that will be hurt byhigher-volatility estimates but that would beneﬁt from a lowerassumption.One fact not obvious from Table 9.6 is that vega is linear. This isillustrated in Figure 9.3, which looks at the value of the 40 strike calland put under a range of volatility assumptions.Figure 9.3 shows that options premiums increase at a constantrate as the volatility estimate goes up: a 10% increase in volatilitywill result in a premium increase equal to 10 times the increasecaused by a 1% rise in volatility.
Table 9.6MMMMCallMMMM
M M M MPutM M M M
Premium
Vega
Premium
Vega
35 Strike
$6.02
0.048
$0.77
0.048
40 Strike
$2.90
0.078
$2.63
0.078
45 Strike
$1.18
0.075
$5.91
0.072
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The Options Strategist
$6Option Value
$5$440 Call40 Put
$3$2$1$00%
20%
40%
60%
Volatility Assumption
Figure 9.3 Option Prices vs. Volatility
The Greeks: RhoThe last of the Greeks is rho, which measures the impact a 1%change in the risk-free rate will have on option prices. Take the following option prices as a starting point:Stock:3-month 50 call:3-month 50 put:
$50$3.69$3.24
The prices above assume a 35% volatility, a 4% risk-free rate, andno dividends. If we recalculate the options’ prices assuming a 5%risk-free rate, we obtain the following:3-month 50 call:3-month 50 put:
$3.75$3.19
Our ﬁrst observation is that higher rates lead to more expensivecalls and lower values for puts. But we know from experience thatwhen short-term rates increase by 1% we have more importantissues to worry about than the value of our options going up ordown by $0.05 or $0.06. Calculating rho may be of interest to market makers and specialists, but for the average investor and trader,this is one Greek lesson they can skip.
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The Relationship Among the GreeksIf you think gamma, delta, and theta spell confusion, hang in therebecause this is where we try to tie it all together. Start with Table 9.7,where we recap which Greek is associated with which option position (we have left out rho because it is of little practical value).Stock positions are relatively easy: long stocks, by deﬁnition, create a long delta position; short stocks result in negative delta. Sincea stock’s delta will always be 1.00, it has 0 gamma (the delta doesn’tchange), and a stock’s price does not incorporate any time premiumso it has neither theta (no time decay) nor vega (no volatilitypremium).It gets interesting when we start comparing the Greeks associatedwith various options positions. Notice that all options positionshave the same sign for gamma and vega; all options positions haveopposite signs for gamma (and vega) and theta. Is this coincidentalor meaningful?A long call position will beneﬁt if the stock is volatile (longgamma) and if the option’s implied volatility increases (long vega),but it will suffer from the passage of time (short theta). The samecan be said for a long put position, even though the call buyer isanticipating volatility to the upside, while the put buyer is forecasting volatility to the downside. The straddle buyer is looking forpure volatility (up or down). He will beneﬁt if implied volatilityincreases and will have to endure the negative impact of time decay.It all ties in. Options buyers are looking for movement in the priceof the underlying and higher option prices due to increased impliedTable 9.7Delta
Gamma
Theta
Vega
Long Stock
⫹
0
0
0
Short Stock
⫺
0
0
0
Long Call
⫹
⫹
⫺
⫹
Short Call
⫺
⫺
⫹
⫺
Long Put
⫺
⫹
⫺
⫹
Short Put
⫹
⫺
⫹
⫺
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The Options Strategist
volatility, but they must be prepared to see the value of their optionserode as expiration nears. Translated into Greek this last sentencebecomes: long option positions result in long gamma, long vega,and short theta.For option writers, a mirror image is created. Writers are hurt byhigh stock volatility (short gamma), but they beneﬁt from decreasedimplied volatility (short vega) and the passage of time (long theta).The Greeks are tied to one another: if the gamma of a position isknown, then so are its vega and theta. A position’s complete picturecan therefore be drawn from its delta (which tells us if it’s bullish,bearish, or neutral) and from its gamma (long volatility or shortvolatility).When traders talk about buying volatility or shorting volatility,they are usually referring to positions that have little or no directional bias (i.e., neither bullish nor bearish), but will either beneﬁtfrom movement in the underlying (buying volatility) or its lackthereof (shorting volatility). A long volatility position is created bybeing net long options, a short volatility position by being net shortoptions. But heed this warning: a position that is net long options—by being long in-the-money calls and short a greater number ofdeep out-of-the-money calls, for instance—can turn into a positionthat is net short options if the underlying moves by a sufﬁcientamount in the wrong direction.
