0-6 calculates that the position delta is zero at35 days to expiration. In Figure 10-6, the 35-day line is the upper lineon the left and the lower line on the right. This line shows that theposition approximately breaks even between stock prices of 68 and 76.The position, however, begins to proﬁt below 68 and to lose above 76.A delta-neutral position therefore is safe from stock-price ﬂuctuationsover a ﬁnite range, not an inﬁnite one. Figure 10-6 clearly illustratesthe potential for loss if the stock price rises sharply.Third, the position gamma, vega, and theta in Table 10-6 arethe same as those risks in Table 10-3. The gamma in both tables is
Managing Position Risk • 329
20, the vega in both is 42, and the theta in both is –170. A deltaneutral position, therefore, is not immune from the risks of changingdelta, changing volatility, or changing time. What a trader might doto manage these risks is discussed next.
Neutralizing Position GreeksNeutralizing position Greeks is explained with two examples that analyze multiple-part positions. The ﬁrst example assumes that interestrates are zero, and the second assumes that they are positive. The zerointerest-rate example is instructive because it demonstrates clearly the“equal and opposite” relationship that exists between some of theGreeks. Also, this example applies generally to options on futures. Positive interest rates apply to options on stocks, where cost of carry is aconcern. Cost of carry is discussed in Chapter 6.The ﬁrst example analyzes the action of a hypothetical trader namedMatthew who wants to neutralize his position risk. In Table 10-7A, thegamma, vega, and theta of Matthew’s initial position are not neutral (≠0).His position is listed at the top of the table and consists of 35 short80 Puts, 60 short 85 Calls, 120 long 90 Calls, and 2,060 short shares.Prices are shown, but they are not necessary to analyze position Greeks.The bottom portion of the table lists the Greeks of the individual options.A position Greek in Table 10-7A is the product of the quantity ofshares represented by the option position and the option’s individualGreek. The delta of the 80 Put position of 885 in row 1, for example, is calculated by multiplying –3,500 and –0.253. The numbers inthe tables that follow were calculated in the Spread Positions screenin Op-Eval Pro. Slight differences are due to rounding.The total Greeks in row 5 are simply the sums of the positionGreeks in columns 3 through 6. Matthew’s position, as illustrated inTable 10-7A, has a total delta of 0, a total gamma of 104, a total vegaof 358, and a total theta of –780. These Greeks indicate that thisposition will proﬁt if there is big stock-price change (positive gamma)and if implied volatility rises (positive vega). It will lose if time passes
330 • Trading Options As a Professional
Table 10-7A
Neutralizing Position Risk—Interest Rate Zero: The InitialPosition
Position: Short 35 80 Puts @ 1.70.Short 60 85 Calls @ 4.73.Long 120 90 Calls @ 2.61.Short 2,060 shares @ 86.10.
Row
Option/Stock
12345
80 put85 call90 callStockTotals
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Price
Qty
Delta
Gamma
Vega
Theta*
1.704.732.6186.10
35601202,060
8853,3974,5722,0600
1072254360104
3687751,5010358
7931,6943,2680780
Individual Greeks
80 Put85 Call90 Call
Delta
Gamma
Vega
Theta*
0.2530.5660.381
3.053.753.63
10.5012.9112.50
22.6828.2427.23
Assumptions: Stock price, 86.10; days to expiration, 53; volatility, 32%; interest rate, 0%; dividends, none.*7-day theta.
