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Teacher, Why Do We Have to Learn This Stuff?

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By this point in the series on Greeks, some of you might have something in common with students asking their teachers why they have to learn the quadratic equation. Those students can't imagine a reason they'll use that equation once they graduate. I can imagine some of you wondering why you should spend a boring few minutes each week learning about the Greeks. You've been trading, choosing your own calls and puts, doing pretty well without even knowing that Greeks existed, or caring, if you did know.

It's fairly obvious why we might want to know about delta since it tells us the probability that an option will be in the money at expiration and how much the option's price will change for each one-point move in the underlying. The Greeks can give us more information, however.

Why do we need to know about them? Risk.

Greeks tell us how much risk we're incurring. For example, let's imagine that October 29, two days before the FOMC's decision, a newbie trader held covered calls on MSFT in her portfolio. For those among you who are also newbies, a covered call is established when a trader or investor owns stock and sells one call contract for each 100 shares of stock owned, with the strike of that call higher than the current price of the stock. The trader collects a small credit for the call that is sold, and that offers a small cushion in case the stock's price drops. However, if the stock rises above the strike of the sold call, the stock can be called away. The trader doesn't participate in any further gains.

We know that stocks and indices can move quite a bit around the time of an FOMC meeting, but perhaps this newbie trader reasoned that if MSFT climbed, she would still benefit up to the price of her sold call. If MSFT dropped, the credit she'd collected from her sold call would cushion the loss. We know that her primary risk from price movement would come if MSFT dropped heavily. How much risk from price movement was she incurring?

Here's the position:

Long 500 Shares MSFT
Short 5 Contracts NOV 37.50 Calls

We learned from an earlier article that long stock positions always have a delta of 1.00/share. What about gamma for stock positions? Gamma, most will remember, tells us how much delta will change for each one-point move in the underlying, MSFT in this case. Since delta doesn't change, ever, for a long stock position, gamma must be zero.

The MSFT shares are then long 500 deltas, and they have a gamma of zero.

What about the 5 contracts of NOV 37.50 calls that were sold? A brokerage reports the following data:

MSFT 37.50 Calls
Delta: 0.09
Gamma: 0.08

Total delta for the 5 sold contracts will then equal -45.00 (-0.09 x 5 x 100). It might make sense that, although delta is positive for calls, it's negative when those calls are sold. Total gamma for those 5 contracts will be -40.00 (-0.08 x 5 x 100). You might remember that although gamma is always positive for both calls and puts, it's negative when those calls or puts are sold.

What are the position delta and gamma for the stock plus short call position?

Position Delta: 500 - 45 = 455 deltas
Position Gamma: 0 - 40 = -40

As of the close on October 29, two days before the FOMC meeting, the covered call position was long 455 deltas, still quite long. For the first one-point drop in MSFT's price, the trader risked a $455.00 loss.

What about gamma, though? What does it mean that gamma is negative? In the words of Lawrence McMillan in OPTIONS AS A STRATEGIC INVESTMENT, it "means that as the underlying security moves, the position will acquire traits opposite to that movement." In other words, if MSFT were to climb, the position becomes long fewer delta. If MSFT were to drop, however, the position becomes longer more delta. It begins to incur even more risk from price movement, because a trader doesn't want to be long more delta as price is dropping. That means that trader loses more for each point MSFT loses.

If you've heard that covered call positions are not the conservative trade that some consider them but are instead quite risky, that negative gamma tells you why. If MSFT were to drop, and the position becomes long more delta, each point loss hurts more. Soon, as the calls drop further out of the money, the gamma of the calls will drop to zero, of course, and the delta of the out-of-the-money sold calls will drop to zero, too. The position delta will then be 500 deltas, the delta of the stock position.

Intuitively, this makes sense. What it means is that the credit that the newbie trader collected for those sold calls is small, and can only cushion a little of the loss. After that cushion is used up, the position loses point for point what the stock position does.

Some calculations might make the covered-call position's risk clearer. We of course don't have to go through these delta and gamma calculations to figure out the risk in a covered call position, but it's an uncomplicated position that most can understand and many have tackled, and the calculations here can be applied to more complicated positions.

