ash and I bought puts as the market went into another swoon. Several upand downswings later, and with some lucky timing, my portfolio was soonback to its September 1 value.I say this not to brag (goodness knows there were many who sufferedin the sell-off), but to illustrate what a powerful weapon options are.Rather than just selling all my stocks and sitting out, nervous about missing out on the next big rally, I was able to keep most of my favorites andoffset their declines with put gains.Since options are so expensive in a downtrending market, it is important to buy and sell them in rhythm with the market. I’m not sure how oneobtains a feel for the rhythm of the market, except through experience. ButI will say that the market is way more predictable during a sell-off. For example, following a sharp break and a 50 percent rebound, it’s a pretty safebet to go short. You could stay short until the market either returns to theprevious low, or has a climactic sell-off day. If you can catch the Nasdaq in aclimactic sell-off day, that’s a beautiful opportunity to sell the puts and letyour stocks go back up unhedged (maybe even buy a few calls for a 24-hourplay), as there most certainly will be a smart rebound. When the rebound isﬁnished, you can think about shopping for puts again. And so it goes.I would not just buy puts and let them sit. This might work for some,but I prefer to time the market. I browse through a lot of stock chartsevery day. If I see a lot of stocks ready for another break, then I ﬁgure themarket is about to break. That’s when I buy puts. Then I sell my puts inthree to six days when it looks like the current sell-off has run its course.I’ll say more about timing the market, especially during a downturn, inChapter 8.
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TheoreticalandPracticalMatters
I
n this chapter we will discuss how options pricing models, volatilitymodels, and other essential tools of the trade are best applied in practice. We’ll see how the different kinds of options pricing models work,and what models were meant to be used with what kinds of options. We’llalso discuss the best ways of estimating volatility—the all-important andmost difﬁcult-to-obtain input to any options pricing model. I’ll talk aboutthe methods and rationale for using slightly different volatilities to compute the fair values of each option on a given underlying. We’ll also have afurther discussion of the greeks, and other practical matters.Rather than delving deeply into mathematics, I will hold the discussion to a practical level and use laymen’s terms. Don’t skip this chapter! Itcontains several important concepts. Options are math-intensive animals.While computer programs exist that can do the math for you, it behoovesthe options trader to know something about what is going on “under thehood,” and not just blindly follow the recommendations of an optionsanalysis program.
OPTIONS PRICING MODELSAn options pricing model is the foundation of an options trader’s work.Not only do fair option values help the trader decide which options to buyor sell, and for what price, the options pricing model is also essential forsimulating the future performance of contemplated positions.127
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We’ll look at the most popular models in use today, discuss their underlying principles, and give you some important information on howthey’re meant to be applied.In nature, random processes tend to distribute themselves in theshape of Gauss’s famous bell-shaped curve, where more probable eventscluster around the center of the curve and less likely events populate thesides. This so-called normal probability distribution seems to apply prettywell in the ﬁnancial markets as well. We ﬁnd that the future price of astock or commodity is predicted fairly well by a bell curve centered overtoday’s price. However, an adjustment is needed to account for the factthat price movements are not linear. An example will serve to illustrate theproblem. Suppose a stock goes up 5 points, then 5 more points, then 5more points. Was each successive 5-point increase equally likely? No. Theﬁnal 5-point increase was an easier achievement than the one preceding it,and that one was easier than the one preceding it. This is because eachsuccessive 5-point increase is a smaller percentage increase.Stated another way, in terms of probability, a stock can double just aseasily as it can go to half its price. For a stock at 100, this means it can goup 100 points or down 50 points with equal probability. The normal probability distribution does not express this. We must call upon a mathematicalfunction called a logarithm to solve the problem for us. It turns out, whenyou take the logarithm of various stock prices, then apply the normal distribution, the results ﬁt reality quite well (not perfectly, but pretty close;more on this later). In other words, make the horizontal axis beneath thebell curve be the logarithm of stock prices, and voilà! You have a curvethat works. This kind of distribution is called a lognormal distribution.The assumption that stocks and other ﬁnancial instruments follow alognormal probability distribution is a principle underlying all of the mostpopular options pricing models.
