using isnot symmetric; it has a measurable bias to the right-hand side, the “upside”in terms of markets. In plain language, the right-hand tail of the lognormalcurve is too fat, implying too high a probability that the market will reachany of the points way out on the curve within a specified time. Not shockingly, the lognormal curve also slightly understates the probability of a sizeable fall in price (hey, the fat tail had to be taken out of some other part ofthe curve, didn’t it?).The works of a French mathematician and an English botanist offer usa reasonable method for adjusting the computation of the probability of amarket attaining any particular price over a known period of time, presumably generating a probability more in line with reality than delta offers. Youmay have read, in chemistry or physics class when you were at school, ofthe phenomenon called “Brownian motion.” This effect originally was theresult of the observation of the total motion of tiny grains of pollen in waterover time. Robert Brown, evidently a remarkably persistent scientist, concluded after long study that this motion was random, and the distance thesegrains moved from their starting point varied with the square root of theamount of time elapsed during the observation.Some decades after Brown’s study, a student of the remarkable mathematician Jules-Henri Poincaré, a fellow named Louis Bachelier, wrote athesis demonstrating that the movements of stock prices on the Paris exchange bore considerable resemblance to the movements of these grains ofpollen. Poincaré, rather surprisingly in retrospect given his brilliance as amathematician, thought Bachelier’s thesis was not up to snuff. Poincaré wasundoubtedly his nation’s leading mathematician at the time; his opinion wasnot to be questioned. Bachelier’s thesis was ignored, and he subsequentlycould not even obtain a university professorship.Time heals many wounds. Half a century later, the Nobel-prize winningeconomist Paul Samuelson rediscovered Bachelier’s paper, and publicizedthe concepts within it, and the notion of random motion varying with the
Profitability 101
67
square root of time became, mirabile dictu, a topic at the forefront of economists’ discussions about the movement of market prices.It’s heartening to see a good piece of research being honored and applied, even if far too late to bring the researcher any benefit, but how do weturn this evidently useful discovery to our advantage? I’ve found it to bepractical, when figuring the expectation in writing (not buying!) WOOM andWTBOOM options, calls particularly, to use a square root of time calculationrather than delta in order to estimate the likelihood of a market going tosuch-and-so a price within a known time frame.This square root of time calculation, SQRTT as I call it, is straightforwardand can be found, again, in the numerous books dealing with the mathematics of trading. (I particularly like McMillan’s discussion of this calculationand its implications in Options as a Strategic Investment, 3rd edition, NewYork Institute of Finance, 1993). SVs and IVs are ordinarily computed as annual values, then reduced according to the actual amount of time we wantto consider. This reduction occurs automatically, as it were, in the calculation of delta, but we must do this minor piece of work ourselves to obtainthe SQRTT figure.Now, back to business. Our negative dollar expectation is the probability that CLZ will end up below 40.00 on option expiration day, times theamount of the loss we’ll realize: 0.9664 × –$125, or –$120.80. Computing ourpositive dollar expectation requires a series of incremental calculations, theresults of which are illustrated in Table 3.3. The probability that CLZ willend up between 40.00 and 41.00 is by definition the probability that CLZmoves at least to 40.00, minus the probability that it moves no higher than41.00. We calculate the profit for this increment by figuring the profit in thetrade at the midpoint of this range, in this case $350 at 40.50. The probability of CLZ ending up between 41.00 and 42.00 is the probability that CLZmoves above 41.00 but no higher than 42.00, the profit for this increment is$1350 at 41.50, and so on, right up the line. We stop calculating when ourSQRTT probability falls below 0.0050.The minimum tick in crude oil and its options is 0.01. How then can wehave option prices such as 0.018? We can’t. Entries like this represent thecomputed theoretical fair value of an option that has not yet traded. There isalso some rounding error in the right-hand column values, which are theproduct of the corresponding values in the two columns to its left. The column showing the delta of the options is just for comparison to the next column, the SQRTT-generated probability that CLZ will be below a specificstriking price on expiration.There’s one more thing to note, too. What about the last 0.0043? Westopped calculating at 0.9957, and isn’t there is more calculable dollarpositive expectation in the remaining tiny chunk of the curve? There is indeed. This seemingly wanton discarding of the last bit of the curve is another
