(This post is longer than usual, but those new to options should find it useful.)
OPTION RISK
The single most important measure of any group of data is the average or
mean. Adding up all the data and dividing by the number of data points gives you the average or mean. After finding the average or mean, the next important question is, How far do all the data points stray from this central point (the mean)? Measuring the amount that data varies from the mean gives us the second most important fact about a distribution. This is called
variance, or
deviation from the mean. Variance is an important statistical term, and it measures deviation from the center. There is another closely related formula that calculates another version of the variance called the
standard deviation.
The mean and variance are the two most critical facts about each and every normal distribution. In fact, if you know those two facts, you have properly identified the whole normal distribution. You don’t need to hunt around for other information because you have it all. The mean tells us where the center of the distribution is, and the variance tells us how wide the distribution is. Change the mean or the variance and you have an entirely different normal distribution. Mean and variance are the fingerprints of each normal distribution.
If you know the variance, then you also know the standard deviation (it’s the square root of the variance). So, knowing one means you know the other. You just have to translate by using the square root. This is very important since option formulas demand that we know the standard deviations of the stocks we want to analyze. In fact, the standard deviation has another name in option analysis:
volatility. Without exaggeration it is the most important part of every option model. It’s a measure of how far the stock price can be expected to move, based on experience. A pretty important measure if you are going to buy an option on that stock.
Volatility
Traders talk about three different categories of volatility: historical, implied, and future. We can best explain them by categorizing them into their time frames: past, present, and future.
Historical volatility is simply a measure of the past, a statistical analysis of what has already happened. To calculate it we take recent data, say 30 days worth, apply the formula that computes standard deviation, and then annualize it. Traders
always look at this data to get a feel and make up their minds as to what should be happening today and what might be happening tomorrow or next week.
Implied volatility is what the market is implying right this minute, the present. To calculate it, we take today’s call premium (
C) and apply the Black-Scholes formula backwards. If we include
C as an unknown, there are then six unknowns in the formula. If we know any five of these, we can readily compute the sixth. Normally we input the five we call SKIT-V* and receive
C (the call premium) as our answer. Alternatively, we can input SKIT-C and derive the value for V instead. This will tell us the volatility that the market is presently trading at. That is, the market is presently implying this volatility, which is why we call it implied volatility.
* SKIT-V is the five inputs required by the Black-Scholes formula:
- S = spot price of stock
- K = strike price of option
- I = interest rate
- T = time to expiry (in years)
- V = volatility of stock, defined as 1 SD of annual price moves
Future volatility is the volatility that will occur over the coming weeks and months. It is the volatility required by the Black-Scholes formula in order to give the correct call premium
C. To calculate it you need a crystal ball. Nobody knows how to calculate future volatility (or if they do, at least they are not sharing the knowledge). Anyone knowing a future volatility for certain could make some awesome trading coups. In fact, the best option traders are those who understand the ways of volatility movements. They exert their edge by being right about certain volatility patterns of the public. They aren’t even close to always being right, but they have a tradeable advantage that nets a profit at year end over many plays. No one knows the future for sure, but knowing a coin is biased 52 percent for heads can make a big difference—casinos have grown rich on similar edges.
The Holy Grail of option trading is the Black-Scholes formula and a knowledge of future volatility. If you have these two, you can become the world’s greatest option wizard. In the search for this Holy Grail many very clever analysts and traders massage the historical and implied volatility information trying desperately to get insight and a glimpse of the future and of future volatility. However, none can agree on the best way to predict future volatility. A general consensus of the research would be that the market’s implied volatility (the opinions of the smartest players) is as good as it gets. And it has been shown again and again that implied volatility is a far better predictor of the future than any simple historical volatility measure.
How to Measure Option Risk
Option risk is defined as any unhedged change in an option’s value. The change in option value is always caused by changes in the components. Change is the only thing that matters. If everything remains constant, then there is no risk, no exposure. But, of course, things do change. There are three key questions in our analysis: What risks show up when things change? What is their rate of change? How can we hedge this?
Let’s use a familiar example: driving your car down the road at 45 miles per hour. To change your speed you must either accelerate or decelerate. In the calculus of motion, speed is the first derivative while the acceleration is the second derivative. For traders, the only second derivative normally used is called
gamma. Traders say the option’s speed is its
delta and the acceleration is its gamma. Just as a car’s power is measured by its ability to accelerate, so too, is an option’s. Traders who don’t pay close attention to gamma are often in shock when they get whiplashed right out of their seats.
Delta is the single most discussed options tool. When people trade IBM options as a substitute for owning IBM stock, the delta puts a simple number on the relative speed of the two. For example, if the delta is 0.45, it means the option moves 45 percent as fast as IBM stock. Deltas always run between 0.00 and 1.00. Then the analysts took it a step further and wanted to know how fast the delta was changing. They used calculus to take the second derivative and named it gamma. You can see the relationships clearly in the following table.
The previous table is an oversimplified example of how deltas and gammas are computed. Notice the column titled Option Premium
C (in the Black-Scholes formula,
C stands for call premium). Each pair of premium values generates only one delta value. Now look at the Delta column. Each pair of deltas generates only one gamma value.
Notice that when the stock price is trading at $101 the option premium is $3.10. However, we expect this $3.10 to become $3.30 if the stock pops up a buck to $102. This $.20 change per $1 spot stock move is the delta. A delta of 0.20 means the option moves 20 percent as fast as the stock between those two points.
That’s about all there is to the way the average person understands and uses delta. Traders, however, spend all their waking hours trying to control and hedge their ever-changing delta exposures. The expected up move
or down move of any option premium should be
almost identical over a small stock price move and not as represented in our table. Also, then, the deltas and gammas should be changing more gradually than in our table. We wanted larger changes than reality usually gives for purposes of the example.