Numerical MethodsThe ﬁrst options pricing models used numerical methods. A numericalmethod is an approach where a process is repeated many times and the results tabulated. For example, to prove that the odds of a coin coming upheads is 50 percent, a numerical method would be to ﬂip a coin manytimes and tally the results. The more iterations, the greater the accuracy.In options pricing, one numerical method, called the Monte Carlo simulation, involves randomly generating future prices for the underlying in away that simulates a lognormal distribution, and where the projected timeframe is the option’s expiration date. For each of these potential prices, wecompute and store the intrinsic value of the option (at expiration). Wethen accumulate these into an average. A sufﬁciently accurate option fair
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value will be obtained after a sufﬁciently large number of samples havebeen taken and tallied.Another numerical approach is called the binomial model. This approach is based on the idea that, from a given point in time, a stock pricemay go to two different new price levels with equal probability. From eachof those prices, the stock price may go to different new prices, and so on,eventually branching out into a large number of possible prices. To showhow this works to simulate the normal probability distribution, I’ll ask thereader to remember from math class a triangular arrangement of pegs (seeFigure 6.1.)If you recall, balls dropped into the top would hit the pegs, trickledown, and collect in columns at the bottom. Under ideal conditions, thedistribution of balls would approximate a normal distribution, especiallywhen a large number of balls are used in a large layout.The binomial model uses the same principle, except that we ﬁddlewith the probabilities of the left and right branching—making it slightlymore likely that the stock price will go higher than lower. This is to simulate a lognormal distribution. Other factors affecting an option’s fair value,such as the potential for early exercise and the way a stock’s price drops at
FIGURE 6.1 Triangular Arrangement of Pegs
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ex-dividend dates, can also be taken into account during the process. Consequently, the binomial model is extremely accurate at high resolutions. Italso works faster than a Monte Carlo simulation.The principle drawback of numerical methods is that they require alarge number of iterations, causing them to be computer intensive. In theearly days, this was a big problem. I remember it taking 10 seconds ormore to compute the fair value of an option in 1982, using one of the veryﬁrst personal computers and the binomial pricing model. That means itmust have taken a minute or more to compute an option’s fair value backin 1973 when listed options ﬁrst began to trade!A minute doesn’t sound too bad if all you need is the fair value of oneoption, but when there are thousands of options, it becomes impossibleto recompute their values fast enough in response to changing conditions during the trading day. That’s why the Black-Scholes model wassuch a godsend.
The Black-Scholes ModelMathematician Fischer Black and economist Myron Scholes, both teaching at the University of Chicago in 1973, developed the ﬁrst closed-form solution to the options-pricing problem. A closed-form solution is simply anequation—a formula. For the ﬁrst time, an option’s fair value could becomputed in a single pass; no iterations. While not a simple formula, theBlack-Scholes formula made it much easier to compute the fair value of anoption, transforming options pricing in the very same year that the optionsexchange opened in Chicago. Thus it played a big part in allowing the derivatives market to ﬂourish.I promised not to delve into mathematics. Nevertheless, I think youshould see what the Black-Scholes formula looks like:C = SN(X1) – Ke–RTN(X 2)where C = the fair value of a call optionS = the current price of the stockK = the strike price of the optionR = the short-term rate of interestT = the duration of the option
(
)
 SLn   + R + .5V 2 T KX1 =VvTX2 = X1 – VvT
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N(X) = the value of the cumulative normal density functionV2 = the variance rate of return on the stockV = the square root of the variance (i.e., volatility)Ln = the natural logarithm functione = the exponential functionSeveral assumptions underlie the Black-Scholes formula:• The markets for options, bonds, and stocks are frictionless.• The risk-free rate is constant over the life of the option and equal to Rper unit of time.• All investors agree that stock prices follow a stochastic process (ref.Section A-4a of Option Pricing, by Jarrow and Rudd1).