closingprice. And likewise, if a commodity future closes at a price Jower thanthe strike price of an at-the-money straddle minus the value of thestraddle premium received, a hypothetical short futures will be initi-ated at that closing price.
This defensive rule is quite arbitrary, but it is not commodityspecific and is chosen to strike a balance between taking toomany premature protective positions and allowing the straddleloss on any one position to increase to a very large number. 
                
BorN AGAINFutures Straddle Exit on. close Bu Exit Sell Exit Net ainprice premium “above below future future future future to water
Sep3 2205 «=.265. 2470 1940 2474 4 53Sep 4 2194 = 255 2449 1939 2474 25 32Sep 5 2221 251 2472 «1970 2474 2 55Sep 6 2259 267 2526 2534 8 109Sep 9 2247 254 2501 7 84sep 10 2282 257 2539 2278 146 122Sep 11 2335 49277 2612 195 195Sep 12 2355 287 2642 225 = 225Sep 13 2315 288 2603 186 =—:186Sep 16 2231 253 2484 -24 67Sep 17 2240 252 “16 75Sep 18 2294-270 147 (147Sep 19 2261 264 ) 108Sep 20 2274 = (261 6 2264 164118Sep 23 2284 257 2278 +144 124Sep 24-2353 205 = 205Sep 25 2383 234 = 234Sep 26 2358 202 202Sep 27 2396 243°0«243Sep30 2377 209. 209Oct 1 2358 180 180Oct 2 2352 175 175Oct 3 2421 239 «= 239Oct 4 2424 22300 «223Oct7 2474 173 173Oct 8 2508 143° 143Oct 9 2467 1720172Oct 10 2395 181 181Oct 11 2437 178 178Oct 14 2521 2303 17 97Oct 15 2502 2303 A 113Oct 16 2478 2278 14 125Oct 17 2500 2303 18 96Oct 18 25: 2335 12 70Oct 21 2 2335 -39 43Oct 22 2553 2335 49 33Oct 23 2486 2303 24 90Oct 24 2450 2278 “19 120Oct 25 2486 2335 4 76Oct 28 2485 2335 10 72Oct 29 2434 2278 «2441 43 120Oct 30 2428 2278 2441 -42 121Oct 31 2335 44 44Novi 2303 2412 3 8Nov 4 2278
Nov 5 2264
Nov6 2269
Nov7 2274
Novs 2359
Novi5 9417
121
FIGURE 8-2. Testing a dynamic option writing strategy involving both options andfutures is a complex procedure. In the example above, hypothetical at-the-money strad-dles are written on December crude oil futures on consecutive trading days betweenSeptember 3 and November 4, 1996,
Each of these straddles has its own unique pair of futures "trigger levels" and mustbe tested against these trigger levels from the time they are written until the momentthey expire to see if a defensive futures position must be initiated. The final columnabove is the net gain to the option writer when the strategy of no follow-up action is
employed.122 OPTION REALITY
At the time a defensive futures position is taken, the straddlewhich it is aimed at protecting will have increased in value, per-haps by as much as 50 percent. Exactly how much it will haveincreased depends on two things: the magnitude of the move inthe commodity future and the time remaining till option expiry atthe time the futures position is taken.
An option writer who is following a defensive strategy basedon buying or selling futures hopes, naturally, that the majority ofthe straddles written never have to be futures protected. Butwhether this happens is entirely beyond the writer's control, forthere is no way that market behavior can be predicted inadvance. The writer also hopes that once a futures position istaken, the momentum in the futures market continues in thesame direction, for, if it does not, if the futures does a suddenabout turn, the writer will have to think about protecting thefutures position as well!
The rationale behind the defensive writing strategy is to limitthe loss that will ensue from an unprotected straddle written ina futures market that has moved sharply, either up or down. Aninevitable consequence of the defensive strategy is that a numberof straddles which were profitable with indiscriminate (unde-fended) writing may now be rendered unprofitable with the pur-chase or sale of a future.
Consider the hypothetical crude oil straddle written onSeptember 3 (Figure 8-2) at the phantom strike price of 2205,for which a premium of 265 points is received. According to theprotocol previously described, the trigger levels on the Decemberfuture will be:
Upper trigger level = 2205 + 265 = 2470Lower trigger level = 2205 —- 265 = 1940
These numbers appear opposite the September 3 futuresprice, in columns 4 and 5. The defensive strategy dictates that if,between September 3 and November 15—when the straddleexpires — the December crude oil future should close above 2470or below 1940, a future will be purchased or sold at that closingprice.BorRN AGAIN 123
On October 7, December crude oil closes at 2474, above theupper trigger level of 2470 established for the straddle written atstrike 2205 on September 3. Assume, therefore, the purchase ofa December crude oil future at 2474. At this point in time, it isclear that a loss (hopefully small) on this overall transaction isinevitable. Consider the possibilities after the futures position istaken. (The straddle itself will be held till option expiry.)
