c04 JWBK147-Smith May 8, 2008 9:48 Char Count= Advanced Option Price Movements 57 Typically, professional traders rebalance their positions whenever the UI moves a certain amount, or sometimes they do it every certain number of time periods. For example, you may want to rebalance the position every time the underlying moves $1 or at the end of every day, whichever comes first. The usually determining factor on the frequency of rebalancing is the transaction costs versus the rebalancing costs. As a result, floor traders can afford to rebalance more frequently than retail traders. NOT EQUIVALENTS Even though the expression is delta neutral, it is important to realize that no combination of long or short options is the equivalent of or a substitute for a position in the UI (except reversals or conversions; see Chapter 23). All the rebalancing and analysis and arbitrage-based pricing models in the world will not make them equal. If they were equal, t here would be no economic need for one of them. Instruments are relatively simple compared to options. With few ex- ceptions, the profit and loss from a UI is strictly related to the price move- ment. An option is subject to many more pressures before expiration, and the profit and loss are nonlinear. The current and future prices of an option are functions of several nonlinear forces. The trader of just UIs is only concerned with the price direction of the UI. An option trader, on the other hand, should take into account price direction, time, volatility, and even dividends and interest rates. As a result, the option strategy may be delta neutral, but the effects of gamma, vega, theta, and even rho may cause profits and losses that are not expected by the delta-neutral trader. The point is to keep monitoring the potential effects of other greeks before and during a trade. c04 JWBK147-Smith May 8, 2008 9:48 Char Count= c05 JWBK147-Smith April 25, 2008 8:42 Char Count= CHAPTER 5 Volatility VOLATILITY AND THE OPTIONS TRADER Volatility is important for the options trader. The expected volatility of the price of the underlying instrument (UI) is a major determinant of the price and value of an option. Some might not consider it important if they are going to hold the po- sition to expiration. They argue that the option will either be in-the-money or it will not. But it is still important for traders to consider volatility be- cause they might be overpaying for the option or miss an opportunity to buy an undervalued option. In addition, by understanding volatility, they might have insights into the potential for the option to expire in-the-money or out-of-the-money. Considering volatility is most important for traders who are not ex- pecting to hold their position to expiration, and it is absolutely critical for traders considering theoretical edge or trading volatility (see Chapter 4 for information on these ideas). One has to know what the implied volatility is before initiating one of these strategies. One has to have an opinion of the future volatility to successfully trade these strategies. It is possible for traders to ignore volatility in their options trading and still be successful, but it is more difficult. Trading options contains more dimensions than trading the UI. Volatility is perhaps the most important additional dimension. 59 c05 JWBK147-Smith April 25, 2008 8:42 Char Count= 60 WHY AND HOW OPTION PRICES MOVE WHAT IS VOLATILITY? Volatility is the width of the distribution of prices around a single point. Usually it is the distribution of past or expected future prices around the current price. Prices go up, and they go down. How far up and how far down is t he volatility of those prices. (Remember that volatility is always expressed as an annualized number, even when the volatility is measured over periods greater or lesser than a year; a formula for de-annualizing volatility is given later in this chapter.) Historical or actual volatility is the annualized volatility of UI prices over a particular period in the past. Were prices highly volatile and moved all over the place or were prices stable and moved within a narrow range? Are prices being checked over the past 10 days? Over the past 20 or 100 days? Or over some period in the past? For example, the annualized volatil- ity of the stock market may have been 10 percent over the past 20 days. Expected volatility that is expected by the option trader is the annu- alized volatility of the UI over some period in the future (usually to the expiration of the option). This is a simple projection or expectation. For example, you might think that the volatility of the stock market will be 20 percent over the next six weeks until expiration of the stock index options. Implied volatility is the volatility implied by the current options price. This can be found by plugging the current price of the option into the Black- Scholes formula (or whatever model is being used) and solving for volatil- ity. Usually, the value for volatility is plugged in and the formula is solved for the value of the option. Here, the situation is reversed—the formula is solved for volatility because the current price is known. BELL CURVES AND STANDARD DEVIATIONS The standard deviation of prices is a description of the distribution of price changes and a good approximation of actual volatility. The mean (commonly called the