%, both options are underpriced. Moreover, the leverage values of the optionsmay not be the same so that their sensitivity to changes in market conditions may, underVolatility 75
some circumstances, make several 102 calls a more desirable purchase than one 98 call.If we ignore these considerations, in relative terms the 98 call is a better value becauseits implied volatility is lower.
While option traders may at times refer to any of the four types of volatilities, twoof these stand out in importance, the future volatility and the implied volatility. Thefuture volatility of an underlying contract determines the value of options on thatcontract. The implied volatility is a reflection of each option’s price. These two numbers,yalue and price, are what all traders, not just option traders, are concerned with. If acontract has a high value and a low price, then a trader will want to be a buyer. Ifacontract has a low value and a high price, then a trader will want to be a seller. For anoption trader this usually means comparing the future volatility with the impliedvolatility. If implied volatility is low with respect to the expected future volatility, atrader will prefer to buy options; if implied volatility is high, a trader will prefer to selloptions. Of course, future volatility is an unknown, so we tend to look at the historicaland forecast volatilities to help us make an intelligent guess about the future. But in thefinal analysis, it is the future volatility which determines an option’s value.
To help the new trader understand the various types of volatility, consider thefollowing analogy to weather forecasting. Suppose a trader living in Chicago gets up ona July morning and must decide what clothes to wear that day. Do you think he willconsider putting on a parka? This is not a logical choice because he knows thathistorically itis not sufficiently cold in Chicago in July to warrant wearing a winter coat.Next, he might turn on the radio or television to listen to the weather forecast. Theforecaster is predicting clear skies with temperatures around 90° (32°C). Based on thisinformation, our trader has reached a decision: he will wear a short sleeve shirt with nosweater or jacket, and he certainly won’t need an umbrella. However, just to be sure,he decides to take a look out the window to see what the people outside are wearing.To his surprise, everyone is wearing a coat and carrying an umbrella. The people outside,through their clothing, are ‘mplving completely different weather. What clothes shouldthe trader then wear? He must make some decision, but whom should he believe, theweather forecaster or the people in the street? There can be no certain answer becausethe trader will not know the future weather until the end of the day.
Much depends on the trader’s knowledge of local conditions. Perhaps the traderlives in an area far removed from where the weather forecaster is located. Then he mustgive added weight to local conditions. On the other hand, perhaps the people in thestreet all listened to a weather forecaster who has a history of playing practical jokes.
The decision on what clothes to wear, just like every trading decision, depends ona great many factors. Not only must the decision be made on the basis of the bestavailable information, but the decision must also be made with consideration for thepossibility of error. What are the benefits of being right? What are the consequences ofbeing wrong? If a trader fails to take an umbrella and it rains, that may be of littleconsequence if the bus picks him up right outside his residence and drops him off rightoutside his place of work. On the other hand, if he must walk several blocks in the rain,he might catch the flu and be away from work for a week. The choices are never easy,and one can only hope to make the decision that will turn out best in the long run.76 Chapter 4
Seasonal Volatility
There is one other type of volatility with which a commodity trader may have to deal.Certain agricultural commodities, such as corn, soybeans, and wheat, are very sensitiveto volatility factors arising from severe seasonal weather conditions. Such conditionsoccur especially in the summer months when drought can destroy major portions of acrop and catise prices to fluctuate wildly. For this reason, grains show a significantincrease in volatility during the months of June, July, and August. Conversely, they showa significant decrease during the early spring months, before American planting hasbegun but after the South American crop has been harvested. Given these factors, onemust automatically assign a higher volatility to an option contract which extendsthrough the summer months. If, in February, a trader has assigned a volatility of 18% toa May soybean contract, he will certainly choose some higher volatility, perhaps 22%,for a November contract. He knows that the November contract includes the summermonths, while the May contract does not. The effect of seasonal volatility on soybeansis shown in Figure 4-10.