Understanding the Black-Scholes OptionPricing EquationThe ﬁrst time you saw the Black-Scholes option pricing model youprobably dismissed it as rocket science and moved on with your life.We can help you make sense of the model in the next few paragraphs.First, it is important to understand that the assumption underlying the Black-Scholes model is that a call option can be replicatedby trading the underlying security. Take a look at the equation,where C is the value of a call option:C⫽SN(d1) ⫺Xe⫺rt N(d2)There are two parts to the main body of the equation. The ﬁrst oneis SN(d1). Buying a call is a bullish position. If one is going to replicate a call option by trading the underlying stock one must purchase
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235
shares of this stock. That is the S in the ﬁrst part of the equation. Howmany shares of stock should be purchased? We have seen that a stockand its options move up and down at different rates, and we havedeﬁned the rate at which options move as delta. N(d1) calculatesdelta. If investor A buys a call option on a stock and investor B buysdelta shares of the same stock, their proﬁt or loss will be the same forsmall moves in the price of the stock. So SN(d1) tells us that to replicate a call, we need to purchase delta shares of stock.But buying delta shares of stock is bound to cost more than purchasing an option. This is where the second part of the equationcomes in. The second part is ⫺Xe⫺rt N(d2). Let’s start with e⫺rt. Thisis the formula used to calculate present value; it takes a given dollar amount in the future and calculates the value of those dollarstoday based on an interest rate (r) and the time period involved (t).X is the option’s exercise price. So Xe⫺rt is the present value of theoption’s exercise price. Think about it this way: if you know with100% certainty that you are going to exercise a call option when itexpires, how many dollars do you need to set aside today to havesufﬁcient funds to pay for the stock upon exercise? The answer isthe number of dollars that will, while earning interest, end up beingequal to the option’s strike price. This is Xe⫺rt.Notice next that there is a negative sign in front of the Xe⫺rt. Thistells us to borrow funds (the positive sign in front of the SN(d1) toldus to invest funds). But the amount to be borrowed is not the fullXe⫺rt rather it is adjusted by N(d2). This last expression calculatesthe probability that the call option will be in-the-money at expiration. So the amount of funds to be borrowed is equal to the presentvalue of the option’s strike price multiplied by the probability thatthe call will end up in the money.To summarize: the Black-Scholes equation tells us that we canreplicate an option by purchasing delta shares of the underlyingstock and by ﬁnancing part of this purchase with borrowed funds.The shares create upside potential, and the borrowed funds give usthe leverage so many options buyers are seeking.As underscored in Chapter 4 and again in this chapter, delta doesnot remain constant. So an investor replicating a call by purchasingdelta shares of stock would have to increase the number of sharesheld by purchasing additional shares as the stock went up (remember, delta is increasing) and decrease the number of shares held byselling part of the original position if the price of the stock fell (delta
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The Options Strategist
is decreasing). If this sounds difﬁcult, it is. Some investors were atone point (mostly before the market break of October 19, 1987) replicating long put positions to insure their stock portfolios using stockindex futures and adjusting their positions by selling more/buyingback some of their short contracts as the market dropped or rallied.The technique was then known as dynamic hedging (or portfolioinsurance), but it proved to be ineffective on the one day they neededit most.