(negative theta). Small stock-price changes will have a nearly zeroimpact on proﬁt (zero delta).If Matthew is concerned that the underlying price will trade in anarrow range, which would expose the position to losses from timedecay, or if he is concerned that implied volatility will fall, then hemust act to reduce these risks. Which Greek, however, should be thefocus of Mathew’s risk-management effort?Table 10-7B demonstrates three alternatives that neutralizeMatthew’s position risk. The purpose of examining three alternativesis to determine the impact on the other Greeks when one is neutralized. Each alternative starts with the position in Table 10-7A andmakes a trade that neutralizes one Greek. Alternative 1 neutralizes the
Managing Position Risk • 331
position gamma. Alternative 2 reduces the position vega to zero, andAlternative 3 focuses on the position theta. All three approaches usethe 85 Calls to neutralize the targeted Greek. However, Matthewcould choose any option.Matthew ﬁrst looks at the position gamma of 104 (Table 10-7A,row 5, column 4). He knows that he must sell some quantity of85 Calls to reduce it to zero. Short calls, remember, have negativegamma, so selling calls reduces the gamma of the position. The quantity of options that neutralizes a position Greek is calculated by dividing the position Greek by the Greek of the individual option. In thiscase, the position gamma of 104 divided by the gamma of the85 Call of 3.75 yields 28. Selling 28 of the 85 Calls therefore reducesthe position gamma to zero. However, simply selling 28 of the 85 Callswill add negative delta to the total position, and Matthew wants hisTable 10-7B
Neutralizing Position Risk—Interest Rate Zero: Selling 85 Callsto Neutralize the Greeks
Alternative 1: Neutralizing Gamma by Selling 85 CallsPosition gammagamma of 85 Callquantity of 85 Calls1043.7527.7 ≈ 28The position gamma is positive, so 85 Calls must be sold.The delta-neutral trade: Sell 28 85 Calls (delta is 0.566).Buy 1,585 shares (2,800  0.57  1,585).Alternative 2: Neutralizing Vega by Selling 85 CallsPosition vegavega of 85 Callquantity of 85 Calls35812.9127.7 ≈ 28The position vega is positive, so 85 Calls must be sold.The delta-neutral trade: Sell 28 85 Calls (delta is 0.566).Buy 1,585 shares (2,800  0.57  1,585).Alternative 3: Neutralizing Theta by Selling 85 CallsPosition thetatheta of 85 Callquantity of 85 Calls78028.2427.6 ≈28The position theta is negative, so 85 Calls must be sold.The delta-neutral trade: Sell 28 85 Calls (delta is 0.566).Buy 1,585 shares (2,800  0.57  1,585)
332 • Trading Options As a Professional
position to remain delta-neutral. In addition to selling 28 calls, therefore, Matthew must buy 1,585 shares of stock because the delta of theshort 85 Call is –0.566 (28  100  0.566  1,585).Alternative 2 in Table 10-7B illustrates a second approach to neutralizing risk by focusing on the position vega. Since the position vegais 358, Matthew must sell options to bring it to zero. Dividing theposition vega of 358 by the vega of the 85 Call of 12.91 yields 28.Selling 28 of the 85 Calls therefore reduces the position vega to zero.Again, however, Matthew wants to keep his position delta at or nearzero, so he buys 1,585 shares of stock. Thus, bringing the position vegato zero required Matthew to make the same delta-neutral trade as inAlternative 1 that neutralized gamma.Neutralizing a negative theta also requires selling options. In thisthird alternative, dividing the position theta of –780 by the theta of the85 Call of –28.24 yields 28, the same number of 85 Calls as in the preceding two trades. The two parts of the delta-neutral trade that neutralize theta therefore are the same as Alternatives 1 and 2. Matthewsells 28 of the 85 Calls and buys 1,585 shares of stock.Matthew’s next step is performed in Table 10-7C. He adds 28 short85 Calls and 1,585 long shares to his initial position and calculatesthe new Greeks. The new position has 88 short 85 Calls (short 28 plusshort 60) and 475 short shares (short 2,060 plus long 1,585). Row 5 ofTable 10-7C presents the Greeks of the new position.As row 5 in Table 10-7C indicates, the new delta is 1, the newgamma is –1, the new vega is –3, and the new theta is 11. None ofthe Greeks is exactly zero owing to rounding, but they are all close tozero. One of the many decisions that market makers must make is howclose is close enough? Based on his own experience, judgment, andcomfort level, Matthew is satisﬁed with these Greeks.The conclusion of Tables 10-7A through 10-7C is that when interest rates are zero, a trade that neutralizes either gamma, vega, or thetaalso neutralizes the other two. Traders therefore do not have to worryabout which Greek to neutralize! When interest rates are positive,however, the situation is different.