A trader who is new to options might initially question whether volatility is reallythat important. He has probably been pursuing directional strategies where volatilitywas not a consideration. It is also possible to pursue a variety of directional strategies inthe option market. But if a trader has a thorough understanding of volatility, he has anadditional variable with which to work. He can, in effect, approach the market fromtwo directions instead of one. Many traders find it easier to work exclusively with
Figure 4-10: Monthly Soybean Volatility (1980-1992)60
 
507
407
307
Annulaized Volatility (%)
“NLU104 ee ee ae
T T T T T ~T T TT —_~T T TJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
 
 
 
—™ high —*- mean =& lowCourtesy of the Chicago Board of Trade.Volatility 77
volatility, rather than try to guess market direction. Moreover, volatility strategies canbe extremely profitable and, when chosen intelligently, can even reduce a trader’s riskexposure. These two variables, market direction and volatility, enable an option traderto pursue many strategies not available to the pure stock or futures trader.
Changing our assumptions about future volatility can have a dramatic effect on theyalue of options. To see this, look at the prices, theoretical values, and implied volatilitiesfor ten-week gold options in Figure 4-11. Note the change in theoretical values asvolatility is increased from 11% to 14% to 17%. The 360 call and put, which areessentially at-the-money, change by approximately 1.85 ($185) for each three-percent-age-point increase in volatility. While out-of-the-money options do not show as great adollar change in value, in percent terms their sensitivity to a change in volatility is evengreater. As volatility increases from 11% to 14%, the 390 call and 330 put more thandouble in value, and double again as volatility increases from 14% to 17%. A three-per-centage-point change in volatility over ten weeks is not at all uncommon. Indeed, thevolatility of gold can show swings of six or seven percentage points in a relatively shortperiod of time. This is evident from the historical volatility of gold shown in Figure 4-12.
Given its importance, it is not surprising that the serious option trader spends aconsiderable amount of time thinking about volatility. Using his knowledge of historical,forecast, implied, and, in the case of agricultural commodities, seasonal volatility, he
Figure 4-11
3 May 1991
August Gold Futures at 358.30Time to expiration = 10 weeksInterest rate = 5.50%
Exerclse Settlement Implled Theoretical Value If...
Price Price Volatility Vol. = 11% Vol. = 14% Vol. = 17%Calls
330 29.20 15.36 28.27 28.84 29.69340 20,60 14.48 19.26 20.40 21.73350 13.20 13.86 11.61 13.28 15.01360 7.80 13.89 6.02 7.87 9.73370 4.20 14,00 2.62 4.20 5.88380 2.40 14.93 94 2.01 3.32390 1.40 15.93 28 86 1.75Puts
330 1.20 15.28 .29 87 1.72340 2.60 14.69 117 2.31 3.64350 5.10 14,04 3.44 5.08 6.80360 9.50 13.92 7.70 9.55 11.41370 15.70 13.88 14.18 15.77 17.45380 23.70 14.57 22.40 23.47 24.78
390 32.50 15.13 31.62 32.20 33.08
edChapter 4
78
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ANNIOA PIOD HoomM-Ol :Z1-p eanb1yVolatility 79
must try to make an intelligent decision about future volatility. From this, he will lookfor option strategies which will be profitable when he is right, but which will not resultin a disastrous loss when he is wrong. Because of the difficulty in predicting volatility,a trader must always look for strategies which will leave the greatest margin for error.No trader will survive very long pursuing strategies based on a future volatility estimateof 15% if such a strategy results in a Joss when volatility actually turns out to be 16%.Given the shifts that occur in volatility, a one-percentage-point margin for error is nomargin for error at all.
We have not yet concluded our discussion of volatility. But before continuing, itwill be useful to look at option characteristics, trading strategies, and risk considerations.We will then be in a better position to examine volatility in greater detail.de 5
Using an Option’sTheoretical Value
From a theoretician’s point of view, the Black-Scholes Model represents a novel solutionto a complex problem. The model requires a limited number of inputs and relativelysimple mathematical calculations. These factors have made the Black-Scholes Model themost popular of all option pricing methods.
