sharper focus still by asking which remaining market, stocks or futures, offers us a better deal in terms of the frequencyof and limits on occasional event risk. The logician’s adage is “anything notprovably impossible is possible,” and this applies fully here. Any outrageous event you care to name will from time to time distort the markets instocks as well as those in grains, meats, energy, interest rates, metals, andcurrencies. The occurrence of this form of event risk is probably equal overthe long term in these two broad markets, but the amount of risk generatedby any specific such event differs markedly between them.Pick up a financial newspaper any day you like. Far more likely than not,you’ll find some number of companies whose share prices have risen orfallen 15% or 25% or even 50% in a single day, due no doubt to news eventsor rumors about the company. Now, sometimes these events are avoidable.When definitive regulatory decisions are announced, for example an FDAdecision to approve or reject a new drug, it’s not uncommon to see a company’s share price change by 60% or 75% or more in a day. Fortunately, thedates of most American regulatory decisions are announced well in advance,
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TRADING OPTIONS TO WIN
and we can dodge this sort of risk, as mentioned previously, by just stayingaway from the affected company’s shares at the indicated time. However,there’s not a single thing the trader can do to mitigate a loss caused by surprise events affecting a company, and these occur far too frequently, withtoo much variety, and in far too many companies’ shares for my tradingtastes. Just repeat “Arthur Andersen” to yourself 30 or 40 times, and if thispoint is not already quite clear, it will become so.By contrast, the very nature of futures markets strongly inhibits thesesevere price shocks over a short period of time, barring the introduction ofnew supply/demand information (which, again, we can avoid by staying outof markets when such new information is known to be on its way). A 10%daily move is gigantic, and quite rare, in futures markets. Any futures marketis by definition a market dealing in a standardized batch of goods, physicalor financial, readily available in large amounts, and of specific quality, quantity, and/or term. These markets are therefore structurally much simplerthan stock markets, thereby holding fewer undefined and opaque risks andsubstantially lower potential for external supply/demand surprises.How so, you ask? Put bluntly, the Kansas City wheat market is not goingto launch a surprise hostile takeover of the Chicago wheat market, ever. TheCenters for Disease Control and Prevention are not going to issue a healthwarning about copper or 30-year bonds, ever. Nor is Indiana Jones ever goingto discover the Lost Warehouse of Cocoa with ten million 62.5-kilo bags sitting inside. Yet any of these types of events, and numerous others, can anddo affect share prices almost every day of the year.The comparative impact of occasional external event risk between stockmarkets and futures markets changes even more when we consider the frequency of event risk stemming from the vagaries of the American legal system. Companies sue companies, people sue companies, governments suecompanies (or indict them, witness again the astonishing Arthur Andersendebacle in 2002, with numerous others likely to follow down the same path).There is also exactly zero doubt that some people are working to allow away for monkeys, cats, trees, duckbill platypuses, and probably even bacteria to sue companies. We traders cannot have any realistic idea, a priori,how such suits will be decided, because the concept of objective law, sadly,is as extinct as the dodo. We suspect, and rightly so in most cases in my experience, that we’re likely to be popped in the chops when a company whoseshares we own gets hauled into court.The Philip Morris company has made cigarettes for a long time, but theseare nowadays suddenly a “defective” product, so says the legal “system,” andtherefore this company should pay billions in “damages.” Hilarious, considering that at least as far back as World War II era films, cigarettes wereroutinely referred to as “coffin nails.” More hilarious still, every cigarettepackage in the United States has had an official government warning printed
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43
right on it since 1968. We traders must admit the risk of such politicallydriven nonsense, and it must be clearly to our advantage to avoid the risksemanating from this curious and extortive legal “system.” As far as we can. . . sigh.Contrarily, no one sues silver or Eurodollars or azuki beans or naturalgas, not even the comically idiotic governor of California, who so pompouslyand corruptly bungled his self-created energy “crisis” in 2000–2001. Legal actions resulting from arguments over genetic modification of grains or fromoccasional paranoia involving supposed conspiracies (gold, crude oil, roundup the usual suspects . . . yawn) may affect futures prices. However, thesesorts of developments have had and will continue to have notably smaller effects on commodity prices per se, than on producers or users of commodityproducts. To be sure, legal flubdubbery will scorch a commodity producer oruser every so often, but producers and users are usually companies, and thisis not our problem unless we’re trading their shares.When we add in what might be called the human destruction factor, theanswer crystallizes as to which broad market is most advantageous (or leastnatively disadvantageous) to trade. Corrupt or incompetent managers havewrecked more companies than you and I can name, as have periodic suddenand negative shifts in public opinion about a company. No company is immunehere. Enron, famously wrecked in November 2001, had at one time the seventhlargest income stream of any American company, and its managers bear, atleast apparently to date, the bulk of the blame for its collapse. WorldCom, oncethe nation’s second-largest long-distance telephone service provider, saw itsshare price fall to pennies and was forced to declare bankruptcy in July 2002when it was discovered that its managers had conducted a colossal and notvery subtle fraud on the shareholders and the public.Between us, we’ve read or heard of hundreds of such trainwrecks. Whilefutures markets undoubtedly have some corrupt major players, and publicopinion is at least as volatile regarding commodity markets, crude oil andgasoline being obvious examples, we will never see any futures market tradeat the equivalent of penny-stock levels. The sudden price moves seen whentraders stampede in futures markets, in either direction, are ultimately restricted because the goods underlying a futures contract have both a demonstrably positive and also demonstrably limited economic value. The only limiton the price of a common stock is zero.