If all subsequent closes of December crude oil remain abovethe original strike price, 2205, there is no need for further action,and the overall loss on the transaction will be limited to thefutures purchase price less the trigger level. To see why this is so,consider these two extreme cases: a futures price at option expiryof 3000 (major bull market) and a futures price at option expiryof 2245 (a reversal in market direction):
With future closing at 3000 at option expiry,
Net gain to writer = premium — payout + gain on future265 — (3000 — 2205) + (3000 — 2474)-4
With the future closing at 2245 at option expiry,
Net gain to writer = premium — payout + gain on future= 265 — (2245 — 2205) + (2245 — 2474)-4
At option expiry, for any futures price above the strike price,the loss on the overall transaction will be limited to 4 points—exactly what the defensive strategy is designed to do in such acase. At option expiry, the December future did in fact close at2417, well above the option strike price of 2205. In checking allthe futures prices occurring after the September 3 straddle iswritten, it is apparent that the price of the future never dipsbelow 2205. But, had it done so, the option writer would be onthe horns of a rather nasty dilemma, for the protective future at124 OPTION REALITY
this juncture would be threatening to backfire, and the writerwould be looking at a loss on the futures position alone greaterthan the total premium received on the straddle. And of coursethe straddle would still be open.
In the same way that option writers have to take protectionagainst the losing side of a straddle, they must also take protec-tion against a future that is incurring too big a loss. The amountof the loss they should be willing to take is again arbitrary, but aconsistent amount would again be the total option premiumreceived. The defensive rule needs to be expanded a little toinclude the case of a future that needs to be liquidated:
A long futures position initiated to protect a call will, itself, be pro-tected if it falls below the straddle strike price, and a short futuresposition initiated to protect a put will, itself, be protected if it risesabove the straddle strike price.
When a defensive futures position is closed out, a loss is imme-diately realized, and the option writer is back in the position whenthe straddle was first written. The premium will have diminisheddue to time decay, but the writer has to be prepared to repeat thedefensive strategy if necessary. It is quite unusual to have to take asecond defensive futures position, but it does happen, and theoption writer must be ready to deal with it when it does.
To appreciate the defensive futures strategy in action, con-sider (Figure 8-2) the crude oil straddle written on September10, at a strike price of 2282 and with trigger levels of 2539 and2025. On October 21, the upper trigger level is exceeded by theDecember crude oil future, and a futures contract must beassumed purchased at 2546, the closing price of the future onthat day. On November 4, the original strike of 2280 isbreached on the downside with a futures close at 2278, indi-cating that the long future has lost enough and warrants liqui-dation. The closing out of the long futures position results in aloss of (2546 — 2278), or 268 points. A second defensive strat-egy proves unnecessary, and when the option expires onNovember 15, with the future at 2417, the overall transactioncan be summed up thus:Born AGAIN 125
Net gain to writer = premium — payout + gain on future257 — (2417 — 2282) — 268— 146 points
By way of contrast, all the straddles written between October14 and November 3 require that short defensive futures positionsbe taken against them, and two of these short futures positionshave, themselves, to be covered before option expiry.
The strategy of using futures in defensive way—in an awk-ward market—can be assessed by comparing the final twocolumns of Figure 8-2. The crude oil price sequence examinedhere is particularly choppy, with many apparent price breakoutswhich go nowhere — precisely the kind of market which makes anoption writer wish he'd gone on vacation. In this particular timecapsule, the defensive strategy compares unfavorably with theundefended strategy. In a more typical period, however, therewould be many fewer futures positions initiated, and the twostrategies would generate similar results. In strongly trendingmarkets, the defensive strategy comes into its own, producingresults that are still slightly negative but vastly superior to thoseof the “sell and hold" strategy.
Great care must be exercised when checking a dynamic trad-ing system, so that no trades are "missed," that is, assumed notto have occurred when they would have occurred. It does nottake many errors of this type to produce a seriously erroneousresult. The reader will appreciate that the testing of 3781 strad-dles, each with its own unique set of contingencies, is a time-consuming and demanding exercise--even with the use of acomputer. The devil, however, really is in the details!