While a trader might appreciate the elegance of the mathematics, it is still the actualperformance of the model in the marketplace which is his primary consideration. Is itreally possible to profit from price discrepancies between model-generated values andthe market prices of options?
To see how the model is intended to function, let us make two assumptions:
1. The price distribution of an underlying contract is accurately represented by alognormal distribution.
2. We actually know the future volatility of an underlying contract.
Clearly, the second assumption is not very realistic since we obviously don’t know thefuture. As we shall see later, the first assumption is also highly questionable. But for thetime being, rather than focus on the assumptions in the model, we want to focus on itsuse by a trader. If we make these assumptions, and the Black-Scholes Model does indeedwork, then we ought to be able to turn any difference between an option’s price and itstheoretical value into a profit. How should we go about doing this?
Suppose that a series of options are available on a certain futures contract with thefollowing conditions:
June futures price = 101.35Interest rate = 8.00%Time to June expiration = 10 weeks
Suppose also that the options are subject to stock-type settlement, requiring full paymentof the premium, as they currently are on U.S. markets.
Given that we can choose any exercise price we want, and that we can choose thetype of option we want (call or put), the only input we still need for accurate evaluationof options on this futures contract is volatility. Since we have made the assumption thatwe actually know the future volatility of the underlying futures contract, let’s imagine
882 Chapter 5
that we have a crystal ball which will teil us the future volatility. When we look intothe crystal ball we see a volatility figure of 18.3% over the next 10 weeks. Now we haveali the necessary inputs for theoretical evaluation, and it only remains to choose a4specific option.
The June 100 call, being close to at-the-money, is likely to be actively traded, solet’s focus on that option. Feeding our inputs into the Black Model (the futures variationof the Black-Scholes Model), we find that the June 100 call has a theoretical value of3.88. When we check its price in the marketplace, we find that it is being offered at3.25. How can we profit from this discrepancy?
Clearly, our first move will be to purchase options since they are underpriced by.63. Can we now walk away from the position and come back at expiration to collectour money?
We noted in Chapter 3 that the purchase or sale of a theoretically mispriced optionrequires us to establish a hedge by taking an opposing position in the underlyingcontract. When this is done correctly, for small changes in the price of the underlying,the increase (decrease) in the value of the option position will exactly offset the decrease{increase) in the value of the opposing position in the underlying contract. Such a hedgeis unbiased, or neutral, as to the direction of the underlying contract.
The number which enables us to establish a neutral hedge under current marketconditions is a by-product of the theoretical pricing model and is known as the hedgeratio or, more commonly, the delta. We will discuss the delta in greater detail in thenext chapter, but certain characteristics will be important in our present example:
1. The delta of a call option is always somewhere between 0 and 1.00.2. The delta of an option can change as market conditions change.3. An underlying contract always has a delta of 1.00.
Option traders commonly drop the decimal point when discussing deltas, aconvention which we will also follow. Therefore, the delta of a call will fall in the rangeof 0 to 100, and an underlying contract will always have a delta of 100.!
Going back to our example, in order to make correct use of the option’s theoreticalvalue, we also need to know the delta, in this case 57 (.57). This means that for eachoption which we purchase we must sell 57% of an underlying contract to establish anunbiased or neutral hedge. Since the purchase or sale of fractional futures contracts isnot permitted, we intend to purchase 100 June 100 calls and sell 57 June futurescontracts. This maintains the proper neutral ratio but allows us to deal in wholenumbers of contracts. We have established the following position:
Contract Contract Delta Delta PositionBuy 100 June 100 calls 57 +5700Sell 57 June futures 100 -5700
 
1. This convention originated In the U.S. stock option market where an underlying contract typicallyconsisted of 100 shares of stock. It became common to equate one delta with each share of stock.Using an Option’s Theoretical Value 83
Note that the delta position on each side of the hedge is the number of contractsmultiplied by the contract delta, keeping in mind that the purchase of a contract isrepresented by a positive sign and the sale of a contract is represented by a negative sign.The option delta position is therefore +100 x 57 = +5700 and the futures position is-57 x 100 = -5700. When we add these two numbers together, +5700 and -5700,the total delta position is zero, and we say that the position is delta neutral Within asmall range, a delta neutral position has no particular preference for either upward ordownward movement in the underlying market. If the total delta position is a positivenumber, this indicates an upward bias; if the total delta position is a negative number,this indicates a downward bias,
Having established a delta neutral hedge we must still deal with the fact that atheoretical value is based on probability. The roulette player who is able to purchase aroulette bet for less than its theoretical value can only expect to show a profit if he isallowed to play many, many times. On any one bet he will most likely lose, since thereis only one way for him to win but 37 ways for him to lose. The same is true of ourhedge. The odds may be on our side because we have purchased undervalued options.But in the short run our hedge might very well result in a loss. Is there anything we cando to offset the possibility of short-term bad luck?