What We Want, We Have . . . And What We HaveIs the FutureSuccessful trading is a matter of understanding and dealing effectively withrisk, managing capital, and selecting favorable markets in which to trade.
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TRADING OPTIONS TO WIN
Risk? When we’re trading to win, we must attempt to minimize it in almostall cases. We’ve examined several risks we must take into account whentrading, and several methods of avoiding or neutralizing or limiting them.We have much more to do here, though. Capital? We must attempt to use itefficiently, and to this end some leverage is essential, but leverage comescomplete with the caveat that its careless use will—not can, but will—sinkour trading ship without a trace. Markets? We want the ability to enter longand short positions freely, the ability to adjust our leverage according to theopportunity at hand, and, without any question, the ability to deal in thebroad market having the lowest available levels of both structural and occasional external event risk. We also insist on being able to take positive action, through the existence of an options market, to protect our capitalwhen we perceive that such action is desirable.Investing in shares over the long term, particularly in U.S. shares, hasbeen an enormously profitable proposition, but trading in shares over theshorter term is simply not the best we can do. Futures markets come muchcloser to offering the full set of conditions we desire for trading in a broadmarket, especially when regarding the essential notion of reduced eventrisk, and this broad market will be our principal focus henceforth. However, until now we’ve only looked at what amounts to a mug shot of risk onthe Post Office wall. Interesting, certainly, but hardly profitable. Let’s beginconsidering how to collect the reward.
CHAPTER 3
Profitability 101Expectation and Options
Statistics are like bikinis. What they reveal is suggestive,but what they conceal is vital.—Aaron Levenstein
’ve used the term expectation frequently here, and now is a good time todefine it properly and to examine how to enlist it firmly in our cause.Have a look at Figures 3.1 and 3.2, will you?These illustrate profit and loss when we simply purchase or sell anyasset to initiate a trading position. Figure 3.1 illustrates the profit and lossfor a purchase and Figure 3.2 for a short sale. In these figures, the horizontal axis measures the change in the price of the asset. The vertical axismeasures our profit and loss. The point labeled Pp is the price at which weprefer, before the fact, to enter the trade. The point labeled Pn is our netentry price, which differs from our preferred entry price by the amount ofour contest risk, the sum of commissions, exchange fees, whatever portionof the bid-ask spread we have to pay, and any slippage that occurs. The areaunder the line, but above the horizontal axis, represents our potentialprofitability, the area above the line and below the horizontal axis, our potential loss. The solid black wedges illustrate our contest risk.There’s something pretty ugly about these diagrams. Sadly, our potentialprofitability is smaller than our potential loss, and this is an extremely undesirable condition in any trade. This excess loss potential, our contest risk,is trading’s equivalent of the house advantage present in all casino games,and we know what happens to gamblers in casinos, don’t we? Our first question must thus be how we can eliminate this initial disadvantage, this initialnegative expectation of the outcome of our trade.
I
45
46
TRADING OPTIONS TO WIN
Profit
Profit
Pp
Pp
Price
Pn
Pn
Loss
Loss
FIGURE 3.1purchase.
Profit and loss for a
Price
FIGURE 3.2
Profit and loss for a
short sale.