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means of compensating for the fat-tail bias in the curve. We’re not beingnegligent by doing this, not at all, because we’ve taken the calculation outto almost three full standard deviations of the curve, most certainly a goodenough approximation to completeness for practical trading purposes.One entry in Table 3.3 requires comment. The range between 40.00 and41.00 technically would encompass partly profit and partly loss. Thebreakeven point for the option buyer in this trade is about 40.16, and, technically, we should perform two calculations for this range: one for the lossfrom 40.00 to 40.16 and a second for the profit from 40.17 to 40.99. Go rightahead if you like. It’s a straightforward proration over the 100 points in thisrange. It’s also a waste of time, pure and simple.We’re not going to exercise our option if CLZ is between 40.00 and40.16. Pay another commission, another chunk of the bid-ask spread, andmaybe more slippage? What are we, crazy? We’ll just sell the option backfor what few points we can get. In short, in calculating the expectation ofthis trade, we’re going to ignore this small range and lump 40.00–41.00 intothe positive dollar column. Purists hate this attitude. That’s all right withme, I find purists a little bit tedious. What they refuse to admit is that we’redealing with a model, an approximation, and 10–15 point ranges, whenthey’re way the devil out on the lognormal curve, don’t mean beans. (As anaside, if we would perform this computation, it would change the total expectation by . . . are you ready? . . . 15.4 cents. Yawn. Let the purists knockthemselves out; we have better things to do.)Well, well, well. Given current market circumstances, if we purchasethe CLZ 40.00 calls tomorrow, we can expect to lose $40.15 each time overa series of identical trades. That’s 32.12% of our initial capital of $125.00!This is a real sweetheart of a deal, isn’t it? Bah! We’d do much better forourselves by visiting a casino. Hell’s bells, we’d do better even by playingsome states’ lotteries.In fairness, we must observe that our calculations only apply to thewhole trade, from date of entry through option expiration. If CLZ spiked upward to, say, $32.00 in 45 days, the premium of the CLZ 40.00 call woulddoubtless increase even if its IV did not (although its IV likely would rise,too). The option buyer would have a chance to cash out a profit at thatpoint. Additionally, the IVs of CLZ call options generally might go higher either with or without a corresponding price move in crude oil, perhaps oneconomic news or increasingly clamorous sentiment about a war in theMiddle East, and the option buyer might be able to turn a profit here also.This type of option purchase is exactly the game I described in Chapter1. As I asked back then, “Do you really want to play this game?” I can’t imagine that any marginally sane trader would, and certainly not when muchbetter alternatives and expectations are readily available. While there aresome chances for profit in this trade, they are pretty remote, and in any
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Profitability 101
case a rather spectacular event must occur, or be rumored to be about tooccur, somewhere along the way. The option buyer in this case is dependenton such an occurrence to put a profit in his pocket; he is hoping for eventrisk to intervene in his favor. Good luck, pal. That way lies, if not madness,then definitely poverty. Luckily, I was inoculated against this type of unprofitable and frankly silly trading long ago (not without, ahem, some cost),and now you are too.Well, so much for those who promote this sort of hype-over-hard-factstrade, but there’s another good question to ask. If buying this sort ofWTBOOM option doesn’t offer us a positive expectation, what about writing it? Can we earn a good profit by selling instead of buying in this situation? In theory at least, a negative expectation present when buying anoption should imply a positive expectation when writing the same optioninstead. Is this really the case? Table 3.4 shows the expectation of writingthis same CLZ 40.00 call if we simply write it and take no action whateverbefore the option expires.This is marginally a better idea because the expectation of the trade isslightly positive, and we have in hand a minor additional advantage in writingthis call. When the option expires worthless, as this one will do in a giganticmajority of cases, our trading costs when exiting the trade are close to zero.