• The underlying security pays no dividends.I wanted you to see that the Black-Scholes formula is not as complicated as you might have thought, especially for a formula that was so incredibly important, and timely, to the ﬁnancial markets.Since the pure Black-Scholes model does not take into account stockdividends, it needs to be modiﬁed when valuing options on dividend-payingstocks. Adjusting the Black-Scholes model for dividends is accomplishedby subtracting from S in the formula the present value of dividends incurred during the life of the option.The Black-Scholes model applies to European-style options, not toAmerican-style options. However, during the years subsequent to 1973,adaptations were developed. Most notably, the Roll model is an early-exercise adjusted model for American-style call options that is based on the factthat, for call options, early-exercise is never economically feasible otherthan right before a dividend. So the Roll model takes the maximum of fairvalues computed at each ex-dividend date using the Black-Scholes model.Puts were a bigger problem, however. It can be economically feasible to exercise a sufﬁciently deep in-the-money option at any time. Theproblem of valuing American-style puts was so challenging that therewas no closed-form solution until my own work in 1984. The Yates modelstarts with a dividend-adjusted Black-Scholes calculation, and adds extravalue because of the early-exercise privilege. I call this extra value theearly-exercise premium. You see, the value of a European-style put goesbelow parity when the option is more than a certain amount in-themoney (see Figure 6.2). For instance, a 10-point in-the-money put optionmight be fairly valued at 9. This happens because an equivalent strategy1
Robert Jarrow and Andrew Rudd, Option Pricing (Homewood: Dow Jones Irwin,1983), Section 8-2.
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FIGURE 6.2 Values of European- and American-style Put Options
to owning a deep in-the-money put is shorting the stock, and professional traders ﬁnd it more economical to short the stock than to buy aput that is valued at parity. Discounting the option brings it into equilibrium. Only then is it equally attractive. Besides market forces, the BlackScholes formula itself will show you that the fair value of a deepin-the-money put is less than its intrinsic value.The effect of the Yates adjustment is equivalent to bending the European-style put valuation curve upward at the left (the in-the-money segment) just enough to lift it above the parity line. Most of the bendingoccurs in the near-the-money area of the curve, with proportionately lessbending as you go farther in or out of the money. The result is an appropriate valuation curve for American-style puts. Coefﬁcients of the modelwere optimized to produce minimal error as compared with the binomialmodel at a high resolution, using a wide variety of historical data. Thus theYates model was fashioned by an engineer driven by necessity, rather thanby a theoretician. It was a pragmatic solution—not unlike when a farmerfaces a challenge and decides it’s time to weld together some parts andcreate a new tool to make his life easier.
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Practical UsesQuick and easy option fair value calculators are available on many ﬁnancialweb sites. Unfortunately, almost all of them indiscriminately use an unadjusted Black-Scholes model. There is no distinguishing of American versusEuropean options, or accounting for dividends. Let the user beware!The correct options pricing model to use for options on futures isthe Black model, developed by the same Fischer Black who helped develop the Black-Scholes model. The Black model is similar to the BlackScholes model, only simpler. The difference is an important one, yetsometimes I see the Black-Scholes model being mistakenly used for options on futures.Besides these problems, there is usually an even bigger difﬁculty withmost fair value calculators. One of the inputs is bound to be volatility, andusually the person has no idea what to ﬁll in there. The slightest differencein volatility can make a difference in the option value, so you’d better use agood number. If you’re guessing at volatility, who cares what model you’reusing? You might as well just try and guess the fair value as well!
COMING UP WITH THE RIGHT VOLATILITYVolatility is a measure of the standard deviation of price changes. It isthe square root of variance. So one way of coming up with a volatility isto apply standard variance formulas from elementary statistics to a recent history of prices. We call volatility measured this way statisticalvolatility (SV).The simplest of these formulas uses only daily closing prices. Other,more powerful formulas make use of other valuable information contained in the daily open, high, and low prices. One difﬁculty is determininghow many days of recent history to include. Usually an arbitrary time period is selected, as the rationale for deciding on a long or a short time period is conﬂicting. While using more data should lead to a better estimate,unfortunately, it is a better estimate for that time period only and it maynot reﬂect the present volatility as well as an estimate based only on themore recent data.So we often choose to use a volatility measured from the underlying’srecent price movements. Of course, ideally, what we really need to feedinto the option fair value calculator is future volatility—how volatile theunderlying will be during the remaining life of the option. However, no onecan know this, so the best we can do is refer to recent volatility measurements and estimate the future volatility.The other way of measuring volatility uses the options themselves.