A Note on BetaThroughout our deciphering of the Greek alphabet no mention hasyet been made of beta. Yet this is probably the ﬁrst Greek letterinvestors learn, in such expressions as “high-beta stock” and “lowbeta portfolio.” How does this letter relate to options?In fact, it doesn’t. Beta is a measure of relative risk, which couldbe taken as a proxy for relative volatility. Beta tells us how risky astock is relative to a given benchmark. Benchmarks can vary. TheStandard & Poor’s 500 Index is sometimes used, but so is the NYSEComposite Index® and the Wilshire Total Market Index. A stockwith a beta of 1.00 can be expected to rise 1% when the benchmarkgoes up 1%, and falls 1% with a market drop of 1%. Stocks withbetas greater than 1.00 are expected to rise and fall more than thebenchmark; those with betas lower than 1.00 will ﬂuctuate less.The problem with beta is that it does not tell you how volatile astock truly is. If High Flying Inc. has a beta of 1.40 we expect it tobe more volatile than the overall market, but what is the overall market doing? Is it staying in a narrow trading range, or is it rising andfalling by a few percent on a daily basis? When we calculated historical volatility in one of the earlier sections, we only needed thestock’s price. We made no reference to any benchmark and werelooking at volatility independently of other stocks or indices.
Option Pricing and Interest RatesWe noted earlier in this chapter the impact changes in interest rateshad on the value of puts and calls (see “The Greeks: Rho”). You will
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237
recall that higher rates mean higher call prices and lower put prices,assuming everything else is held constant. You also will recall thatchanges in interest rates have a relatively minor impact on premiums: a change from 4% to 5% in the risk-free rate increased thevalue of the at-the-money 50 call by $0.06, from $3.69 to $3.75. Mostinvestors will consider this to be small change.There are, however, some option market participants for whominterest rates are central: specialists and market makers. Take theYAWN 45-day calls and puts, which in Chapter 4 (see the section oninterest rate and dividends) we valued at $1.81 and $1.55 respectively, based on a 4% interest rate assumption. Let’s round the callprice to $1.80 to make things more true to life. And ﬁnally, let’sassume that a market maker purchases 100 shares of YAWN at $60,buys one 45-day 60 put at $1.55, and writes one 45-day 60 call at$1.80. What is this market maker’s risk and proﬁt potential?There is no risk to the downside because the market maker is longthe 60 put. If the price of the stock drops, the put can be exercisedand the stock sold for $60. There is also no upside potential. IfYAWN rises in price, the 60 call will be assigned and the marketmaker will be forced to sell 100 shares at $60. If the stock closes atexactly $60 at option expiration, then the call and put will bothexpire worthless and the market maker will keep the 100 shares,which will be worth $60.So, with YAWN moving lower, the ﬁnal value of the position is$60. If YAWN rises, the ﬁnal value of the position will be $60; dittoif YAWN remains unchanged. No matter what the price action inthe stock, this position will be worth $60 at option expiration. Is thisgood or bad? The net cost to establish the position was the $60 paidfor the stock, plus the $1.55 to purchase the put, less the $1.80received from writing the call, or a net cost of $59.75. The marketmaker realizes a proﬁt of $0.25, which represents a 0.42% return,which in turn annualizes to 3.39%. Not great, but in line with therisk-free rate of 4%.Yet market makers usually post two-sided markets that include abid-ask spread. So the market on the 60 call could be $1.75 to $1.85and the market on the 60 put could be $1.50 to $1.60. What happens if the market maker is able to purchase the put at his bid priceof $1.50 and write the call at his offering price of $1.85? The totalposition can then be initiated for $59.65, will generate a proﬁt of$0.35 and return 0.59%, or 4.76% annualized. The market maker is
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The Options Strategist
now earning more than the risk-free rate on his “risk-free” optionposition. Not bad, even if our numbers do not take into account themarket maker’s (very low) transaction costs on options and stock.There are two lessons to be learned here. The ﬁrst is that it is possible to build no-risk, or low-risk positions using options. This isuseful for specialists and market makers whose primary goal is toremain in business. (The secondary goal is to make money, but it isvery difﬁcult to make money if you have gone out of business.) Thesecond lesson is that there is more than one way to skin a cat. Letus illustrate.The position built by the market maker was:⫹Stock⫹Put⫺CallThis turned out to have a ﬁnal value of $60 no matter what theprice of the stock at option expiration. This position is known as aconversion. It was, in fact, a $60 “Treasury bill.” So the positionreally was:⫹Stock⫹Put⫺Call⫽T-BillBuying a stock and a