It would not be practical to document all the details here, sothe overall results, by commodity, are presented in Figure 8-3,from which it is immediately apparent that the effect of employ-ing the defensive strategy is to reduce the variability of resultsacross commodities. The trending markets, which scored heavilyagainst the option writer under the sell-and-hold strategy are nowmuch less unfavorable and even favor the writer in some126 OPTION REALITY
instances. The trendless markets which formerly favored thewriter, are now substantially less favorable. Meanwhile, the over-all edge moves in the writer's favor, as the payout ratio drops from1.000 to 0.902. Whether this shift in the ratio will be significantin the light of expected transaction costs remains to be seen.
The result is not unexpected, since the strategy of protectingan option against a runaway trend in the underlying future is amanifestation of the well-known market truism that lossesshould be cut and profits left to run. It will be interesting to seeif this writer's edge—established for a defensive strategy-can beimproved upon by any other means.
Taking dynamic action to limit large losses is an obvious strat-egy that must surely have occurred to every trader who has everwritten a straddle that has seriously backfired. By dynamicaction, I mean using futures for protection, covering the optionwith an identical offsetting trade, or covering the option withanother option on the same future, all of which—the authoravows — amount to the same thing in the long run. Running awayto fight another day is a common-sense discipline (more spokenof than followed I might add), and it is welcome to see its use-fulness confirmed, if only on a statistical basis.
*
A second strategy an option writer might employ to increasehis edge is also fairly obvious, but much less easy to implementor test. This is the strategy of being selective about which strad-dles to write in the first place. If a writer can come up with a con-sistent method of comparing market volatility (measured) withimplied volatility (computed), he may logically choose to writeoptions only when they appear to be overvalued. Option valua-tion comparisons are accomplished by using the option volatilityformulae given at the end of the previous chapter. As always,when calculating market volatility, the choice of time base isarbitrary.
To test for "overvaluation,” I compared a simple reading ofmarket volatility-calculated from the mean absolute deviation,MAD, over 30 days—with the implied volatility for each of theBorn AGAIN 127
acing, Average Average Fayou eos
FINANCIALSS&P 500 254 2459 2430 0.988 ‘onT-Bonds 252 298 302 1.03 (0.950)Swiss franc 254 219 188 0.849 (1.146)Yen 254 228 197 0.864 (0.974)RESOURCEGold 252 107 102 0.953 (1.252)Silver 252 369 293 0.794 (0.707)Crude oil 251 167 176 1.064 (1.389)Cotton 249 466 325 0.697 (0.704)FOODSoybeans 254 452 443 0.980 (1.153)Wheat 254 364 425 1.168 (1.420)Com 254 281 288 1.025 (1.949)Cattle 254 272 223 0.820 (0.555)Cocoa 249 86 68 0.791 (0.779)Coffee 249 1348 1114 0.826 (1.240)Sugar 249 78 55 0.705 (0.500)Total —-® 3781 Average -~—t 0.902 (1.000)
* Different exchanges operate on slightlydifferent business schedules
FIGURE 8-3. Comparing the final two columns of the table above, it is apparentthat the effect of employing a defensive futures strategy when writing straddles is toshift the edge in the writer's favor. Commodities such as the grains and coffee, whichexperienced large trends during 1996 and which previously, when unprotected, createdlarge losses for the writer now generate much more favorable results. However, optionwriting results from the trendless markets, which previously favored the writer, are nowconsiderably less favorable.
The number that truly matters is the overall payout-to-premium ratio, which isnow 0,902, indicating that the trading edge has shifted in favor of the writer. There areadditional costs associated with a defensive futures strategy, since there will be someincrease in the number of transactions. Notwithstanding, the shift in the payout ratiodoes strike the author as significant.
at-the-money straddles in the data base. If the implied volatilityexceeded the market volatility, I assumed the straddle written;otherwise, I excluded it from the summation. There is no specialsignificance about choosing 30 days as the time base, other thanthat it falls in the general range of time intervals that strike a rea-sonable balance between long-term volatility and short-termvolatility, and between data which go too far back in time and are128 OPTION REALITY
possibly nonrepresentative and recent data which are certainlyup-to-date but of rather small sample size. The important thingabout choosing 30 is that I chose it in advance—not after testing25 and 35 and finding that 30 gives the “optimum” result. If anyresults are in error here, it is most assuredly not on account ofconfirmation bias.