Managing Position Risk • 333
Table 10-7C
Neutralizing Position Risk—Interest Rate Zero: The NeutralizedPosition
Position: Short 35 80 Puts @ 1.70.Short 88 85 Calls @ 4.73.Long 120 90 Calls @ 2.61.Short 475 shares @ 86.10.Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Col 7
Row
Optionor Stock
Price
Qty
Delta
Gamma
Vega
Theta*
12345
80 Put85 Call90 CallStockTotals
1.704.732.6186.10
3588120475
8854,9814,5724751
10733043601
3687931,136 2,4851,501 3,26800311
Individual Greeks
80 Put85 Call90 Call
Delta
Gamma
0.2530.5660.381
3.053.753.63
Vega
Theta*
10.50 22.6812.91 28.2412.50 27.23
Assumptions: Stock price, 86.10; days to expiration, 53; volatility, 32%; interest rate, 0%; dividends, none.*7-day theta.Conclusion: When interest rates are zero, if one Greek is neutralized, then all Greeks are neutralized.
Neutralizing Greeks when InterestRates Are PositiveTables 10-8A through 10-8C present a three-part example similar toTables 10-7A through 10-7C, except that the short-term interest rate is5 percent. Table 10-8A presents an initial delta-neutral position (delta –2), with a gamma of –92, a vega of –143, and a theta of 326.Hypothetical trader Matthew again confronts three possibleapproaches to neutralizing the Greeks in his initial position. The threealternatives in Table 10-8B all involve buying the 60 Puts, but eachtrade targets a different Greek. In Alternative 1, Matthew neutralizesthe gamma by buying 15 of the 60 Puts (92  5.97 ≈ 15). He also buys
334 • Trading Options As a Professional
Neutralizing Position Risk—Interest Rate Positive: The InitialPosition
Table 10-8A
Position: Short 25 55 Puts @ 1.18.Long 30 60 Puts @ 3.48.Short 35 65 Calls @ 0.60.Long 1,700 shares @ 58.00.
OptionRow or Stock12345
55 Put60 Put65 CallStockTotals
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Price
Qty
Delta
Gamma
Vega
Theta*
1.183.480.6658.00
2530351,700
6881,7026681,7002
127179145092
1062772240143
2883613990326
Individual Greeks
55 Put60 Put65 Call
Delta
Gamma
Vega
Theta*
0.2750.5670.191
5.075.974.13
 7.85 9.25 6.40
11.5112.0211.40
Assumptions: Stock price, 58.00; says to exp., 60; volatility, 28%; interest rate, 5%; dividends, none.*7-day theta.