TABLE 3.4
Expectation of Writing CLZ 40.00 Call on May 17, 2002
StrikePrice
OptionPrice
Delta× 100
ProbabilityBelow*
Probability ofRange
Profit at RangeMidpoint
Expectation inRange
40.0041.0042.0043.00
0.1000.0600.0400.035
4.402.932.171.93
0.96640.97780.98370.9857
0.96640.01140.00590.0020
85.00–430.00–1430.00–2430.00
82.14–4.90–8.43–4.86
44.0045.0046.0047.00
0.0320.0300.0230.018
1.801.671.331.14
0.98680.98790.99050.9919
0.00110.00110.00260.0014
–3430.00–4430.00–5430.00–6430.00
–3.77–4.87–14.11–9.00
48.0049.0050.0051.00
0.0140.0120.0100.009
0.950.850.730.71
0.99330.99410.99500.9952
0.00140.00080.00090.0002
–7430.00–8430.00–9430.00–10430.00
–10.40–6.74–8.48–2.08
52.00
0.008
0.64
0.9957
0.0005
–11430.00
–5.71
Overall expectation:Notes: CLZ = 26.15; Z options expire 11/17/2002.
*Probability of being below the striking price on the same line.
$1.79
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What would have been the exiting contest risk expense stays in our pocket,where it belongs. (Side note: this is not entirely true when we write optionsin markets on some exchanges. These exchanges have the curious practiceof charging a small fee just for removing the expired option from their books.We should expect this sort of practice to become more common over time,as exchanges increasingly become profit-centered corporations instead ofmutual/partnership organizations.)Unfortunately, writing the CLZ 40.00 call here isn’t a good trade either.First, we have exactly zero interest in entering a trade with a positive expectation of, basically, pennies. Even if the expectation were, say, +$80.00,this still wouldn’t be a worthwhile effort. Why? We haven’t yet considered,have we, how much capital we we’ll be required to commit to this trade. Wehaven’t looked at what our return on capital (which I’ll call ROC in future)might be. When we write the CLZ 40.00 call for 0.10, $100.00, less $15.00contest risk, the NYMEX exchange will require an initial margin of $288, or$864 when we multiply it by 3 to provide our required cushion. Our notionalROC is 85 / 864 = 9.84% over six months’ time, a mere 1.64% per month. Thiscalculation is notional, because it doesn’t include the additional capitalwe’ll have to commit if CLZ would move sharply higher during the trade.We won’t be writing these options, period, because we can and shoulddo much better with our capital. And we will . . . but by using other strategies and different tactics. Irrespective of the strategies and tactics we willsubsequently examine, keeping the expectation of a trade in our favor mustbe our first concern.
First a Foundation, Then the FirstBrick in the WallThere are several reasonable measures we can employ to evaluate the potential desirability of an option trade. Expectation is not unique to this task;it is, however, the one I prefer and use. Adopting a useful measuring stickfor risk and reward, though, is only the first part of creating a profitabletrading methodology.We’ve long since agreed, haven’t we, that one factor necessary to successful trading is the acquisition of information about markets. There isn’t,as far as I know, any yardstick for evaluating the usefulness and potentiallyprofitable implications of a single specific piece of information other thanexperience, but there is an extremely useful question we can ask ourselvesconcerning information in general. What happens if some of the information we acquire is bad, pure and simple? Also, can we determine if such information is bad, in a general way? One good way to do this is . . . well,come along and see for yourself.
CHAPTER 4
Just the Facts,Ma’amAvoiding Moonshine, Morons,and Myths
No one stays here by faking reality in any manner whatever.—Ayn Rand, (Atlas Shrugged)
epending on whose figures you believe, somewhere between 75 and95% of all traders lose when they trade, in any market you care toname. They lose capital over time! No matter what the exact percentage is, this is a horribly high figure and a completely unacceptable result. It’sa perfectly believable proportion, mind you, if we assume that traders (as opposed to longer-term investors) are all gamblers at heart. However, it’s an utterly unbelievable figure if we assume that this same huge majority of tradersapply any sort of favorable methodology with some amount of discipline. Youand I must conclude, therefore, that a very large proportion of all traders either use consistently inferior methods or lack consistent discipline, or both.This chapter isn’t about method. It’s about acquiring one essential partof a trading discipline. Specifically, we must avoid or ignore or reject outrightthe bad information, the sundry misconceptions, misperceptions, and mythsthat a sizeable number of market participants, would-be gurus, and selfstyled experts, including, of course, the alleged financial media, attempt tofoist upon us. This class of so-called information, in broad, represents asub-type of the pseudo-information risk we saw in Chapter 2.