put and writing a call (assuming the options’strike prices and expiration dates are the same) looks like a T-bill.This equation is known as the put-call parity equation. It shows howto build a risk-free position, but it also shows how to build equivalent positions. Take the case of an investor who is long a stock andpurchases a put option to protect it. She is long stock, long put, andby rearranging the terms of the above equation we get the following:⫹Stock⫹Put⫽T-Bill⫹CallThis investor could replicate her long stock/long put position bypurchasing a Treasury bill and a call option. Note that the positionsare only equivalent if the T-bill’s maturity value is equal to theoption’s strike price and if the expiration dates and exercise pricesof the put and the call are identical. Think back to some of the payoff diagrams in earlier chapters. How does the buy call diagramcompare to the protective put diagram? They both have limited riskto the downside and unlimited proﬁt potential to the upside. Thisis not surprising since they are equivalent strategies. (Compare alsobullish credit and debit spreads in Chapter 6.)Note that equivalent strategies are not identical. The main difference is usually in the risks of early assignment. A strategy short in-
Options 901: Advanced Theoretical Notions
239
the-money options will have a greater probability of early assignment than the equivalent strategy with long in-the-money options.Still, it is useful to know that a short stock/long call position (wherea trader buys a call to protect a short stock position) is equivalent toa long put position (an issue we discussed in Chapter 6). The latterposition is easier to initiate (only one trade), has lower transactioncosts (one option commission versus one stock and one option commission), does not require borrowing stock, and will be more readily approved by most brokers.
Extra! Options Offered Below Intrinsic Value!The above headline was inspired by the following quotes:SPX:Mar 1250 put:
999.44$245.80–$247.80
Back to basics: an option’s intrinsic value is equal to its in-themoney amount, that is, the difference between the value of theunderlying and the option’s strike price. By doing the math on theMarch 1250 put we see that this six-month option has an intrinsicvalue of $250.56, yet it is offered at $247.80. Is this, as the numerousofferings in our mailbox keep telling us, a “free gift”?Not quite. Options on the Standard & Poor’s 500 Index (SPX) areEuropean style, only exercisable immediately prior to expiration. Ifthis option were American style, could it not be purchased and exercised immediately to capture its full intrinsic value? Yes, and no,and there are two lessons to be learned from the above.The ﬁrst point is that deep-in-the-money European-style putoptions frequently trade at a discount to their intrinsic value. Thereason is as follows: if a trader purchases this put at $247.80 becauseit is trading at a discount to its intrinsic, he must wait a full sixmonths before exercising it. Obviously holding this option for sixmonths creates a major market risk. The SPX could rally and theoption would be in-the-money by only 200 or even 100 points byexpiration date. (Yes, as the bulls keep reminding us, it could alsobe out-of-the-money). To capture the option’s discount the traderwould have to hedge his long put position by buying the component stocks of the S & P 500 Index. Any loss realized on the put as
240
The Options Strategist
the market rallied would then be made up by gains on the stocks.This would tie up a lot of money, although this trader would endup collecting all of the dividends paid by the component stocks overthe coming six months.If we take the midpoint of the option’s bid-ask spread, $246.80,the discount to intrinsic value is $3.76. This represents 0.38% of theindex’s value, or a 0.76% annualized discount (since this is a sixmonth put). This 0.76% represents the difference between the costof borrowing funds and the yield on the Standard & Poor’s 500Index. Unfortunately, there is no free lunch: the discount on the putsimply reﬂects the net cost of carrying the index’s component stocksuntil option expiration. Whatever is gained by buying the option ata discount is taken away through the cost of the hedge.The second lesson is that even if this put were an American-styleoption, buying it at $247.80 would not necessarily lock in a risk-freeproﬁt. Remember that an American-style index option will settle, ifexercised, based on the closing value of the index on the day it isexercised. If this option is purchased with a few hours left in thetrading day and is later exercised, the amount obtained upon exercise may end up being worth less than its purchase price. Occasionally, you will see American-style index options trading at adiscount to intrinsic. This usually happens when the market is moving strongly in one direction and option traders start pricing optionsbased on where they expect the market to be in a few minutes,rather than where it is at the present.So the next time you see an option quote that looks too good tobe true, it probably will be.