Statistically, expectation is a completely straightforward concept. It’ssimply the probability of success multiplied by the gain realized when successful, minus the probability of failure multiplied by the loss realized whenunsuccessful, added up for all possible outcomes in a game or enterprise. Inactivities such as most dice games, expectation can be calculated very easily. In others, for example, bridge and backgammon, such a computation isconsiderably more difficult, but still usually available given a bit of effort.Sometimes, as in trading, expectation cannot be precisely calculated at alland we must instead settle for an approximation. If the result of our calculations turns out to be a positive number, we say we have a positive expectation, and vice versa.Traders use an amazing variety of methods in an attempt to eliminatethe initial negative expectation when buying and selling assets. Some tradersanalyze economic data to try to anticipate a market’s likely future movement and trust that their analytical skills will overcome the initial negativeexpectation represented by the contest risk. Some traders watch for one oranother pattern of price movement to appear on a chart, arguing that whena certain pattern appeared previously, a market subsequently moved insuch-and-so a fashion, and will likely do so again. Some watch the activitiesof the traders on the exchange floors or those who manage funds, with notions of riding the coattails of the big boys. Some look for demonstrable errors in the price of an asset and try to construct an arbitrage between themispriced asset and another one.If there are eight million stories in the Naked City, there are at least thatmany ways to attempt to gain an edge in the markets. There is nothing innately wrong with any of these methods, provided only that the trader canmake them work to his steady profit, but most methods share a common
Profitability 101
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weakness in my view. They tend to begin a trade with a known and measurable negative expectation and depend on future events to overcome this initial risk. Is there some rule etched in stone commanding that our only choicewhen we trade is to depend on future events for a profit? Not a bit of it. Wecan easily erase that nasty little wedge of contest risk when we want to, andthis is where options come onto our playing field.
Flexibility—A Powerful EdgeTrading in pure assets is not the most flexible of activities. When we buy anasset, we profit if the price rises and we lose if it falls, and the opposite result occurs when we sell short. There is little the pure asset trader can doto protect himself from loss during or in anticipation of an adverse move inthe asset’s price, aside from exiting the market. He might attempt to useother assets to obtain some insurance, paired stocks and intramarketspreads in futures being two common insuring tactics. These tactics are,overall, a tricky proposition because the “protective” asset may be subjectto its own adverse price move for reasons unrelated to the original asset,and the relation in the price of the two assets may itself change adverselyover time for a wide variety of reasons.From the standpoint of flexibility, the pure asset trader is similar to afootball quarterback whose playbook has only four plays: buy, sell, sellshort, and short cover. No matter how skillfully we may trade, having theflexibility to protect and enhance the expectation of our trading positionswhen and as we please must be nothing but advantageous to us. Optionsoffer us this flexibility. More usefully still, options offer us the opportunity,literally, to design trades with desirably high positive expectation.Suppose the price of some asset, a stock or a future, is 60. Suppose alsothat the asset has a regularly traded and reasonably liquid market in its options, and further that, through whatever type of market analysis we prefer,we believe this asset’s price is likely either to rise or to do nothing in thenear future. Having looked at all these conditions, we decide to buy thisasset. In the interest of eliminating our contest risk and moderately expanding our range of profitability, we might consider selling (“writing” is the usualterm) a 65-strike call option expiring at some time in the future. Supposethat we do write this option and receive, arbitrarily, a premium of 4 for it.Figure 3.3 shows the comparable profitability curves of this trade versus asimple asset purchase, on the date the option expires. The hypotheticaldollar profit/loss on the vertical scale is what would result if we would execute such a trade in the “C” coffee futures traded on the New York Boardof Trade (NYBOT). The straight diagonal line is the same as the line throughthe point Pn in Figure 3.1, the profitability curve of an asset purchase. The
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TRADING OPTIONS TO WIN
4000
3000
2000
Profit/Loss
1000
050
55
60
65
70
–1000
–2000
–3000
–4000
PriceFIGURE 3.3 Profitability curves for buying an asset and writing a call(flattened diagonal line) versus a simple asset purchase (straight diagonal line).
other line, which begins as a diagonal and then flattens out, is the profitability curve of buying an asset and writing a call option against it, as in thepresent discussion.
First Things First—Reduce the RiskThroughout the text, we’ll assume that we pay a one-way or half-turn commission of $15.00 for all our futures and options trading. For the sake of discussion, we’ll also assume we’ve paid 5 points of the bid-ask spread in thefutures contract and 10–20 points of the bid-ask spread when writing an option. Bid-ask spreads in futures options can be sizeable, because liquidity inthese options tends to be relatively low, so we are correct to assume a conservative, that is, high, bid-ask cost when trading these options. Just for fun,we’ll assume no slippage (lucky us) from our intended purchase and saleprices respectively.