D
Staying Out of the Trickbag—Reality RulesPhilosophers have argued for millennia over the nature of the real world,over whether reality is an objective truth, a subjective one, some combination of the two, mystical, or merely an illusion. As traders, we don’t have a71
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dog in this fight, but we must nonetheless form a world-view about trading.We can accept any of these viewpoints we like and design a methodology inaccordance with our chosen one, but we cannot even hope to prosper intrading by embracing a false perception of trading reality, that is, attempting to delude ourselves or allowing others to practice fakery on us.Whatever our own world-view may be, trading itself is largely a subjective activity. Please observe that it must be so. If there were a hard, objective,mechanical method to employ in order to guarantee profitable trading,those who know or knew of such a method would have applied it to markets by now. Objective knowledge is impossible to hide over a long periodof time (consider all those throughout history who have tried and failed atthis endeavor). Perforce then, such a method would at some point have become widely known, and it would have come to be ineffective at some point,as traders who formerly lost to the users of such a method changed theirown methods over time. Ergo, there isn’t any such purely objective method.In trading, there are few certainties and far too many opinions, maybes,possiblys, and perhapses. One of the few hard and objective realities in trading is our net equity, our bottom line after each day’s market movements.Does it seem reasonable that you and I should entrust our objective tradingresults to the tender mercies of supposition, superstition, and in many casesplain stupidity?Well, that was a no-brainer, wasn’t it? Now, tell me why so many tradersdo exactly this, because I haven’t a clue (no jokes, please). Why do lots oftraders believe in and trade according to half-truths, delusions, the diktat ofself-anointed gurus, and outright myths? Don’t get me wrong here; I surelydon’t have and don’t pretend to have all or even most of the answers to theproblem of trading in a consistently profitable fashion over the long haul,but I didn’t just fall off the watermelon truck last Tuesday, either. I’ll guarantee you this much; shedding all this type of mental baggage in favor ofhard fact and disciplined methods can do absolutely nothing but improveour trading results. As the old satiric songwriter Tom Lehrer used to say,with just a trace of irony, “I have a few modest examples right here.”
Models Are Useful, but . . . There’s NoHoly GrailThe first myth we must dispose of is that any model of future price distribution is consistently accurate. Even though this statement may sound contradictory given our discussion previously, it is not. In the first place, nomodel can predict the unpredictable, and the unpredictable event occurswith disturbingly higher frequency than any model indicates. Even asidefrom obvious examples such as weather calamities and wars, the number of
Just the Facts, Ma’am
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possible unpredictable or imponderable events is enormous, and some ofthese will—not might, but will—intrude on our trades from time to time. Nomodel can warn us that such events impend, and the trader who believesotherwise is headed for an appointment with a bankruptcy judge.Looking at the matter objectively, all we traders can really do is to keepour likelihood of success, our expectation, in the plus column. Models areuseful because they provide a means of calculating our approximate expectation in a trading position. Sometimes the calculated figure is rather rough,sometimes it’s very close, but it is always only an approximation. This approximation can be dead wrong at times, too. When our trading model indicates one thing but our own eyes perceive something else entirely, we mustresolutely junk the model, at least temporarily, and not bother with a visitto the optometrist. Part of Long-Term Capital Management’s downfall wasdue to their outright refusal to discard certain of their risk and pricing models that were being proven more and more inaccurate with each passingday by the markets’ movements. This is an error we must avoid: falling inlove with a model is just deadly to our profitability (no disrespect to ElleMacPherson or Cindy Crawford, of course).Aside from the plainly egomaniacal notion that we can somehow modelfuture price movements with near precision, there are other reasons wemust use models cautiously. Many famous models, Black-Scholes-Mertonand Rubenstein-Leland (better known as “portfolio insurance”) to name two,include one or more highly dubious assumptions. Please understand herethat I’m not in any way criticizing any of these wonderfully talented theoreticians. I’d pay hard cash, any time at their convenience, to sit down withRobert Merton or Mark Rubenstein or Myron Scholes over a long lunch anddiscuss modeling and trading with any of