on the next dollar move, and more on the next dollar. If spy moves up far enough, her option will eventually have the highest delta it can have – 100% – so that she will then make a dollar for every additional dollar move in SPY.
Thorne, being short the option, has negative gamma. Upward price movement hurts him at an accelerating rate, just as it helps Rosie (since her profit comes out of his pocket). Downward price movement will help him, but at a decelerating rate. Delta measures his profit as price declines. Since it gets smaller, each dollar of price drop brings him a little less profit than the last. The first dollar’s drop makes him $.54, the second $.44, and then less and less. If SPY moves down so far that the probability of its ending up above $173 at the October expiration becomes zero, then the 173 call’s value will be zero, and so will its delta. At that point Thorne will have gotten all the profit out of the trade that he possibly could have, which is the $1.87 credit he originally took in by selling the option. Any further drop will not help him. Nor will it hurt Rosie any more. She will have made her maximum loss, which is that same $1.87.
Next week we’ll check back in and see how Rosie and Thorne’s trades are fairing.
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