Profitability 101
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The commission portion of our contest risk is just $30.00, so let’s compute the bid-ask portion. The New York “C” coffee contract involves 37,500pounds of Arabica coffee and is priced in cents per pound. Again, by assumption, our cost of the bid-ask spread on the future is 5 points. In the coffee market, there are 100 points to the penny, so the bid-ask cost on thefuture works out to $0.0005 × 37,500, or $18.75, and our bid-ask cost on theoption is four times that, or $75.00. Summing these figures, our total contestrisk is $123.75. Against this, we sold the call option for 4 cents: $0.04 ×37,500, or $1,500.00. Less our contest risk over the whole trade, this puts$1,376.25 net in our pocket, and the brokerage will require us to leave thissum in our trading account. However, these dollars earn interest (or if theydon’t right now, we can and should arrange matters so that they will), sothis requirement is not entirely unfavorable.At this point, after we’ve written the call option, our initial contest riskhas vanished. We have dollars in hand and are sitting in a trade we like atprices we wanted. Balancing this good news is our necessary acceptance ofa now limited profit potential. If the price rises above 65.00 prior to the expiration of the call option we wrote, we will still make the 5 cents profit,$1,875.00, less a $15.00 exit commission, $1,860.00, due to the price movement from 60 to 65. We’ll keep the net premium, $1,376.25, we received fromwriting the option. When the market has moved above 65.00 cents, the traderwho bought the option we wrote will, of course, exercise it, call our coffeecontract away from us at 65.00 cents, and claim all the subsequent profit ifthe price of coffee continues to rise. That’s perfectly all right with us, for wegranted the other trader this right when we wrote the option.By writing the call option, we have in reality made three trades: one inthe asset itself, a second in the call option, and a third involving risk. We’vetraded some amount of possible future profit for a direct reduction in therisk of the position. Whether the market moves in the fashion the othertrader prefers or in the fashion we prefer, we’ve laid off two chunks of risk.The contest risk is gone completely, and we can now also suffer some adverse movement in the price of the asset without incurring any loss of ourinitial capital. Put very baldly, for us to end up with a loss on this trade, theother market participants must now come and take money out of our pocketinstead of the usual case of us voluntarily paying them to enter the trade.This is a hugely useful advantage, especially when considered over a longtime frame involving a large number of trades.This hypothetical but perfectly plausible trade requires a little commentary regarding capacity risk and capital. How much capital have we applied to the trade? The exchange, at tonight’s close, will allow us to depositjust $1,806.00 plus our entry contest risk as our initial capital commitment,our initial “margin” . . . if we wish to. While we certainly could enter this coffee buy-write trade by having just $1,806.00 plus actual contest risk in our
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TRADING OPTIONS TO WIN
trading account, we would be thoroughly and irredeemably irresponsible todo so, for our capacity risk if we would do so is absolutely off the charts. Ifwe would employ just this minimum amount of capital, we would essentiallybe stating that our market view is correct, and it’s correct right this minute,and it won’t ever become incorrect over the term of the trade. This wouldbe pure insanity on our part, and some considerable number of times wouldlead to our getting absolutely pummeled by an adverse move in the marketprice. I won’t speak for you, but if I have to get beaten up, I’d much soonerstep into the ring against the heavyweight boxing champion. It’s much moreprofitable.Minimum margin is a sign of minimum intelligence, correct? Let’s demonstrate that we’re well above minimum intelligence simply by making a firmpolicy of committing a good deal more capital to each trade than the exchange may require. As a starting point for controlling the capacity risk present in this or any futures/options position, committing 300% of the requiredinitial margin is an excellent idea.Sometimes, and it’s well worth your time to ask your broker about this,your brokerage will agree to formally set your account’s margin requirementat some multiple of SPAN. Whether they’ll do so or not depends ultimatelyon their back-office computer systems. Brokerages have every positive incentive to do this, though. Offering such a disciplined auto-caution margining policy is good marketing, aids the trader in proper money management,and undoubtedly increases customer satisfaction, all for just pennies incost. In fact, the brokerage should actually turn a profit by offering this sortof margining policy because they’ll earn interest on the nominal excessamount of margin we will apply to every trade.Diversification of our trading positions may allow us to reduce this300% figure somewhat. If we should subsequently enter a number of otherpositions in different markets, we may very reasonably consider that all themarkets we’re trading are unlikely to move against our several positions inconcert on any particular day. Diversification, depending on our selectionof markets, might reduce our gross capacity risk sufficiently to allow us toreduce our capital commitment to perhaps two or two and a half times theinitial SPAN requirement. Whether well diversified or not, we cannot takeany reduction in our capacity risk for granted. Market factors, suddenchanges in supply or demand, weather, governmental action, or an emotionaltidal wave striking other participants in one of our markets may indicatethat we should prudently allot four or five times the SPAN requirement forour trade in that market, conceivably even more.The comfortable ratio of capital to margin requirement in futures markets isn’t easily definable. It varies with the markets we’re trading, with ourprior experience in these particular markets, and with our personal emotional makeup, but when it comes to capacity risk, erring on the side of cau-
Profitability 101
51
tion can never be wrong. Put another way, nature saw to it that your butt iswell padded because you’ll fall down on occasion and the padding will lessenthe pain. Keep your trading account well padded, too, for the very samereason.