average) of the prices being examined is basically the middle of the distribution. In the option world, the standard devi- ation is annualized so that various volatilities can be compared on the same scale. Standard deviation is easier to understand with a diagram and a little more explanation. Figure 5.1 shows the closing prices for a particular in- strument, Widgets of America, for the past 60 days. You can see that the prices move around $50 during that period of time. c05 JWBK147-Smith April 25, 2008 8:42 Char Count= Volatility 61 60 50 40 30 20 10 0 1357911131517192123252729313335373941434547495153555759 Days Price FIGURE 5.1 Daily Prices Standard statistics can be used to calculate the mean and the standard deviation. The standard deviation is simply the statistical description of the variability around the mean. In this case, the mean is $49.47, and the standard deviation is $3.13. This shows that roughly two-thirds of prices will fall within $3.13 of the mean, $49.47. In other words, two-thirds of the time, prices can be expected to range between $46.34 and $52.60. Volatility in the option world is defined as this one standard deviation. A volatility of 20 percent says that the price will vary 20 percent around the mean 68 percent of the time on an annualized basis. The data for examining actual or historical volatility can be precise be- cause they are known. The actual mean and standard deviation can be cal- culated. The data for expected volatility must be assumptions: that the cur- rent price is the mean of the distribution and that prices will be distributed around this mean. It makes sense to assume that prices will be randomly distributed in the future around the current price (the truthfulness of the concept of random price action is discussed later in this chapter). However, the current price of the instrument should not be the actual current price but actually the forward price at expiration of the option. The carrying charges from now until expiration must be taken into account be- cause carrying charges will cause a drift in the current price to the forward price. This is necessary because the forward price is the economic equiva- lent of the current price carried forward to the expiration date. The forward price of the instrument is the price that has such carrying costs/benefits as dividends and interest payments built in. Fortunately, carrying charges are built into the Black-Scholes Model. Statistically, the first standard deviation of prices contains roughly 68 percent of all prices, two standard deviations contain nearly 95 percent c05 JWBK147-Smith April 25, 2008 8:42 Char Count= 62 WHY AND HOW OPTION PRICES MOVE of prices, and three deviations contain nearly 100 percent of prices. Just because the price of the UI eventually moves beyond the third standard deviation does not mean that the model or standard deviation was wrong. The standard deviation simply tells what the expectation for the future is as based on the past. This is usually good enough but might not be. Volatil- ity deals with probabilities, not certainties, so traders must make do with standard deviations and assume that occasionally the bizarre will happen. The standard deviation is based on a sample of t he total universe of possible prices and, therefore, is an ultimately inaccurate though reason- able estimate of the attributes of the whole universe. Still, it provides a good working guide because absolute accuracy is not necessary for trad- ing profits. The standard deviation can be wide or narrow. High volatility means wide distribution, which is illustrated by a wide bell curve. High volatility means that the chances are greater that prices significantly away from the mean will be hit. Low volatility means narrow distribution, which is illus- trated by a narrow bell curve. Low volatility means that the chances are less that far away prices will be hit. Figure 5.2 shows a wide distribution that would mean that prices are expected to cover a lot of territory. Figure 5.3 shows a prices series that is going nowhere. And, of course, Figure 5.4 shows a normal amount of range. One of the critical attributes of a normal distribution is that you can describe all normal distributions knowing only the mean and standard de- viation. This is obviously an important advantage for computational speed. high volatility distribution Exercise price Present price of UI FIGURE 5.2 High Volatility c05 JWBK147-Smith April 25, 2008 8:42 Char Count= Volatility 63 Present price of UI low volatility distribution Exercise price VOLATILITY FIGURE 5.3 Low Volatility Present price of UI normal volatility distribution Exercise price FIGURE 5.4 Normal Volatility c05 JWBK147-Smith April 25, 2008 8:42 Char Count= 64 WHY AND HOW OPTION PRICES MOVE However, the normal distribution is far too inaccurate for options pricing. A lognormal distribution is needed instead. PROBABILITY DISTRIBUTION Built into the options-pricing-model equation is an assumption of how the price of the UI will move in the future. The model does not predict the future price behavior but does assume what the probable distribution of those prices will be. It is critical to know what the possible future prices are for the UI. Absolute knowledge that the price of the UI will be at $55 at expiration would be invaluable. Knowing that, for instance, the probabili- ties