From $102 to $103 there is a bigger move (0.30) in option value. This change in delta, from 0.20 to 0.30, must be protected against. Note the gamma between those two deltas. It is 0.10 which represents the delta move we just identified. That’s why we need gamma, to point out the risk of potential changes in delta.
The measurements of all the other multidimensional types of option risk require similar analysis. Whether we wish to measure volatility risk* (vega =
∂C/∂V) time risk (theta =
∂C/∂T), or interest rate risk (rho =
∂C/∂I), we are always measuring the option premium change, which is
∂C, per small move of the component.
Delta and Gamma Hedging
So, we’ve introduced the standard measurements of option risk. Now, how do we eliminate the risk after we’ve identified and measured it? The answer is that, we must offset the risk with an appropriate hedging vehicle.
We define hedging vehicles as any and all tradable assets in the spot, forward, futures, or options markets. It is up to us to find the best asset to neutralize our risk. The hedging vehicle must offset the driving force of the risk you have. Clearly, all option risks are not driven solely by the spot price. Traders worry not just about S, but about the big three, STV. Hedges must be selected for their ability to offset T (time) and V (volatility) and the lesser risks (rhos), too. If we fail to hedge all the multidimensional risks, there is some market movement that will ambush us in the end.
Let’s start by talking about delta exposure. Imagine you have bought 1000 call options on IBM. This represents the right to call 100,000 shares of IBM stock. The option delta happens to be 0.15. Your exposure is calculated as delta underlying shares, which is, 0.15 x 100,000 = +15,000. This means your exposure is identical to owning 15,000 shares of IBM right now. Said another way, your daily P&L will move up and down in value as if you were the proud owner of 15,000 shares of IBM. To offset this risk you must sell short 15,000 shares of IBM against it. If you did this, your portfolio would show a position of long 1000 IBM options and short 15,000 shares IBM stock. For the time being your risk would be neutralized,
vis-à-vis IBM’s spot price.
This is called being
delta neutral. That’s how traders hedge their delta risk. They buy or sell IBM shares in spot, forward, or futures markets trying to stay flat. But as various SKIT-V components change, the delta of 0.15 will shift relentlessly up and down, say to 0.20, then to 0.17, and so forth. This makes the portfolio longer or shorter IBM without the traders doing any tinkering at all. If the traders run a new exposure report, it will indicate the equivalent shares of IBM that they are long or short as of that moment. To stay delta neutral the traders must sell excess long positions or buy back excess short positions. This is known as
dynamic hedging (also called rebalancing or rehedging). The traders hope everything stays quiet and they won’t have to rehedge very often. But the various components of SKIT-V never seem to keep still very long. For a fleeting second you think you understand exactly what must be hedged, but then minutes later the whole thing has slithered off into an entirely different risk profile.
Once we’ve neutralized the delta, what do we do about gamma? The purpose of gamma is to foretell potential moves in delta. That is, it helps you anticipate the increases and decreases in delta as IBM spot prices move. There are two trading approaches to gamma.
The first approach is simple and passive: You ignore the gamma and simply rebalance the delta hedge as needed. Any cost to your P&L is chalked up to the contrariness of options. The second approach is more aggressive: You try to neutralize the gamma to make it go away. But you cannot make gamma go away by buying or selling shares of IBM. This is a perfect example of an inappropriate hedge. You must hedge with other options to lower your exposure to gamma.
Here are a few quick hints about hedging gamma. At-themoney options have the most gamma. As you move away from being at-the-money, in either direction, the gamma falls toward zero. There is very little gamma to deep in-the-money or deep out-of-the-money options. So, at-the-money options are best hedged with other at-the-money options, and so on.
Typically, a portfolio that is long options has positive gamma risk. So you would need to sell other options to lower this risk. Conversely, a portfolio that is net short options tends to have negative gamma risk and you must buy other options to reduce the risk. Hedging gamma is a relatively advanced concept which we can’t do justice to here. Suffice it to say that only the more experienced option people are knowledgeable on the topic. The others avoid it completely using the first approach (benign neglect), but their ostrich-like tendencies do not make the risk disappear. They try to hedge their gamma risk by constantly readjusting their delta neutral positions and are surprised when they can’t ever keep up.
There is a cost for constantly rebalancing the hedge when you are gamma negative. You must buy some IBM every time it rallies and sell when the price goes lower. You are at risk in choppy markets. But you are being paid to render this service to the market, if you think about it. The reason you are gamma negative is that you have sold options to the marketplace. Therefore, you have already been paid the premiums for taking on this risk.
Think of it like this: you are wearing a new fashion creation for traders, the option hedging poncho. It’s one of those hooded pullovers with a single-pocket pouch in front to keep your hands warm (it’s continuous and goes all the way through). When you sell options, you immediately earn money and you stuff this cash into the pouch. But you must pull it out later in dribs and drabs as you gradually lose money from delta neutral hedging because you are always buying high and selling low. On average the sum of the dribs and drabs will exactly offset any money you earn from selling options—if you sell at fair prices. After thousands of cases, the poncho will be empty. Any interest earned on the money is included in this calculation.
The reverse is true if you are gamma positive. Because you start by buying options you must pay the money up front, so you will have to borrow it at first. You will regain this money in dribs and drabs as you sell high and buy low while delta neutral hedging. The sum of all the dribs and drabs you are able to stuff into the poncho will allow you to pay back your borrowings and any interest due. On average, after many cases, you will break even and the poncho will once again be empty.
This break-even philosophy requires, as the Black-Scholes formula does, that we can hedge without fees and bid/ask spreads, and so forth. In the real world, however, we will lose any such expenses when we try to dynamically delta hedge to stay neutral (flat).