We know that in the long run the laws of probability are on our side. We can befairly certain of making a profit if we are given the chance to make many bets at thesame favorable odds. The more bets we make, the better our chances of making a profitidentical to that predicted by the theoretical pricing model. One way to accomplish thisis to approach the initial hedge as a continuing series of small bets. We can replicate thelong-term probability by reassessing our position at regular intervals, and then makingappropriate changes in the position so that each new interval represents a new bet.
Suppose that one week later the price of the June futures contract has moved upto 102.26. At this point we can feed the new market conditions into our theoreticalpricing model:
June futures price = 102.26Interest rate = 8.00%Time to June expiration = 9 weeksVolatility = 18.3%
Note that we have made no change in the interest rate or volatility. The theoreticalpricing model we are using, the Black model, assumes that these two inputs remainconstant over the life of the option. Based on the new inputs we can calculate the newdeita for the 100 call, in this case 62. The delta position is now:
Contract Contract Delta Delta PositionLong 100 June 100 calls 62 +6200Short 57 June futures 100 -5700
Our total delta position is now +500. This is the end of one bet, with another about tobegin.84 Chapter 5
Whenever we begin a new bet we are required to return to a delta neutral position,In our example it will be necessary to reduce our position by 500 deltas. As we shall seelater, there are a number of different ways to do this. But to keep our presentcalculations as simple as possible we will make the necessary trades in the underlyingfutures market, since an underlying contract always has a delta of 100.
Here we wish to sell off 500 deltas, and we can do this by selling 5 futures contracts.Our delta position is now:
Contract Contract Delta Delta PositionLong 100 June 100 calls 62 +6200Short 62 June futures 100 6200
We are again delta neutral and about to begin a new bet. As before, ournew bet dependsonly on the volatility of the underlying futures contract, and not on its direction.
The extra five futures contracts we sold were an adjustment to our position.Adjustments are trades which are not necessarily made for the purpose of adding to thetheoretical edge, although they may also have that effect. Rather, they are madeprimarily to ensure that a position remains delta neutral. In our case, the sale of the fiveextra futures contracts has no effect on our theoretical edge since futures contracts haveno theoretical value. The trade was made solely for the purpose of adjusting our hedgeto remain delta neutral.
The steps we have thus far taken illustrate the correct procedure in using anoption’s theoretical value:
1, Purchase (sell) undervalued (overvalued) options.2. Establish a delta neutral hedge against the underlying contract.3. Adjust the hedge at regular intervals to remain delta neutral.
Since volatility is assumed to compound continuously, theoretical pricing modelsassume that adjustments are also made continuously, and that the hedge is beingadjusted at every moment in time. Such continuous adjustments are not possible in thereal world, since a trader can only trade at discrete intervals. By making adjustments atregular intervals, we are conforming as best we can to the principles of the theoreticalpricing model.