are greater than 67 percent that the price will be between $45 and $65 would be valuable but not as valuable as knowing the exact closing price. Thus, the options traders will want to know what the potential future range is of the options that they are trading. This is based on the potential future range of the UI. The range of prices shown by the usual bell curve in Figure 5.4 is a normal distribution. One major feature of a normal distribution is that it is symmetrical. Each side of the curve is identical to the other side. The nor- mal distribution is wrong in the real world. There are two main reasons: First, it allows absurd situations such as negative prices. It seems rather reasonable to assume a 50 percent chance of prices going from 50 to either 49 or 51. But a normal distribution also assumes that the price could just as easily go from 1 to −9 as from 1 to 11. This is not in line with reality. Second, prices of financial instruments are not equally and randomly dis- tributed about the midpoint (the randomness of prices is discussed later). Instead, each instrument has a unique distribution pattern. For example, the price of stock index and stock options has an upward drift to it. Stock prices have moved erratically higher since stocks began trading under the buttonwood tree in Manhattan. Bond prices are mean reverting around par because the bond will mature at 100. Most other instruments, such as cur- rencies and futures, tend to have essentially symmetrical distributions. LOGNORMAL DISTRIBUTION Prices of financial instruments follow more closely what is called a lognor- mal distribution. A lognormal distribution does not consider the changes in the absolute points of the underlying distribution but rather the rates of return. For example, a normal distribution outlines the probability of price c05 JWBK147-Smith April 25, 2008 8:42 Char Count= Volatility 65 0123456789101112131415 lognormal distribution normal distribution FIGURE 5.5 Lognormal and Normal Distributions changes from 50 up to 51 or down to 49. A lognormal distribution, on the other hand, looks at this same price movement as a rate of return. It would instead say that there is an equal probability of prices climbing 10 percent from 50 as it is to drop 10 percent from 50. (The Black-Scholes Model uses a lognormal distribution, which is a fairly good assumption for most instru- ments except bonds.) Figure 5.5 shows the difference between a normal distribution and a lognormal distribution. But note that this is quite different from looking at absolute changes in price. It is looking at relative changes from the last price. For example, assume that prices drop 10 percent from 50. That would be 45. A lognor- mal distribution still assumes that prices have an equal chance of climbing or declining 10 percent. Assume that they rise 10 percent from the now current price of 45. The result would be 49.50. The distribution is now no longer symmetrical as far as absolute price changes are concerned, though it is symmetrical as far as percent changes are concerned. Note also that prices can never drop below zero. Subtracting 10 per- cent, for example, over and over again from any price will move the price closer and closer to zero but never cause the price to decline below zero. On the upper end, there is clearly no such boundary as zero. c05 JWBK147-Smith April 25, 2008 8:42 Char Count= 66 WHY AND HOW OPTION PRICES MOVE Another aspect of a lognormal distribution is that it means that high strike options will always be worth more than low strike options, even when they are equidistant from the price of the UI. This is due to the fact that the lognormal distribution allows for the price of the UI to go to great heights but to never go below zero. There are, therefore, greater chances of hitting a higher price than a lower price. This skew, or assymetry, means that the 55 call should have a greater theoretical value than the 45 put with the UI price at 50 and assuming all other factors are worthless. A lognormal distribution is a probable distribution of prices that is very reasonable. In addition, it is also very easy to describe on a computer, thus making it quick and easy to calculate. THE REALITY OF PRICE DISTRIBUTIONS It is important to realize that even the lognormal distribution does not cor- respond to reality. There are two main problems. The first problem is that empirical studies of actual prices show that price distributions tend to have more extreme prices and more prices clus- tered around the mean and fewer prices in the intermediate ranges. In ef- fect, the real-world distribution is higher near the center and on the ex- treme tails but lower in the midrange. The second problem is that prices are discontinuous. What this means is that prices jump around, sometimes leaving large gaps between one price and the next. A piece of news comes out and the prices of the instru- ments jump. The Black-Scholes Model, and most other models, assumes that prices are continuous. This means that prices flow logically one after the other. Prices will go from 56.50 to 56.51 without jumping up to 56.52. Of course, this is not true in the real world. There are price gaps, particularly during highly volatile times. The net result is that the models make assumptions about the real world that are not true. The question is: Does it matter? For most traders, the difference between the assumptions in the Black-Scholes Model and the real world is trivial. Typically, the transaction costs will be greater than the difference implied by the discrepancies in the Black-Scholes Model. The difference will be more important to professional traders and market makers. Much of their trading styles and, hence, profits comes from looking for small discrepancies between what they perceive to be the fair value of the option and the current price. They are very concerned with the concept of theoretical edge that was discussed in the previous chapter. Knowing the most accurate value of the option is critical to this type of trading. [...]