800 shares of stock so that his ending position remains delta-neutral.In Alternative 2, he buys 15 of the 60 Puts, which neutralizes theposition vega (143  9.25 ≈ 15) and he buys 800 shares to maintaindelta-neutrality. Alternative 2 is the same as Alternative 1.Alternative 3 in Table 10-8B, which targets theta, however, is different. Bringing the position theta of 326 to 0 requires Matthew tobuy 27 of the 60 Puts (326  12.02 ≈ 27). The question then is, “Whythe difference?”Table 10-8C has three parts that answer this question. The upperpart of the table illustrates Matthew’s new position after he buys 15 ofthe 60 Puts and 800 shares of stock. His four-part position now consists of 25 short 55 Puts, 45 long 60 Puts (up from 30), 35 short
Managing Position Risk • 335
Table 10-8B
Neutralizing Position Risk—Interest Rate Positive: Buying 60Puts to Neutralize the Greeks
Alternative 1: Neutralizing Gamma by Buying 60 PutsPosition gammagamma of 60 Putquantity of 60 Puts92 5.9715.4 ≈15The position gamma is negative, so 60 Puts must be purchased.The delta-neutral trade: Buy 15 60 Puts (delta is 0.55).Buy 800 shares (1,500  0.567  850)Alternative 2: Neutralizing Vega by Buying 60 PutsPosition vegavega of 60 Putquantity of 60 Puts1439.2515.4 ≈15The position vega is negative, so 60 Puts must be purchased.The delta-neutral trade: Buy 15 60 Puts (delta is 0.567).Buy 800shares (1,500  0.567  850)Alternative 3: Neutralizing Theta by Buying 60 PutsPosition thetatheta of 60 Putquantity of 60 Puts32612.0227.1 ≈27The position theta is positive, so 60 puts must be purchased.The delta-neutral trade: Buy 27 60 Puts (delta is 0.55).Buy 1,485 shares (2,700  0.55  1,485)Note: When interest rates are positive, neutralizing theta requires a different number of contracts thanwhen neutralizing gamma and vega.
65 Calls, and long 2,500 shares (up from long 1,700). The positiondelta (–3), the position gamma (–3), and the position vega (–4) are allnearly zero, but the position theta is not. At 146, the position thetaestimates that $146 will be made in one week if the other factorsremain constant. The middle and lower parts of Table 10-8C explainwhy the position theta is 146.The middle part of the table calculates the value of the position asa net debit of $158,300. Debit means that establishing a positionrequires a net payment. Funding the position therefore requiresMatthew to pay interest on borrowed funds or to forego interest on hisown equity capital that could be invested elsewhere.The lower part of Table 10-8C calculates the amount of interestrequired to ﬁnance the position. At 5 percent, interest for one week
336 • Trading Options As a Professional
Table 10-8C
Neutralizing Position Risk—Interest Rate Positive:The Neutralized Position
Position: Short 25 55 Puts @ 1.18.Long 45 60 Puts @ 3.48.Short 35 65 Calls @ 0.60.Long 2,550 shares @ 58.00.
OptionRow or Stock12345
55 Put60 Put65 CallStockTotals
Col 1
Col 2
Col 3
Col 4
Col 5
Col 6
Price
Qty
Delta
Gamma
Vega
Theta*
1.183.480.6658.00
2545352,550
6682,5536682,5503
12726914503
20641622404
1885413990146
Qty

Price

Value
Value of 55 Put position251.182,950 creditValue of 60 Put position453.4815,660 debitValue of 65 Call position350.662,310 creditValue of stock position2,55058.00147,900 debitValue of total Position158,300 debitInterest for one week at 5% on value of total position 158,300  0.05  52  152 ≈ 146 (146  position theta)Conclusion: When interest rates are positive, if either gamma or vega is neutralized, then the other isalso neutralized. The position theta, which is not neutral, offsets the impact of interest. If theta ispositive, it offsets the interest cost. If theta is negative, it offsets the interest income.*7-day theta.
on $158,300 is $152, which is approximately the weekly amountearned from time decay, as indicated by the position theta of 146.The difference of $6 between the calculated interest and the theta is dueto rounding.The conclusion, therefore, from Tables 10-8A through 10-8C hastwo parts. First, when interest rates are positive, if either gamma orvega is neutralized, then the other is also neutralized. Second, theposition theta, which is not neutral, will offset the impact of intereston the position. If the theta is positive, as in this case, it offsets the
Managing Position Risk • 337
interest expense of carrying a debit position. If theta is negative, it offsets the interest income from the invested cash from a credit position.The addition of interest rates therefore means that Matthew mustneutralize either gamma or vega, which will neutralize the other,whereas theta will remain nonneutral and offset the interest factor.The conclusions about theta and a position’s interest factor fromTables 10-8A through 10-8C apply only when there is a signiﬁcantstock position. As discussed below, theta cannot be related to interestwhen there is no stock component in a position. In such positions,theta risk must be viewed differently.