The results of the "selection by valuation” test—the kind oftest which the advent of large spreadsheet computer programsnow renders practicable — are documented in Figure 8-4. There islittle to be read into the changes in the payout ratio by commod-ity, because falling sample sizes at the individual commodity levelare bound to affect the results due to pure chance. There is alsolittle significance to be attached to the wide disparity in the num-ber of "overvaluations" observed in each commodity. For example,the disparity between gold (238 observations) and wheat (80observations) is understandable in light of the atypical price vari-ations occurring in the gold and wheat markets during 1996.
By the volatility comparison test, gold options were substan-tially overvalued on almost every trading day of 1996, even thoughimplied volatilities were registering the lowest values seen in 20years. The low implied volatilities were naturally tied to the day-to-day price variability in the gold futures market at that time.But, low as they were, implied volatilities refused to mirror short-term market volatility levels, because the option market wasalways expecting price variations to regress to their historicalnorms—and sooner rather than later. Reality is that “overvalua-tion” by mathematical calculation in a very quiet market is notreally overvaluation at all, and a market registering unusually lowlevels of implied volatility will often seem falsely overvalued.
In contrast, the wheat option market of 1996 was falselyundervalued, since the option market was constantly (and notunreasonably) expecting the unusually high price variability toregress toward its historical mean. In a high-priced, volatile mar-ket, implied volatility normally stays below short-term marketvolatility, so that "undervaluation” goes hand in hand with highimplied volatility.
Returning to Figure 8-4, it appears that selectivity by valua-tion has marginally improved the overall writer's edge from 0.902to 0.884—not a great deal, perhaps, but at least a move in theBorn AGAIN 129
* Tradit Average Average Number Payout
days 19! remium out_in sample _ PremiumFINANCIALSS&P 500 254 2544 2597 189 1.021T-Bonds 252 293 287 173 0.980Swiss franc 254 226 199 221 0.881Yen 254 245 198 198 0.808RESOURCEGold 252 105 103 238 0.981Silver 252 371 291 243 0.784Crude oil 251 175 174 90 0.994Cotton 249 476 337 195 0.708FOODSoybeans 254 515 484 127 0.940Wheat 254 359 476 80 1.326Com 254 302 298 137 0.987Cattle 254 271 224 175 0.827Cocoa 249 88 73 186 0.830Coffee 249 1355 1118 165 0,825Sugar 249 78 59 213 0.756
Total ——-® 3781 Average ——B 2627 0.884
* Different exchanges operate on slightlydifferent business schedules
FIGURE 8-4. As described in Figure 8-3, the effect of introducing a defensivefutures strategy when writing straddles is to shift the payout ratio from 1.000 to 0.902.
The effect of selecting as writing candidates only those options that are overvalued(con.paring market volatility with implied volatility) is to shift the payout ratio furtherin the writer's favor, but only by a small amount, from 0.902 to 0.884. The sample sizedrops from 3781 to 2627, and the number of sample observations is no longer con-stant by commodity. In the computation of the overall payout ratio, equal weighting isgiven to each observation, rather than each commodity.
Note the smaller sample size reduces somewhat the confidence level that can beplaced on the overall result.
right direction. After eliminating all the "undervalued" straddlesfrom the payout test, the number of observations drops from3781 to 2627. It is a little bit of a puzzle that overvaluations andundervaluations don't occur in equal proportion, since the netoutcome of writing all the straddles is already known to result infair value.
In grasping for an explanation for this anomaly, I wondered ifthe small sample of 30 observations could be yielding a biasedestimate of the mean absolute deviation, MAD. (Market volatil-130 OPTION REALITY
ity was calculated from Eq. 5-2, or 20 x MAD.) I also reconsid-ered a lingering reservation I have about using futures closingprices as the only basis for defining variability. It is true that thedaily high price and daily low price of a future ought, logically, toinfluence the calculation of volatility. After all, a future may havea huge daily range, close unchanged, and count for zero in avolatility definition that encompasses only closing prices. Dailyranges are never used in volatility calculations because they can-not, as yet, be handled by any mathematical theory. (Now here isa problem are for an applied mathematician to direct his or hertalents.) It is generally accepted—by omission, perhaps — thatclosing prices averaged over a sufficiently long period will workequally as well, because options, at expiry, are valued at a specificclosing price. There is no doubt, however, that the failure to usehighs and lows in a volatility calculation does represent a loss ofinformation of unknown (hopefully small) dimension.