them, or with numerous others. Iadmire these gents enormously, but notwithstanding that, some of the assumptions included in pricing models are quite suspect.The primary suspicious assumption is that markets are continuous.Models many times (please keep in mind that I’m not familiar with all themodels now in use) assume the mathematical continuity of markets, that is,that prices in markets move in small amounts, 0.01 or 0.05 or 1⁄8 or something similar. It would be delightful if this were the case, but it isn’t at all,and you and I and anyone who has traded for longer than a month knowthis to be a fact. Markets frequently have ticks with large jumps, or gaps—the interjection “Duh!” must come in here someplace—and this awkwardfact damages the usefulness of any model that assumes mathematicallycontinuous markets.These gaps are by definition discontinuities, and the effects they mayhave on our trading results are simply not included in any model of which I’maware. If I were feeling bold, I’d say that they can’t be included. How couldanyone model the likelihood of occurrence of a 10-cent gap in soybeans, say,
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next Thursday? Or a 30-cent gap? The frequency of such occurrences we caneasily calculate by looking at historical price movements, but there is no wayto model when the next such gap will occur.Can we compensate for the risk this and other dubious assumptionsembedded in models pose to our capital? Indeed we can, and we can do soin at least two distinct ways. We’ve touched on both of them before, but wecan now appreciate another reason to apply these tactics. Diversification ofour trading positions is desirable from the standpoint of limiting capacityrisk, and it becomes even more so as a means of dodging the risk that ourtrading model will be spectacularly wrong at some point. Certainly it mustpose much less risk to us to have our model become radically inaccurate regarding one of our positions if we hold 10 or 12 positions than if we holdonly 2 or 3.We can also compensate in another manner. In addition to our firmcommitment to avoiding the use of more than a fraction of the total leverage available to us, we can take another bite out of the potential risk whena model becomes erratic by keeping the size of each of our positions small.Old advice, you say, nothing new here? Not really, for there’s a verypractical twist to this second tactic within the context of models. The SPANmargin requirement is derived from the lognormal model; therefore, unliketraditional keep-your-positions-small advice, we can actually quantify theanswer to the question of “How small is small?” Here, “small” is, to a limiteddegree, a matter of personal preference, but there’s a nice, easy rule of thumbto help us quantify the notion of “small.” Assume that we’re considering entering a position in a pure asset trade. For the sake of caution, let’s postulate that a sizeable adverse price move will occur within a few days or acouple of weeks, in the market we’re considering. We can generate quite adecent real-world estimated maximum size of the initial position to take inthat market.Suppose we’re trading an account of $50,000, we have nine differentpositions in various markets, and our current margin requirement, as computed by the SPAN procedure, is $9,200 (yes, my skeptical friend, this is entirely possible, even normal for cautious traders). By our own rule fromChapter 2, we’ll treble the actual margin requirement to $27,600, leaving$22,400 in theoretical free capital. Suppose that we’re interested, for whatever reason, in selling April unleaded gasoline (ticker symbol HUJ), whichhas an initial SPAN margin requirement of $2,700 as I write this. Given oursituation, what is a reasonable and risk-controlled maximum number ofHUJ contracts for us to trade?We can almost certainly agree that a move of 7.00 cents in one week’stime is a significant move for a contract now trading at 60.00 cents. A 10%or larger move in a week’s time? Oh, yes, we can agree on this, I do believe.Fine, the initial margin requirement on HUJ tonight is $2,700, times 3 for
Just the Facts, Ma’am
75
our initial safety cushion, or $8,100, plus 700 points, or $2,940, of postulatedpotential adversity risk, giving a total of $11,040. According to our free capital figure of $22,400, we would just have room to sell two contracts, whichwould involve $22,080 of assumed capital commitment encompassing theexchange margin figure of $5,400, plus the 1400-point postulated loss, plusthe extra cushion we give ourselves when entering the trade.However, we aren’t going to sell two contracts of HUJ in this hypothetical situation; we’ll sell exactly one. The other way to say this is that we willdivide our current free capital by 2 before beginning our calculation. I can already hear the objections to this properly conservative notion of initiallyselling just one lot of HUJ. “You joker, I’m not going to sit for 700 points riskin HUJ!” and “How can you allot $11,000 for one lousy contract of gasolineand claim you’re