C
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10A
P
T
E
R
TACTICALCONSIDERATIONS
T
he preceding chapters in this book focused on strategiesthat can be easily implemented. In this last chapter wereview some of the tactical considerations that complement those strategies.
Risk ManagementSome investors have sworn off options and other derivatives. Theyargue that if Long-Term Capital Management (with its star-studdedstaff that included a Nobel Prize winner), Enron (M.B.A.s from thebest schools, political connections, and top-ﬂight accountants), andtheir cousin Jeff (who devised this “can’t lose” trading system) allwent up in smoke trading options and other derivative securities,241Copyright 2003 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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The Options Strategist
why should they stick their necks out only to have their headshanded to them on a platter?In presenting the strategies in this book, we made every effort toexplain the associated risks. To these we would like to add threegeneral risk management guidelines that can minimize your risksof implosion.The “It’ll Never Go There” AttitudeUpon learning of an option strategy that has a 99% chance of beingproﬁtable you: (a) hit the speed-dial to call your broker because themarkets are closing in 12 minutes; (b) dismiss it out of hand becauseit sounds too good to be true; (c) investigate further because youwant to know what happens the 1% of the time when it is notproﬁtable.Obviously, you chose (c) and learned that if things go the wrongway, your trading account would get wiped out. But there is just a1% chance that this will happen, so what do you do? Do you goahead and risk it? Before making a decision, look at the probabilityof getting wiped out (POGWO) as a function of the number of timesthis strategy is initiated.Strategy used onceStrategy used 4 times(quarterly over one year):Strategy used 12 times(monthly over one year):Strategy used 50 times(weekly over one year):Strategy used 250 times(daily over one year):
POGWO: 1%POGWO: 3.9%POGWO: 11.4%POGWO: 39.5%POGWO: 91.9%
If you played Russian roulette with a 100-chamber gun and onlyone of the chambers contained a bullet, the probabilities of blowingyour brains out would be identical to the ones above.Certain aggressive options strategies are nothing more than agame of Russian roulette. We do not know why traders enter intothese strategies—if it’s the thrill of pulling the trigger and hearingthe hammer fall on an empty chamber, or if it’s the profound con-
Tactical Considerations
243
viction that “it will never happen to me” because “the stock willnever go there”—but we do know that the odds have a tendency tocatch up with these players.Writing deep out-of-the-money options appears to be the weaponof choice for those who like Russian roulette. Although short callpositions have theoretically unlimited risk, most of the damage hasbeen caused by short puts, both equity and index options. Thelargest binge of aggressive put writing in a bull market ended inOctober of 1987 with bodies littered all over the street. As a result,brokerage ﬁrms tightened up their options trading requirements,but we still come across traders holding guns to their heads. Theonly way to reduce one’s POGWO to zero is to throw away the gun.LeverageOptions are often blamed for various ﬁnancial disasters when thetrue culprit is leverage. It is possible, for instance, to wipe out anaccount’s total value by purchasing U.S. Treasury bonds (fully guaranteed by the U.S. government) if the position is leveragedsufﬁciently.Chapter 3 showed how leverage works both ways. Does thismean it should simply be avoided? We do not think so. Take theexample of an investor with $100,000 who has decided to allocatethis sum to an option writing program. He plans to write puts onstocks he is willing to own and, if assigned on the puts, to write covered calls on these stock positions. All positions will be of equal dollar value, ﬁxed at $10,000. If this investor writes puts on 10 stocks,no leverage will ever be involved: if all 10 put series are assigned,he will be forced to purchase $100,000 worth of stocks for which hehas the cash.This investor also could decide to write puts on 11 stocks. His reasoning is as follows: if the stocks are properly diversiﬁed the oddsare relatively low that all 11 puts would be assigned at the sametime. If some stocks are purchased on assignment at the ﬁrst optionexpiration, calls will be written immediately. By the second optionexpiration more puts may be assigned, but there is also a chance thatsome of the calls will be assigned and some of the stock holdingssold. For all 11 puts to be assigned would probably require the overall market to correct substantially, in which case he would be forced