Establishing Risk LimitsThe preceding exercises calculated position risks, tracked how theychange, and explained how they can be neutralized. The exercises donot, however, address how much risk is acceptable. While there areno scientiﬁc answers to questions about acceptable risk, there are someguidelines that traders can use to determine their own limits.Fluctuation in position value is inevitable. Therefore, traders mustdecide, ﬁrst, how much of an adverse ﬂuctuation is tolerable. “CanI withstand a $500 swing or a $5,000 swing in my account equity andstill be able to trade rationally?” Some traders focus on risk limits indollar amounts, and some prefer percentages. Either way, thissubjective and personal decision is fundamental to good risk management. It forms the basis for every risk limit. After a trader establishesthis limit, then position size—quantities of stocks and options—naturally follows and can be determined using the Greeks.Three types of positions and appropriate risk limits are discussednext: delta-neutral positions with a stock component, delta-neutralpositions without a stock component, and directional positions.
Delta-Neutral Positions with a Stock ComponentWhen analyzing delta-neutral positions with a stock component, onecan infer from the exercise in Tables 10-8A through 10-8C that
338 • Trading Options As a Professional
concerns about risk should focus on either gamma or vega. This inference stems from two observations. First, since reducing either gammaor vega also reduces the other, there is no need to focus on both.Second, theta seems to take care of itself because it is related to theinterest component of a position, at least when there is a stock positioninvolved.Position Greeks provide concrete estimates of the proﬁt or loss thatwill result from a one-unit change in the related component. Traderstherefore should study both the historic and implied volatilities of theunderlying and make a subjective decision about “normal ranges.”Does the implied volatility “normally” change by three percentagepoints in a few days or by eight percentage points? A subjective answerto this question leads to a risk limit based on vega. In Table 10-8A, thevega of –143, for example, estimates that $143 will be lost if impliedvolatility rises by 1 percent. If a trader predicts that implied volatility“normally” changes less than 3 percentage points in a few days, thenthe position in Table 10-8A has a normal risk of three times $143, or$429. For a trader with a chosen risk limit of $1,000, this positionseems “acceptable” because a sudden loss of $1,000 requires a7 percent rise in implied volatility, a change that lies outside the rangeconsidered “normal.”The $1,000 risk limit and the belief that a 3 percentage pointchange in implied volatility is “normal” leads to a vega limit of 333.If a position has vega of 333, then a decrease in implied volatility of3 percent would cause a $1,000 loss, and a rise of 3 percent wouldcause the same loss for a position with a vega of –333. Consequently,a trader with this limit would calculate the vega daily and act accordingly to keep the position within the limit.
Delta-Neutral Positions without a Stock ComponentWhen a position does not include stock, the position theta cannotbe related to interest, as in Table 10-8C. However, the position thetacan be related to the position vega and gamma because they have
Managing Position Risk • 339
opposite signs. Assuming the same expiration for all options, a position with positive vega and positive gamma will have negative theta,and vice versa. The setting of risk limits therefore requires a choice.A trader must ask, “Do I hope to proﬁt from time decay at the risk oflosing from rising implied volatility or from a big stock-price change?”or “Do I hope to proﬁt from rising implied volatility or from a bigstock-price change at the risk of losing from time decay?” In otherwords, “Do I want to be net long or net short options?” Once a traderdecides this issue, then limits on vega or theta can be established andmonitored.