The empirical evidence based on the result of equality ofexpectations does suggest that Eq. 5-2 underestimates marketvolatility, or at least that estimate of market volatility which oughtproperly to be compared with implied volatility. If the overalloptions market is fairly priced, these volatilities should averageout the same over a long time period and a large number of inde-pendent observations.
Could there be some way to modify Eq. 5-2 in the light of theempirical evidence that it estimates low? I think there is. Thequestion can be formulated this way: If MAD multiplied by 20yields 2627 "overvalueds” out of a total sample size of 3781, whatmultiplier of MAD would yield 1890 (exactly half of the total)“overvalueds"? The answer is a multiplier of 22, so that Eq. 5-2when modified becomes:
Market volatility = 22 x MAD
This is a wholly pragmatic definition of market volatility, butone that works where it matters; in a comparison with impliedvolatility. Rocket science this is not, but who cares? —if the thingflies.
It is less of a puzzle that a comprehensive test using valuationas the selection criterion should yield only a modest improve-Born AGAIN 131
ment in the overall payout ratio. The small improvement in thewriter's edge, from 0.902 to 0.884 is, none the less, an unbiasedestimate and deserves to be taken at face value. Maybe it is unre-alistic to expect a larger improvement. As discussed above in ref-erence to gold and wheat, option valuations are going to be outof whack for long periods in very atypical markets, since expec-tation of regression to the mean is certainly going to dominate attimes, as are special factors governing uncertainty that cannotpossibly be reflected in historical prices.
Is a writer's edge of 0.884 the best that can be expectedthrough selectivity? Hard to say. Figure 8-5 shows the effect ofselecting only those straddles where implied volatility is at leastfifty percent higher than market volatility. The payout ratio heredrops from 0.884 to 0.851, but the sample size is too small andthe observations concentrated in too few commodities for theresult to be considered reliable. One thing is clear from the val-uation test; having an after-the-fact defensive strategy in placewill be a lot more important to an option writer than having a sys-tem to spot which options to write in the first place.
In the testing of the valuation strategy above, MAD is calcu-lated as a simple average of the 30 most recent price changes.This basic calculation may not necessarily give the most consis-tent and logical estimates. For example, a simple average ascribesthe same weighting to each reading of price change whether thatprice change occurs 1 day or 30 days back in time. When a verylarge price change gets to be 30 days old, and drops off the list ofreadings to be averaged, the MAD can drop rather abruptly, andsomewhat illogically.
A more dynamic, and more easily maintained running esti-mate of MAD is achievable via a calculation known as exponen-tial smoothing, in which recent observations are given moreimport than distant observations. Whether a technique like expo-nential smoothing would change the outcome much is not aquestion I wish to probe (it probably improves it a little), for thisis getting into boutique science of a kind that can lead to falselyoptimized results. While in practice I recommend using expo-nential smoothing, as far as estimated payouts are concerned, I'llbe happy to stick with the conservative no-frills estimate I got onthe first pass.132 OPTION REALITY
ani Average Average Number Payout
days 1 premium payout insample PremiumFINANCIALSS&P 500 254 2525 2967 22 1.175T-Bonds 252 248 254 9 1.024Swiss franc 254 299 253 40 0.846Yen 254 330 181 46 0.548RESOURCEGold 252 105 103 88 0.981Silver 252 492 96 22 0.195Crude oil 251 143 233 15 1.629Cotton 249 0FOODSoybeans 254 681 100 7 0.147Wheat 254 364 0 0 0.000Com 254 324 375 26 1.157Cattle 254 291 346 2 1.189Cocoa 249 0Coffee 249 1593 1057 38 0.664Sugar 249 °
Total ——® 3781 Average ——B 315 0.851
* Different exchanges operate on slightlydifferent business schedules
FIGURE 8-5. When straddles with extremely high apparent overvaluations (a ratioof implied volatility to market volatility greater than 1.5) are selected as a subgroup,the sample size drops to 315, distributed very irregularly across the various commodi-ties. For example, gold is highly overvalued 88 times, cotton, wheat, cocoa, and sugarnot at all.
Although the payout ratio drops further, from 0.884 to 0.851, this result cannotbe considered reliable because of the skewed weightings by commodity and the lowoverall sample size.
*
Since dynamic valuation using short-term market volatilityappears to contribute so modestly to the writer's edge, it seemsnatural to ask whether comparing implied volatility with long-term market volatility is likely to be a superior selectivity strategy.If so, it is tantamount to saying that regression to the mean is amore powerful option valuation factor than current price actionin the futures pit.
This question, however, has to remain unanswered — at leastby me