Using the proper procedure, how would the adjustment process look if we were tocarry the hedge to expiration? The results are shown in Figure 5-1. In our exampleadjustments were made at weekly intervals. At the end of each interval the delta of theJune 100 call was recalculated from the time remaining to expiration, the current priceof the underlying futures contract, the fixed interest rate of 8.00%, and the knownvolatility of 18.3%. Note that we have not changed the volatility, even though othermarket conditions may have changed. Volatility, like interest rates, are assumed to be
2. While a futures trader might talk about the theoretical value of a futures contract, from an optiontrader’s point of view the underlying contract has no theoretical value. The theoretical value iswhatever price the trader thinks he can trade the contract at.Using an Option’s Theoretical Value 85
‘ constant over the life of the option. In practice, a trader may, and often does, changehis opinion about volatility.What will we do with our position at the end of ten weeks when the options expire?At that time we plan to close out the position by:
1. Letting any out-of-the money options expire worthless.
2. Selling any in-themoney options at parity (intrinsic value) or, equivalently,exercising them and offsetting them against the underlying futures contract.
3. Liquidating any outstanding futures contracts at the market price.
Let’s go through this procedure step by step and see what the complete results of ourhedge were.
Original Hedge: At June expiration (week 10) we can liquidate our June 100 calls byeither selling them at 2.54, or by selling futures at 102.54 and exercising the calls. Eithermethod will result in a credit of 2.54 to our account. Since we originally paid 3.25 foreach option, we have effectively lost .71 per option, and our total loss on the options is100 x -—.71 = -71,00.
As part of our original hedge we also sold 57 June futures at 101.35. At expirationwe had to buy them back at 102.54, fora loss of 1.19 per contract. Our total loss onthe futures is therefore 57 x -1.19 = -67.83. Adding this to our option loss, the totalloss on the original hedge is -71.00 -67.83 = -138.83. This certainly does not appearto have been successful. We expected to make money on the hedge, and yet we havea sizable loss.
Adjustments: The original hedge was not our only transaction. In order to remain deltaneutral, we were forced to buy and sell futures contracts over the ten-week life of theoption. At the end of week one we were 500 deltas long, so we had to sell 5 futures at
Figure 5-1Total Total InterestFutures Delta of Delta Adjustment  Futuree on
Week Price 100Cali Poeltlon In Futures Adjustment Variation Variation
0 101.35 57 0 0 0 0 0
1 102.26 62 +500 sell 5 5 ~51.87 —72
2 99.07 46 —1,600 buy 16 +11 +197.78 +2.43
3 100.39 53 +700 sell 7 +4 ~60.72 —.65
4 100.76 56 +300 sell 3 +1 ~19.61 —.18
5 403.59 74 +1,800 sell 18 -17 —158.48 -1.22
6 99.26 45 ~2,900 buy 29 +12 +320.42 +1.97
7 98,28 35 —1,000 buy 10 +22 +44,10 +.20
8 99,98 50 +1,500 sell 15 +7 59.50 -.18
9 403.78 93 +4,300 sell 43 -36 —190.00 —.29
10 402.54 buy 3686 Chapter 5
102.26, At the end of week two we were 1600 deltas short, so we had to buy 16 futuresat 99.07; and so on each week until the end of week ten. At expiration we were shortan extra 36 futures contracts, and we bought these in at the closing price of 102.54,Note that each time the futures price rose our delta position became positive, so thatwe were forced to sell futures; and each time the futures price fell our delta positionbecame negative, so that we were forced to buy futures. Because our adjustmentsdepended on the delta position, we were forced to do what every trader wants to do:buy low and sell high.
What was the result of all the adjustments required to maintain a delta neutralposition? In fact the resuit was a profit of 205.27. (The reader may wish to confirm thisby adding up the cash flow from all the trades in the adjustment column in Figure 5-1.)This profit more than offset the losses incurred from the original hedge.
Carrying Costs: What else will effect our final profit or loss? Originally we bought callsand sold futures. While futures are subject to futures-type settlement and require noinitial cash outlay, the options are subject to stock-type settlement and require fullpayment. We bought 100 calls for 3.25 each, for a total outlay of 325.00. Based on ourassumed 8.00% interest rate, the carrying cost on 325,00 for 10 weeks is .08 x 70/365x 325.00 = 4.99. We will have to include this debit of 4.99 in our final calculations.