Directional PositionsA directional position is a position that intentionally has stock-pricerisk, and the main risk of such a position, of course, is its delta. Theﬁrst determination therefore is, “How much delta can I take on?”Again, there is no scientiﬁc answer to this question. While the maximum risk of a long option is the total price paid plus commissions,traders practicing good risk management generally set predeterminedlimits, known as stop-loss points, at which they close a position andtake a loss that is less than the possible maximum loss. Stop-loss pointscan be stated in a dollar amount, the price of an option, or the priceof the underlying stock. Regardless of how they are expressed, tradersset stop-loss points individually. It is a subjective decision.In the case of short options, the position delta does not fully statethe risk because a stock-price change against the position will causethe position delta to increase adversely, generating a loss that grows atan increasing rate. This effect is known as negative gamma. Stop-losspoints therefore are especially important for short option positions,which also must be monitored continuously.The gamma, vega, and theta risks of directional trades can beadjusted, but the general impact of such adjustments is to also adjustthe delta. The comparison at the beginning of this chapter illustratedthat the call spreads in Table 10-3 had lower Greeks than the outright
340 • Trading Options As a Professional
long calls in Table 10-1. A trader who sells calls with a higher strikeprice in an effort to reduce the vega risk of some existing long callsalso reduces the delta. Since delta is almost always the bigger risk,reducing vega may have little value.Another strategy that reduces the theta risk of long calls is sellingputs with a lower strike price. The positive theta of the short putsreduces the time-decay risk of the long calls, but short puts alsoincrease the position delta, and the increased delta increases positionrisk. Again, the value of reducing theta risk while increasing delta andposition risk is questionable. Ultimately, directional positions mustfocus on delta risk, and the amount of delta risk to assume is a subjective decision.
SummaryThere are both quantitative and subjective elements in managing risk.Every strategy from outright long options to vertical spreads and ratiospreads has unique tradeoffs of proﬁts versus risks. Managing thoserisks requires an understanding of how the risks of individual optionschange and how they interact with each other as market conditionschange. The task of managing risk is also different for directionaltraders and delta-neutral traders.When trading with a directional forecast with long options, it issometimes possible to increase proﬁts and decrease risk at the sametime by managing a position’s delta.Calculating the Greeks of multiple-option positions is straightforward. First, the Greeks of individual options are multiplied by the number of contracts. Second, the Greeks of the total position are the sumof the Greeks of the individual option positions. It is not always easyto anticipate how position Greeks will change because the Greeks ofindividual options change at varying rates as time passes, as volatilityrises or falls, and as the stock price ﬂuctuates above and below thestrike price.
Managing Position Risk • 341
When interest rates are zero, neutralizing either gamma, vega, ortheta also neutralizes the other two. When interest rates are positive,however, neutralizing gamma or vega will neutralize the other, butthe theta will remain nonneutral. With a stock position as part of adelta-neutral position, the position theta is related to the interest component, either the cost of carry or the interest income.Although position risk can be quantiﬁed, only traders can choose arisk limit, which can be stated in dollar terms or in terms of one of theGreeks. If a trader understands the potentials—both good and bad—and combines that knowledge with the stock-price ranges estimatedby historic and implied volatility, then the chances of earning proﬁtsare improved.
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EPILOGUEhis book has discussed several topics that both market makers andadvanced individual traders need to master. Computers assist market makers in trade execution, adjusting prices, and monitoring position risk, and they greatly increase efﬁciency. Computers, however, donot make decisions, and they do not replace the human element inmaking markets. Option market makers still need to be well versed inoptions price behavior, volatility, synthetic relationships and arbitrage,delta-neutral trading, setting bid-ask prices, and managing position risk.For individual traders, the goal of learning how market makers thinkis to improve your skills in entering orders and in anticipating strategyperformance. Knowing that market makers are in a unique business,not in competition with investors or speculators; knowing that they areonly one participant in the market, not the market; knowing that theyface tough decisions, just as you do; and knowing that they make orlose money by assuming risk should give you conﬁdence in tradingwith them.While market forecasting remains an art, an increased understanding of volatility should help you in estimating stock-price ranges,picking price targets, and anticipating how option strategies will perform. Traders of all stripes must be