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Each option, by virtue of its very price, is making a statement about howvolatile its underlying is. Volatility measured this way is called impliedvolatility. The higher the option’s price (all else being equal), the higher theimplied volatility (IV). To understand how IV is computed, you can think ofIV as the options pricing model being worked backward to determinevolatility, normally one of the key inputs to the formula.In truth, pricing models cannot be worked backward. IV has to be derived by working the pricing model forward repeatedly through trial anderror, experimenting intelligently with different volatilities until the theoretical option price converges with the actual market price.Using the implied volatility measured from a particular option to compute the fair value of the same option would be meaningless, as it wouldalways return a fair value equal to the current price. So in practice, we average together the IVs from many, if not all, of the options of a particularasset, often giving more weight to the options of greater trading volume.We also keep a brief, recent daily history of these average IVs and averagethose into one grand average IV.Now, we can feed this grand average IV into the option fair value calculator. Or would some kind of proportionate mix of SV and IV be appropriate? You see, coming up with a good volatility number to use for anoption’s fair value calculation is a challenge! I believe IV is better to usethan SV because IV embodies the collective expectation of all market participants as to the future volatility of this asset. To use IV is to hitch a rideon the shirt tails of all the others who have (presumably) struggled tocome up with an appropriate volatility to use.But let us say, after weighing the various methods, we have ﬁnally chosen our number. Unfortunately, there is a problem with applying a singlenumber, whether it be SV or the grand average IV, to all the options of aparticular asset. With many assets, the options are systematically valued atdifferent volatility levels, not only as you move up and down the strikes(volatility skew), but also as you move across the various expirationmonths (intermonth volatility differences). Since this can make signiﬁcantdifferences in options values, let’s take a closer look at how the optionspricing model can be adjusted to reﬂect this.
VOLATILITY SKEWWith many assets, the options of various strikes are priced using steadilyincreasing or decreasing volatilities as you move from strike to strike. Inmost index call options, for example, volatilities decrease as you movefrom the in-the-money (ITM) strikes toward the out-of-the-money (OTM)
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strikes. In many commodity call options, the reverse is true: Volatilities increase in the calls as you go farther out-of-the-money.The fact that volatility skew is systematic means that we can try todeal with it in our models. Fortunately, one ﬁnds that when the IVs of allthe options of a particular month and type (e.g., the March calls) are plotted, they form either a straight line or a smooth curve, as demonstrated inFigure 6.3. In this graph, the vertical axis represents IV and the horizontalaxis represents how far in- or out-of-the-money each option is in terms of aconvenient formula: KLn  SSqrt ( t )where S stands for the price of the underlying, K stands for the strikeprice, Ln is the natural logarithm function, and Sqrt(t) means the squareroot of time until expiration. Mapping IVs into this formula for the horizontal axis allows us to plot current IV readings in a framework for easy viewing and comparison of multiple skew curves.
FIGURE 6.3 Plotting IVs of One Month and Type
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Such a curve (or even a straight line, if that’s what it is) is easilymodeled using a parabolic formula. You may recall from high schoolmath class that the formula for a parabola isy = ax2 + bx + cIn this application, y would be volatility and x would be the degree to whicheach option is in- or out-of-the-money. An algorithm can determine the constants a, b, and c that cause the curve to ﬁt the data points most closely.Once this curve is ﬁtted, we can use the curve to give us an appropriate volatility number for any of the March OEX calls, not only at the current OEX price, but also at other projected OEX price levels.The curve also provides us with a context in which to judge the fairness of current option prices. For example, in the ﬁgure, notice that thesecond data point from the right lies above the line, while the third datapoint from the right lies beneath the line. Since each of these data pointsrepresents an option, this is saying that the one option is currently overvalued, while the other option is undervalued.By ﬁtting a different curve to each group of options (of the samemonth and type), this approach deals with the intermonth volatility differences and models the skew at the same time.