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to buy $110,000 worth of stock. Under this scenario he would haveto borrow $10,000 from his broker. That’s a conservative use of margin and a comfortable degree of leverage.Now think of this same investor writing puts on 12 stocks, or 15,or 20, 25, 30, or even 50 stocks. At some point he would becomeuncomfortable. If not, he would end up short puts on 50 differentstocks and have to assume the obligation to purchase $500,000worth of shares with only $100,000 to back him up. This, by the way,is about as far as he could leverage his put writing, assuming hisbroker’s margin requirements are the exchange minimum.Holding 50 short put positions will not be a problem as long asthe market (or at least the relevant stocks) cooperates, i.e., remainsstable or rises. But any downturn will generate a margin call, evenif none of the puts has been assigned. There is no room to maneuver. This investor will have to meet the margin call by covering someof his short puts or selling stocks he may have bought on optionassignments. He is being forced out of his position at what is probably the worst time to liquidate. He is selling low instead of sellinghigh.Leverage can be a useful tool, but many traders and investorshave been forced to liquidate because they had leveraged positionsto the last nickel. As options strategists at a brokerage ﬁrm wewould get phone calls from brokers that went like this: “I have aclient with $22,544 of excess margin. How many of the XYX December 40 puts can he write?” These clients—and their brokers—wereusing our ﬁrm’s margin requirements as a trading guideline: “If mybroker lets me write 12, it must be okay to write 12.” Remember thatyour brokerage ﬁrm’s margin requirement is for their protection,not yours.Overleverage proved to be the downfall of Long-Term CapitalManagement. The company was correct in predicting that thespread between different debt instruments would narrow. The problem was that it was leveraged to such an extent that when thespreads started to widen the company did not have the necessarystaying power and had to call it quits. Part of their problem mayhave been of the “it’ll never go there” type (i.e., these spreads are athistorically high levels—they cannot get any wider). Don’t waituntil you are awarded a Nobel Prize to learn from this experience.
Tactical Considerations
245
DiversiﬁcationIn the late 1990s, none of us held enough tech stocks as they repeatedly hit new highs. Then, during the tech bust and the demise of thedot-coms, we all held too many tech stocks. It takes discipline notto pile into the sector of the month—to hold energy stocks whenbiotech is all the rage, or banking stocks when health care is the onlyoutperforming sector.Even conservative investors who limit their options trading tobuy/writes should diversify across sectors. There is nothing newhere, but how often is this pillar of sound investing ignored?
Earning Your StripesMost brokerage ﬁrms have different levels of approval for optionstrading. A representative example would be:■
level 1: covered writing
■
level 2: buying calls and puts
■
level 3: spreading (usually within certain limits)
■
level 4: writing equity puts
■
level 5: writing uncovered calls (sometimes two levels—one forequity options, one for index options)
If you are new to options trading you will have to earn yourstripes. Don’t expect more than a level 2 approval until you havegained some experience. And when you are refused approval at aspeciﬁc level, don’t take it personally. When we returned to the U.S.after a few years offshore, we moved our option account to a nationally recognized ﬁrm. Our application was rejected for writing putsfor lack of trading experience (although they did permit us to writecovered calls). We had at that point been trading options for closeto two decades and had, in fact, trained the said ﬁrm’s brokers inthe uses of options. A follow-up letter straightened things out, butdon’t forget that most ﬁrms will err on the side of caution whenlooking at your option account application.
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Options in Tax-Deferred PlansAt some point you may want to know what you can and cannot dowith options in your tax-deferred account (IRA, 401(k), SEP, etc.).This line of inquiry will make getting approved for options in yourregular account look like a piece of cake. The only constant we havefound among brokerage ﬁrms is that they all will let you write covered calls. Besides this strategy, anything goes: some ﬁrms let youbuy protective puts, others don’t; some ﬁrms let you buy calloptions, others don’t; some ﬁrms permit the writing of equity puts,others don’t. There is even one ﬁrm that completely disallowsLEAPS.You should also be aware that some ﬁrms have two sets of rulesfor their tax-deferred accounts: there are the general rules, whichpermit covered writing and maybe the purchase of puts and calls,and the preferred client rules, which allow more strategies. A preferred client can be someone who has an established trading recordwith the brokerage or a sufﬁciently large portfolio. It may be worthasking your broker what it takes to get upgraded if you are looking