Part 2: A Zero-Sum Game
In any single trade, the option buyer and seller are fundamentally at odds. Both types of player (referred to as options long and options short) make their money in opposite ways, and at the expense of the other. The long players expect to make more money scalping Gamma than they lose in Theta over the option’s life. The Short players bet that the Theta they collect will outweigh the market movement and the negative Gamma they incur, most poetically described as “wishing for death”.
To understand how and why markets sometimes get “managed” at expiration it would make sense to first understand the Option Greeks. This combined with who the players actually are, and understanding the regulatory inconsistencies will tell the full tale on why markets are ripe for manipulation near options expiration.
Keeping Score
Managing Options risk is a complex task. We are going to focus here on only three of the “Greeks” used to quantify and manage risk, Delta, Gamma, and Theta. These are the most important ones affecting an option trader’s behavior as expiration approaches and the market is hovering near a strike. We’ll attempt to explain them plainly and simply through examples. For these explanations we must assume that all other Greek parameters: like volatility, rho, etc remain static to better isolate the effects of delta, gamma, and theta on risk.
Delta
In physics Delta means rate of change. In calculus Delta is the tangent of the trajectory. But Delta actually has 3 definitions in the practical trading world. These definitions largely overlap but are not necessarily the same for the whole life of the option.
1. Correlation with the underlying: a Call has a 20 delta. The model generating that delta assumes the Call’s value will change by 20% of what the underlying changes. E.g. Crude Oil goes up by $1.00. The Call will go up by $0.20 assuming other Greeks remain the same.
2. Hedge Ratio: The long 20 delta call would be directionally neutralized if it had a hedge of short 0.20 futures per long options contract. E.g. I am long 100 Crude Oil calls with a 20 delta. I will sell 20 futures to hedge myself directionally. Therefore I will (theoretically) neither make nor lose money in either direction due to underlying movement. I am directionally “flat”
3. Probability of Expiring in-the-money: according to the model, said 20 delta call has a 20% chance of expiring in-the-money. e.g. an option with 30 days to expiry at this volatility has an implied probability of a 20% chance of expiring in-the-money.[1]
Gamma:
Gamma is the second derivative of the option. In physics, it is the rate-of-change of the rate-of-change. In calc it is the tangent of the velocity. For our purposes it is simply how much a delta itself will change (Correlation, Hedge ratio, or Probability), given a change in the underlying price.
Using our Crude Oil 20 delta call option again: Crude rallies from $90.00 to $91.00. In our example, the option has a 20 delta and its correlation/hedge ratio/probability all point to a change in the option’s value of $0.20. But that cannot be entirely correct if one measures the option’s value at the end of the $1.00 move in crude.
Because the market has moved higher, the option has an increased probability of going in the money. Therefore its Correlation, Hedge Ratio and In-The-Money Expiration Probability must increase. In our example, we use our model to re-calculate the delta of the call and find that its delta has gone from 20 to 25. This difference of 5 deltas over a $1.00 move is its Gamma.
Therefore we now have the ability to sell 5 more futures against our 100 calls if we wish to rebalance our directional risk. We get to “Sell High”. And if the market drops back down to $90.00, the option’s delta will once again become 20. We will get to “Buy Low”. Such is the virtue of being long Gamma. The ability to sell when something goes up, and buy it back when it comes down. Provided of course your model is right, and as we’ve said multiple times other Greeks don’t change. Gamma however comes with a cost called Theta.
Theta
The rate at which an out-of-the-money option loses its value over time is Theta. In short, it is the rate at which your long lottery ticket wastes away. As time goes to zero, your out-of-the-money option’s chances of expiring in the money go to zero as well. It is not unlike having tickets to an event that you wish to sell. If interest is tepid in the event (Jethro Tull : Bore ‘em at the Forum) and you can’t get face value for them from someone, you are said to be out-of-the-money. You will lower your price as we get closer to the event itself in the hopes of unloading them. That is an imperfect example of Theta.
Using our 20 delta call again: if it has a Theta of .05. That means it will lose 5 cents of value per day from the march of time, again assuming all those other Greeks we are not talking about remain the same. So as a holder of that Crude Oil call with a 20 delta, you are in a race against time. If you cannot make more than 5 cents per day from delta readjustments (aka Gamma) after the underlying moves, you will be a net loser of money. Put another way, you must “scalp your Gamma” to profit by 5 cents daily just to break even on your option investment. More than 5 cents and you profit, less than that and you lose.
Options Yin and Yang
Gamma and Theta are opposite sides of the same coin. These risks and how they are managed by opposing counterparties, combined with the asymmetric setup in the system are the key to the reasons for why so many option expirations get “pinned” at a strike with large open interest. And also why rarely but more sensationally, markets blow through strikes with big open interest.