only using $9,200 for nine other positions, what a con job!”Think it through, my friend. Neither of us will often tolerate anythinglike 700 points risk in the real world given our choice, but we might nothave the choice. We’re talking about model failure here, the possibility ofoccurrence of large gaps in a market, typically resulting from some outrageous or highly improbable event. Will you guarantee me that some terrorist loony-tune doesn’t blow up a refinery or a pipeline, or that a for-realfull-bore live shooting war doesn’t erupt in the Middle East, either of whichmight result in much more than a 700 point move in HUJ, even perhapswithin one day?No, you won’t do this, because you can’t. Such events are possible, andyour guarantee would be worthless. We’ll be both conservative and practical to acknowledge these possibilities, and others unnamed, and to insureright now, if and as we can, against the risk their occurrence would pose toour capital. In this instance we can, and the insurance is so reasonablypriced that we’d be fools to decline it. All we need do is refuse to take tooheavy a position.The point here is that if the model, and hence our trade, fails becauseof external event risk, I’ll still be here to trade another day. You might havedifficulty surviving such an event if you would trade 8 or 10 lots, which younotionally could do in our hypothetical account. One more time. It’s calledcapacity risk, and to avoid it we must make as certain as we possibly canthat we have the capital capacity to deal with a random outrageous eventand not get blown out of the water like a torpedoed rowboat.If you protested against the theoretical capital allotment of $11,040, ascompared to the margin figure stated for our hypothetical pre-existing positions, please check the discussion again. By adding a position of 1 contractin HUJ, our initial margin as required by the exchanges, per the SPANmargining procedure, will only rise to $11,900. The rest of the figure we calculated is cushion, that amount of the insurance we voluntarily and gladlyaccept in order to provide against a potential failure of our model. The
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gambler in Chapter 1 may lose his stake due to external event risk, but hecannot lose more. We are using a considerable amount of leverage and wecan lose more, a lot more, and we cannot breezily dismiss the possibility ofoccurrence, however remote, of the outrageous sub-event. As our assortedpositions begin to move in our favor, we can and definitely should reducethis per-trade cushion, in the interest of efficient usage of capital, but wequite literally cannot afford to undertake such a reduction before we enterthe trade.
So, You Don’t Deal in Options, Eh?Some traders apparently believe a number of myths about options. Theybreak into cold sweats and begin twitching uncontrollably when the subjectof options is raised. This reaction is remarkable to watch and difficult to understand, because every American who has capital or good credit, whetherhe trades in any market or not, implicitly deals with options almost everyday of his life, one way or another.Do you own a car? Do you own your home? Do you own rental property?Unless you bought these assets for cash, you deal or have dealt in options.Don’t shake your head at me, you do deal in options. Can you pay down, orpay off, your loan on these assets any time you wish? If not, you have my sympathy, and I retract the preceding statements (and you really should havesomeone advise you, before taking out another such loan). If so, however,whichever lender financed your purchase of any of these assets also sold youan option to prepay all or part of the loan you made. It’s your choice to prepay or not, correct? This is the precise definition of an option.In this case, the lender sold you a put option on your interest payments,and a very interesting one, too, because it’s a variable option. To exerciseyour option, you don’t have to pay off the whole amount of your loan; youmay do so in pieces, month on month as you find convenient, merely bypaying an amount above the required monthly payment. When you elect toprepay a loan, you lower your capital debt immediately, but you also effectively sell back some amount of future interest to the lender for the intriguing price of only the time value of money, the expected interest rateover the remaining term of the loan. If interest rates, which by definition arethe time value of money, are beginning to increase generally at some point,and if you’ve no other use for your capital for the moment, you might wellconsider prepaying part or all of your loan. Naturally, your decision to prepay will be influenced by the current interest rate charged on the loan, andwhether this interest rate is fixed or variable, but you absolutely own an option here, because it’s your choice to prepay or not.
Just the Facts, Ma’am
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You’ve paid for this option, certainly; it’s not a free lunch at all. Lendersused to charge