Approximating ExpectationIn our hypothetical trade here, as in any other, we have to look at what willhappen if the market moves against us. We’ve partially insured against sucha move. Our dollars in hand from the start of the trade cushion us fromsome adverse movement in the underlying asset’s price, but there’s nothingat all to prevent our asset’s price moving sharply lower, much lower thanthe amount of insurance we hold. The question now becomes, “How muchinsurance is enough?” and the answer is: we don’t have sufficient information to deal with this question—yet.By definition, expectation is not only a matter of profit and loss, but includes the probability of occurrence of the profit and the loss. So, what isthe probability that our asset’s price moves from 60 to 65 in 60 days’ time,or from 60 to 40? Good question. It’s so good a question, in fact, that it andrelated questions baffled mathematicians for centuries. Finally, beginningin the 1950s, a number of exceptionally talented mathematicians and statisticians figured out a few methods of answering this question. The originalreasoning toward these early solutions proceeds approximately as follows.In any market, the next “tick” of the price of any asset is random. Mightbe up, might be down, might be neutral, and, other things being equal, if wetake a huge sample of a market’s ticks, we will likely find out that the countsof upticks and downticks will be close to equal at the end of the trials. Thisrandomness is intuitively appealing. The next tick on Ford Motor stockmight be due to the Ford Foundation offering a big block of stock andthereby taking out all the bids on the specialist’s book, thus pushing theprice lower. Or, a mutual fund might come in and bid for a million or moreshares, swallowing up whatever shares are offered at that time and bumping the price higher. We have no way of knowing which of these or otherevents will occur before any particular tick. We can therefore argue that thedirection of a single tick is random, a straight fifty-fifty proposition if wedisregard no-change ticks.What about a large series of ticks? What about the next 100,000 ticks in,say, gold futures? If the next single tick is random, then the result of a longseries of ticks should logically be some sort of sum of all possible outcomesof these random events. Indeed it is, but we must be satisfied with an approximate answer, a model, rather than an absolute chalkboard sum. Oneof the most frequently used models for the summing up of large numbers of
52
TRADING OPTIONS TO WIN
demonstrably random events is the Gaussian or normal distribution, a.k.a.the famous bell curve. There’s a problem with using the normal distributionfor our model in trading, though.The normal distribution curve is symmetric and extends infinitely inboth directions, as illustrated in Figure 3.4. But, in the world of trading, itrepresents a distribution of linear price changes, for example, +7 or –5points. This representation is useful, but not quite what we want. We are interested in the ratio of price changes. By way of example, consider an assethaving a price of 100. A 10-point move upward and a 10-point move downward are not equal here, for modeling purposes. These are linear price moves(or additive, if you prefer), and that’s fine, but we must compare apples onlywith apples. When dealing in ratios, we must apply multiplication, not addition, to the upside and division, not subtraction, to the downside. A 10%upward price move in our asset gives us the proper ratio of change, 100 times1.10, or 110, thus a 10-point gain. The price at the end of an equal downwardmove, in the model, is 100 divided by 1.10, or 90.909, a loss of only 9.0909points.Cutting through the mathematical implications of all this, the normaldistribution curve isn’t the one we want; it is suited to linear changes, and10 points is 10 points is 10 points as far as the normal curve is concerned.What we require is its first cousin, the lognormal distribution curve, whichreflects the ratios of price change that we need. It is not a symmetric curve,and rightly so, since we surely agree that 10 is not equal to 9.0909. To further illustrate that we’re on the right track, the left-hand side of the lognormal curve terminates at zero. This is obviously reasonable, since asset pricescannot trade below zero (although we’ve likely enough occasionally ownedshares that should have traded below zero; someone should have paid us to
FIGURE 3.4
A normal distribution curve.