WHAT CAUSES VOLATILITY SKEW?There are at least two good reasons for the existence of volatility skew.The ﬁrst is CEV (constant elasticity of volatility). When we discussed CEVin Chapter 3, we noted that CEV is the name for the observed tendency ofvolatility to change when the price of the underlying changes. For example, with most stocks and stock indexes, as prices increase, volatility declines; when prices decline, volatility increases. This negative correlationwould be indicated with a negative CEV factor.CEV largely explains why index call options trade at lower IVs as you gofarther out-of-the-money. The options market is anticipating that if the market moves higher, volatility will decrease, working against the increase inout-of-the-money call option values. Thus it values those calls cheaper now.Likewise, puts, trading at higher IVs as you go out-of-the-money, are alreadypriced to anticipate higher volatility levels if the market were to fall.CEV is an important factor in other markets as well. For example, incommodities, where volatility spikes happen more often when the priceof the commodity goes up, the correlation is positive and the CEV factoris positive. This causes OTM call options to be valued at progressivelyhigher IVs.
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Note that CEV is systematic only in broad, general terms. Not all pricechanges are accompanied by a volatility change, and even when they are,the relative magnitudes of the resulting volatility changes can vary. Inpractice, however, we ﬁnd that it is appropriate to model CEV as an integral part of our options analysis. Also, we ﬁnd that a linear CEV model issufﬁcient. In other words, we can assume that volatility changes proportionately with price, at least for small to moderate price changes.
LeptokurtosisAnother source of volatility skew is leptokurtosis. This may sound like thename of a disease. However, it is a term for how the behavior of real pricesseems to deviate from the lognormal distribution. Studies have shown thatstock prices, as well as the prices of many other ﬁnancial instruments, follow a lognormal distribution with some degree of leptokurtosis2.One feature of leptokurtosis is that extreme price moves happen morefrequently than they should. Therefore, out at the extreme ends (the “tails”)of the bell curve, the line is lifted and extends farther out. The other featureis that the center of the curve is higher, indicating a greater propensity forprices to remain where they are. To balance out these increased probabilities, the leptokurtic curve is lower midway out on either side, just beyondthe left and right “shoulders” of the curve. (See Figure 6.4.)The options markets are more or less aware of leptokurtosis and
FIGURE 6.4 Leptokurtic Curve
2
Eugene Fama, “The Behavior of Stock Market Prices,” Journal of Business 38, 1(January 1965), pp. 34–105.
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value options accordingly. This accounts for much of the volatility skewthat we see.One wonders why more options pricing models based on leptokurtosis haven’t been developed and put into use. Perhaps it is because of difﬁculties modeling the leptokurtic distribution, or perhaps it is becausetraders had already become so accustomed to using the ﬁrst options pricing models—those based on a lognormal distribution. In any case, a standard options pricing model, together with a good volatility skew model,turns out to be an excellent practical solution. Note that an options pricingmodel based on leptokurtosis would not do away with the need to modelvolatility skew. Other causes of the skew, including CEV and other deviations from the lognormal distribution, such as a real or perceived priceﬂoor or ceiling, would still need to be taken into account.
Irregular Empirical DistributionsAs we have said, stocks and other asset types exhibit price changes according to an imperfect lognormal distribution. Besides leptokurtosis, Iexpect that many stocks and other assets have individual systematic deviations from the lognormal distribution. Someday I intend to develop thecapability of computing and displaying an asset’s empirical distribution—its historical distribution based on the past several years’ price changes.With such a tool we may ﬁnd, for example, that a particular stock has apropensity for sudden, radical price drops (along with, perhaps, gradualincreases). The empirical distribution for this stock would be lopsided. Itwould have a fat tail on the left. The options market might remember thisand build an appropriate premium into the put prices. On the other hand, itmight not. If not, this would present an opportunity, assuming the stock’spattern of crashing from time to time continues. If a sufﬁcient number ofstocks can be found that have lumpy or oddly shaped empirical distributions, it might be possible to set up a trading system that systematicallytakes advantage of these using various options strategies.