The first point that MSFT drops, the loss is the position delta or $455. How did I figure that? The 500 shares of stock lost $500, of course. That's easy. During this one-point loss, the value of the sold call has theoretically dropped by its delta. That benefits the seller of the call, cushioning her loss in the stock's value by 45 deltas. The trader with the covered call position has theoretically lost $455.00. So, for the first point drop, the trader has lost the position delta that was calculated earlier.

The second point that MSFT drops, the stock position loses another $500, of course. What about the option? Since it's a call, it drops in value again for the second point, but this time when we figure out how much it drops, we have to consider how gamma impacts delta.

For each one-point drop in MSFT, gamma tells us that delta will change by 0.08. The negative gamma tells us that for each one-point change, delta reduces by 0.08 since the gamma is negative. Delta will then be 0.01 (original 0.9 - 0.08). Since there are 5 contracts, the option position loses another $5.00 (5 x 100 x $0.01). Since the holder of the covered-call position sold that call, it benefits the holder for the call to lose that $5.00, so that helps cushion the loss from the stock position. The total loss for that second point drop in MSFT is then $495.00.

Another way of thinking about it is that for the second one-point drop, the position loses the position delta we calculated earlier plus the gamma or $455.00 + $40.00.

The total loss for the two-point drop has been $455 + $495, so the position has lost $950.00. For those who prefer formulas, the total loss for a 2-point drop in MSFT = 2 x $455 (position delta) + $40.00 (gamma) = $950.00.

As should be obvious, the downside protection offered by the sold call is soon depleted as the delta of that sold call approaches zero. The call is moving far enough out of the money that the position is left long 500 deltas, which means it will lose $500 for each one-point drop in MSFT.

Other positions can get into even more trouble. Studying delta and gamma exposure can help traders determine their risk from price movement. We didn't need all the complicated calculations to tell us that the covered-call position wouldn't provide much of a cushion if MSFT should drop severely, but in other cases, performing such calculations can be eye opening.

Some traders attempt to neutralize the risk from price movement. In some cases, they attempt to do so by establishing delta-neutral positions. Remember the anecdote that started this series of articles, when the former market maker talked about the time he had shorted 25,000 shares of stock to neutralize the 25,000 deltas he was long when he bought 500 contracts of a call with a delta of 0.50? What would happen as the stock moved? The fact that the delta was 0.50 tells us that the option was an at-the-money one. Gamma is largest for at-the-money positions. That means that delta will be changing most rapidly as the position moves away from the at-the-money position.

The position that was delta-neutral will soon be anything but. If the market maker hadn't already closed the position, adjustments would be needed. Depending on which way the stock price moved, the market maker would be required to either short more stock or buy-to-cover some he already owned.

Some experienced traders go further and establish gamma/delta neutral positions. Such positions can be complicated and varied, of course, but are always started by first neutralizing the gamma and then the delta. For example, McMillan details a position that included hypothetical OCT 60 calls with a gamma of 0.05 and OCT 70 calls with a gamma of 0.025. To neutralize gamma, two OCT 70 calls would be sold for each OCT 60 call purchased.

With gammas neutralized, the trader can then neutralize deltas. The deltas of the two options are 0.60 for the OCT 60 call and 0.25 for the OCT 70 call. McMillan set up the comparison to show 200 OCT 70's have been sold and 100 OCT 60 purchased. The delta of this position would be +1000 (100 x 0.60 x 100 + 200 x -0.25 x 100 = 1000). The holder of this position would then need to short 1000 shares of this stock to neutralize the delta. This would not impact gamma since the gamma of a stock position is always zero. The position would be neutralized with respect to both delta and gamma.

Of course, gamma would change as the position moved and an adjustment would be needed, but for small movements, the position is neutralized with respect to price movements. Such positions may not be neutralized against other risks, however. That position might either benefit from or suffer from a change in volatility or the passage of time, for example.

An Internet search for gamma/delta neutral positions turns up potential trades that neutralize gamma and delta with the intention of benefiting from either time decay or a change in volatility. Many, as in the example from McMillan's book, require the selling of naked options, sometimes establishing some kind of ratio spread. Being somewhat squeamish about naked short options positions, I have not personally attempted such positions, and perhaps some of you won't, either. However, that doesn't mean that we can't all benefit from understanding what position delta and gamma tell us about the risk we're incurring from price movement.

Market makers seek to neutralize as much risk as possible in their positions. Next week, we'll hear from a former market maker who talks about risk as well as answering other questions about what market makers do.
 

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