53
Profitability 101
own these dubious assets).The lognormal distribution, as illustrated in Figure 3.5, and another statistical relative of the normal distribution called the binomial distribution,are the bases for a sizeable number of risk and pricing models. Good. Wenow have at least a theoretically sensible way to begin our attempt to measure the probability of the price of an asset moving from A to B over astated period of time.In Figure 3.5, P0 marks the current price of an asset. Pf is some possibleprice in the future about which we’re curious. Now, the probability of anyevent ranges between zero (it’s impossible) and one (it’s a certainty), so thewhole area under the curve is equal to 1, for purposes of probability. Theprobability that the asset’s price will be at or below price Pf at some time inthe future is represented by the shaded area, taken as a decimal fraction ofone. The size of the shaded area is easily computed by using the cumulativenormal density function, which formula can be found in any statistical textbook or in the many fine books dealing with the mathematics of trading.No matter what pricing model we choose, the purpose of the model isto make the calculation of this probability available to us, and the lognormal model certainly accomplishes this end. Even if we don’t want to takethe trouble to crunch a bunch of numbers in order to calculate this probability ourselves, there are lots of ways to obtain it, whether from commercial services, numerous Internet sites, our brokerage, or our fellow traders.We’re not out of the modeling woods just yet. We’re looking at the lognormal curve as a way to represent the distribution of future prices of an
P0FIGURE 3.5
Pf
A lognormal distribution curve.
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TRADING OPTIONS TO WIN
asset after a very large number of ticks, perhaps 100,000 or more. That’s allvery well, but in the real world we aren’t likely at all to see, let alone hold atrade through, a hundred thousand ticks in any market except, rarely, thevery most active ones. The COMEX/NYMEX market in gold futures, for example, only sees several hundred ticks on a normal day. We must ask ourselves, then, whether it is valid to assume that any model, and particularlyin our case the lognormal distribution, realistically represents the probabilityof price movement over only the next 5,000 or 10,000 ticks? After thesesmaller sets of ticks, the distribution of the real-world prices will differ bysome amount from those indicated by the lognormal model, not a doubt inthe world, but the lognormal curve will still be a reasonable approximationof market reality, and so is worth our attention.
Applying a ModelWhile this is an interesting topic, what we really want to know is how togain an advantage by applying this (or any) model of potential future pricedistribution to our trading. Taking the line of least resistance, let’s justcombine a profitability curve with the lognormal price distribution curve,as in Tables 3.1 and 3.2, and see what we might discover. These tables arederived from real-world prices in the New York coffee market on June 17,2002. December coffee futures settled at 52.10 and the December coffee 55strike call option at 3.10 cents, worth therefore $1,162.50 before considering contest risk.These are tables describing large portions of two expectation curves. Ifwe graphed them, they would be pictorial representations of the definitionof expectation. Each line in the tables contains the profit or loss at pointson the curve (fourth column) multiplied by the probability (third column)that the indicated price will be reached. In theory, such tables would haveone line for each 0.01 change in price, but such detail is very difficult to display in a readable fashion. Please pardon the abbreviation. Completely unsurprisingly, Table 3.1 confirms that negative expectation is present in thesimple purchase or sale of an asset, and also illustrates the asymmetry ofthe lognormal curve. Table 3.2 shows the shift in the trade’s expectationwhen we write a mildly out-of-the-money call option against the purchase ofan asset. These tables are useful enough for indicating our approximate expectation in a couple of types of trades, but they hardly tell the whole story.Embedded within the numbers in the tables is a vital assumption.The values in expectation tables, whether we consider them proportionally or as absolutes, are not constant from market to market. The other wayto say this is that the price distribution curves, both real-world ones and modeled ones, vary significantly among different markets and in any single market over time. These variations are due to the fact that the volatility of any
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Profitability 101
TABLE 3.1
Expectation for a Purchase
ObjectPrice
ProbabilityBelow Object Price
Probabilityof Range
Profit/Loss atRange