Variation Costs: Finally, we must take into consideration the variation costs requiredto maintain our futures position. As futures move up or down in price, cash is eithercredited to, or debited from, a trader’s account. In theory, a trader can earn interest ona cash credit, but must pay interest on a cash debit. For example, we initially sold 57futures contracts at 101.35. One week later the futures price rose to 102.26, so ouraccount was debited 57 x (101.35 - 102.26) = -51.87. Financing this debit for nineweeks at 8.00% cost us—51.87 x .08 x 63/365 =~.72. In order to remain delta neutralat the end of week one we sold an additional five futures contracts, for a new total ofshort 62 futures. One week later (week 2) the futures price fell to 99.07, so our accountwas credited 62 x (102.26 - 99.07) = +197.78. The amount of interest we were ableto earn on this credit for 8 weeks at 8.00% was +197.78 x .08 x 56/365 = +2.43. Thecash flow resulting from futures variation is shown in Figure 5-1 under the “variation”column, and the resulting interest is shown under the “interest on variation” column.The total interest is +1.36.
We can now summarize all the profit and loss components resulting from ourposition:
Original Hedge Adjustments Option Carrying Costs Variation Costs-138.83 +205.27 ~4,99 +1.36
The total profit is -138.83 +205.27 -4.99 +1.36 = +62.81. How much did thetheoretical pricing model predict we would make? We bought 100 calls worth 3.88 for3.25 each, for a total theoretical edge of 100 x (3.88 — 3.25) = 100 x .63 = +63.00. Inother words, the theoretical model came very close to predicting the actual profit fromthe position.Using an Option’s Theoretical Value 87
In our example, the profit and loss was made up of four components. Two of thesewere profitable (the adjustments and the variation costs} while two were unprofitable(the original hedge and the option carrying costs). Is this always the case? It is impossibleto determine beforehand which components will be profitable and which will not. Onecould just as easily construct an example where the original hedge was profitable andthe adjustments were not. The important point here is that if a trader’s inputs arecorrect, in some combination he can expect to show a profit (or loss) approximatelyequal to that predicted by the theoretical pricing model.
Of all the inputs, volatility is the only one which is not directly observable. Wheredid our volatility figure of 18.3% come from? Obviously, one can’t know the futurevolatility, but in this case the author took the ten price changes in Figure 5-1 andcalculated the annualized standard deviation of the logarithmic changes (the volatility}.Hence, the volatility of 18.3% was the correct volatility for this series of price changes.The complete calculations are given in Appendix B.
In the foregoing example we assumed that the market was frictionless, that noexternal factors affected the total profit or loss. This assumption is basic to manyeconomic models, including the Black-Scholes Model. In a frictionless market weassume that:
1, Traders can freely buy or sell the underlying contract without restriction2. All traders can borrow and lend money at the same rate.
3. Transaction costs are zero.
4, There are no tax considerations.
A trader will immediately realize that option markets are not frictionless, since eachof the above assumptions is violated to a greater or lesser degree in the real world. Forexample, in certain futures markets there is a daily limit on the amount of allowableprice movement. When this limit ts reached, the market is said to be locked, and nofurther trading can take place until the price of the futures contract comes off its limit.>Clearly, in such a market the underlying contract cannot always be freely bought or sold.
Additionally, individual traders cannot generally borrow or lend money at the samerate as large financial institutions. If a trader has a debit balance, it will cost him moreto carry that debit; if he has a credit balance, he will not earn as much on that credit.There is a spread, and perhaps a fairly large one, between a trader’s borrowing andlending rate. Fortunately, as we discussed in Chapter 3, the interest rate component isusually the least important of the inputs into a theoretical pricing model. Even thoughthe applicable interest rate may vary from trader to trader, in general it will only causesmall changes in the total profit or loss in relation to the profit or ioss resulting fromother inputs.
3. It is sometimes possible to circumvent the problem of a locked market by either buying or selling thephysical commodity rather than the