... Count= Volatility 67 RANDOM PRICES The Black-Scholes Model and other models assume that prices are random within the constraints of the lognormal distribution Prices must be considered random for a model but might not be random in the real world Prices must be random or else the arbitrage condition inherent in most models will not hold However, prices must not be considered random, or you will never... to twist the expiration and strike prices to fit your outlook For example, a straddle is constructed by buying a put and a call with the same strike price That is the plain vanilla But you can change the strike prices by, say, buying an out-of-the-money put and an out-of-the-money call and create what is called a strangle Or why not buy the call for nearby expiration but the put for far expiration? The... profits and losses So being able to identify the correct valuation and the future direction of implied volatility is of crucial concern to an option trader c05 JWBK147-Smith April 25, 2008 8:42 Char Count= 69 Volatility There are several methods for predicting the future direction of implied volatility There are classic forecasting techniques, such as regression analysis, time series analysis, and even... volatility for IBM Jun Jul Aug Sep 10.0% Oct c05 JWBK147-Smith 70 April 25, 2008 8:42 Char Count= WHY AND HOW OPTION PRICES MOVE Let’s take a look at an example Figure 5.6 shows a chart of the implied and 30-day historical volatility for IBM Notice how both the implied and historical volatilities have moved within ranges until very recently Actually, the ranges had extended back more than a year before the... structural change in the underlying instrument before you will see a permanent significant change in the range of implied volatility For example, a major acquisition could permanently change the character of the company’s business and therefore change the range of implied and historical volatility c05 JWBK147-Smith 72 April 25, 2008 8:42 Char Count= WHY AND HOW OPTION PRICES MOVE There are times when... significant losses for the investor This strategy assumes that the edge that the investor has in stock selection is so superior that he can withstand a lot of headwinds caused by trading an option or options that have a lot of edges against him For example, what if the investor is buying a near dated call on U.S Widget? But what if the options is overvalued and there is little gamma and the time decay... is easy to see that prices are not random Academic tests of randomness set up straw men and then knock them down On the other hand, there is extensive evidence of seasonality of prices and of implied volatility Furthermore, bond prices are not random Bond prices eventually have to revert to par or 100 at maturity This means that prices are random when the bond is at 100 but will have a strong negative... bullish or bearish on the implied volatility and therefore on the options Figure 6.2 shows the current situation with implied and historical volatility for IBM I think that this situation would suggest that we should be on the short side of options We now have decided on the two most important factors affecting options prices We want to be long the stock and short volatility We can now start to construct... CHAPTER 6 Selecting a Strategy ptions allow the investor to sculpt the returns in their portfolio When you buy a stock and the price rises $1, you make $1 You lose $1 if the price declines $1 Your profits are linear and directly related to only the change in the price of the stock Interest and dividends will make a slight change to the outcome though these factors are also linear Options blow apart this... range when it is at the high end of its range and above the historical The wider the divergence between the implied and the historical the better r We are looking for implied volatility to climb to at least mid-range or even to the top of the range when it is at the low end of its range and below the historical The wider the divergence between the implied and the historical the better r We are neutral . 61 60 50 40 30 20 10 0 13579111315171 921 2 325 2 729 313335373941434547495153555759 Days Price FIGURE 5.1 Daily Prices Standard statistics can be used to calculate the mean and the standard deviation. The standard. JWBK147-Smith May 8, 20 08 9:48 Char Count= c05 JWBK147-Smith April 25 , 20 08 8: 42 Char Count= CHAPTER 5 Volatility VOLATILITY AND THE OPTIONS TRADER Volatility is important for the options trader above the historical volatility and bullish if it is significantly below the historical volatility. 27 .5% 25 .0% 22 .5% 20 .0% 17.5% 15.0% 12. 5% 10.0% OctSepAugJulJunMayAprMarFebJanDecNov 30D HV IV Index