Intermonth Volatility DifferencesOne often sees nearby options trading at higher levels of implied volatilitythan the farther out options. What causes this?Remember that one of the underlying assumptions of the BlackScholes model is that markets are smooth and continuous. In real life, markets can, and often do, jump. Sometimes, in response to surprising news ora sudden order imbalance, the price of a stock can jump several pointsfrom one trade to the next, allowing investors no opportunity to make a decision or trade at any intermediate price levels. Indeed, this kind of jump of-
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ten happens from the close of trading one day to the opening of trading thenext, because signiﬁcant company news is often released during off hours.The effect of potential price jumps is to bolster the values of all options. However, nearby options are bolstered more because a price jumpwould affect them most. The farther out options have more time in whichto smooth out the effects of price jumps.
BALANCING FACTORSIn options, a lot of things balance out. The markets are more efﬁcient thanthey may seem at times. Apparent discrepancies or IV differentials are usually there for good reason. For example, the volatility skew in index calloptions would seem to work in favor of call buyers looking for a rally.However, the CEV effect will lower implied volatilities across the board asthe index advances, with the result that a call option might end up tradingat very nearly the same implied volatility after the rally. The same might betrue of buying index puts in expectation of a sell-off, except in reverse:Volatility skew would hurt but CEV would help.When the market is expecting a big change in the price of a stock, witheither direction just as probable, the nearby options will usually trade athigher implied volatility levels than the farther out options. This would, onthe surface, attract traders to put on calendar spreads—buying the fartherout options and selling the nearbys. However, big price action in the underlying would hurt calendar spreads. So making the nearby options moreexpensive is the market’s way of laying odds that calendar spreads won’tbe proﬁtable—again, balancing things out.As a ﬁnal example, covered writing and naked put selling are equivalent strategies in the sense that their performance diagrams are the sameshape—meaning their risk/reward characteristics are the same. Comparing the two strategies, put selling is better from the standpoints of fewertransactions and the fact that your collateral gets to earn interest. However, covered writers get to sell the slightly more expensive call options(stock call options are more expensive than stock put options, for reasonsgiven in the next section). Also, covered writers receive stock dividends.All factors considered, the two strategies are nearly identical.
PUT/CALL PARITYWith a potential news announcement that could cause a stock to jump inprice, one might think that the stock’s calls would reﬂect the excitement
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with extra premium, but not the puts. As it turns out, the puts will be moreexpensive as well.This happens not only because the news might somehow turn out tobe disappointing, but also because of a strategy used by professionalscalled a conversion. A conversion consists of buying the stock, selling acall, and buying a put. The call and the put are of the same strike price andexpiration date. You can think of a conversion as a covered write plus aprotective put purchase. Or you can think of it as long stock plus shortsynthetic stock.A conversion is theoretically a risk-free position. Its performancegraph is a ﬂat horizontal line. That is because no matter where the price ofthe stock goes, the outcome is the same, as the three elements of the conversion perfectly offset each other. Since the outcome is certain, the onlyway to make a proﬁt with a conversion is if you can put it on for a bargain.If a call is expensive and its corresponding put (same strike price andexpiration) is cheap, traders will buy conversions using that call and thatput. Their trading will cause downward pressure on the calls and upwardpressure on the puts, until both are driven to parity, at which point it is nolonger worthwhile to buy any more conversions.A similar thing happens if a put is temporarily more expensive than itscorresponding call. Traders will do reverse conversions—shorting stock,buying calls, and selling puts—again driving call and put prices back intoparity. Thus, calls and puts of the same strike price and expiration alwaystrade at close to the same IV.Again,