TomTom, the maker of car navigation systems, is dueto come out with a statement which will be a real sharemover. However, the investor does not know whether itwill affect the share price positively or negatively. Forthat reason he decides to buy a 6-month at-the-moneystraddle on TomTom. Combining the two payoff profilesof a long at-the-money call and a long at-the-money putone will get Figure 10.4. To show the difference betweenthe payoff of buying an at-the-money straddle on TomTom and the profit of buying this straddle, suppose theinvestor paid $3 for the at-the-money straddle on TomTom and the share price is currently trading at $20. Theprofit profile at maturity of buying an at-the-moneystraddle on TomTom would look like Figure 10.5.
10.5
STRANGLE
The strangle is very similar to the straddle with the difference that the call and the put do not have the samestrike. So, a strangle consists of a call with a higher strikeand a put with a lower strike. Therefore, like the straddle,the strangle is also of use to investors who think that theunderlying share will move away from current levels.However, an investor might prefer a 90%/110% strangle
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SEVERAL OPTION STRATEGIES
0TomTom price
At the money
Profit
Figure 10.4 Payoff profile at maturity for a long at-the-money straddle on TomTom.
0TomTom price−3
17
20
23
Figure 10.5 Payoff profile at maturity for a long at-the-money straddle on TomTom.
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to an at-the-money straddle because he thinks that theshare price will move much more than 10% from currentlevels but he wants to minimize his loss in case the shareprice does not move at all. By using the strangle to supporthis view his initial investment is much less than whenbuying an at-the-money straddle. If the share does movemuch more than 10% from current levels he will profitalmost as much as with an at-the-money straddle, but hewill have paid much less than in the case of the straddle,which makes his strategy less risky.As an example consider an investor who thinks thatTomTom’s share price will either move away from itscurrent $20 level by at least 20% in the next 6 months ornot do anything at all. Instead of buying an at-the-moneystraddle where the initial investment is large and – in thecase of a big move – the profit at maturity is large as well,the investor decides to buy a 6-months 90%/110% strangle where the initial investment is small and – in the caseof a big move – the profit is almost as big as for thestraddle. The payoff profile at maturity for this strategyis shown in Figure 10.6. To see that this strategy is lessrisky than buying a 6-month at-the-money straddle,suppose the investor paid $1:50 for the strangle and, asper the example in Section 10.4, he would have to pay $3for the at-the-money straddle. If the share price were tomove down by 20% TomTom’s share price will be $16.The strangle would therefore make a profit of $0:50 (thestrike of the 90% put is $18 and therefore makes $2 on thisput) and the straddle would make a profit of $1. The twodifferent profit profiles of the straddle and strangle areshown in Figure 10.7. Figure 10.7 clearly shows that the
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Payoff
SEVERAL OPTION STRATEGIES
0TomTom price
90%
ATM
110%
Profit
Figure 10.6 Profit profile at maturity for a long 90%/110% strangleon TomTom.
Straddle
016.5
23.5
Strangle
−1.5TomTom price
−317
18
20
22
23
Figure 10.7 Profit profiles at maturity for a long 90%/110% strangleand an at-the-money straddle on TomTom.
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AN INTRODUCTION TO OPTIONS TRADING
strangle is a less risky strategy, but the potential profit isless as well. However, for very big moves the difference inprofit between the straddle and the strangle is very smallin percentage terms.
Chapter
11DIFFERENT OPTIONSTRATEGIES ANDWHY INVESTORSEXECUTE THEM
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O
ther than traders who trade options proprietary,there are basically two types of option investors,portfolio managers and corporates. In general, portfoliomanagers execute strategies to get a good return on theirinvestments whereas corporates tend to invest in optionsto hedge a certain exposure. This chapter discusses whichstrategies are typical for the aforementioned investorsand what they are looking to achieve.
11.1
THE PORTFOLIOMANAGER’S APPROACHTO OPTIONS
The difference between a portfolio manager and an optiontrader is that a portfolio manager usually has a view onthe underlying share and buys or sells options to give thisview some gearing whereas an option trader usually onlyhas a view on the volatility of the underlying share.However, as shown in Section 8.2 it is important thatthe trader has a view on the direction of the stock whentrading long-dated options.Consider the situation when a pension fund sees a lot ofupside potential for the stock Daimler Chrysler (DCX) inthe next 2 years, and suppose DCX is currently trading at$40. Suppose that the pension fund thinks that DCX willbe trading at $50 in 2 years’ time. Rather than just buyingthe share DCX, the pension fund could buy a call optionon DCX with a strike price of $40 for a price of $5 toimprove the return-on-investment ratio. The reasoningbeing that if the pension fund buys 1,000 shares today it
DIFFERENT OPTION STRATEGIES
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will cost the pension fund $40,000 to give an expectedpayoff of $50,000, whereas buying 5,000 options DCXwith a strike price of $40 for $5 each will give the sameexpected payoff of $50,000 having only cost $25,000. Inthe first case the return on the investment is 25% over2 years; in the second one it is 100% over 2 years.
11.2
OPTIONS AND CORPORATESWITH CROSS-HOLDINGS
The majority of large companies have cross-holdings –that is to say, they own shares in other companies. Thereason for this is that companies want to diversify theirrisk and at the same time they are looking for a goodreturn on this investment. What usually happens is that alarge company buys a certain number of shares in anothercompany already knowing at what level it wants to dispose its stake in that company. As a real life example,Daimler Chrysler (DCX) thinks that Asia will be thebiggest growth area for the car industry in the next5 years. Since DCX sells cars to customers at the topend of the range and the average income per person inAsia is still far below that in the U.S. and Europe, it isunlikely that DCX will benefit much from the potentialgrowth in Asia. However, other car manufacturers arelikely to see big growth in Asia which could dampenDCX’s overall market share in the world’s car industry.This could even jeopardize DCX’s market share in theluxury car industry since other car manufacturers will geta stronger balance sheet than DCX’s. This means it is ofcrucial importance that DCX breaks into Asia. Since
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DCX’s cars are too expensive for the average Asian, theyneed to buy a stake in another car manufacturer. So, asDCX actually did, it bought a stake in Mitsubishi Motors.Suppose DCX is looking to get a 50% return on this investment over the next 5 years. One strategy it could haveexecuted is to sell 150% strike calls expiring in 5 years onthese Mitsubishi shares.1 This is a very attractive strategy to DCX since, if Mitsubishi shares do not go up by50%, DCX will still have received the premium of the calloptions. However, if the Mitsubishi shares go up morethan 50%, DCX will only receive 50% since the call options will get exercised against DCX; however, this is stillno problem to DCX because it was only looking to make areturn of 50%. So, this strategy ensures that DCX willstill get some money even if the Mitsubishi shares do notgo up by the desired 50%.
11.3
OPTIONS IN THE EVENT OFA TAKEOVER
Options can be very useful in a takeover situation, sincethe shareholders of the company being acquired areusually at least partially paid in shares of the acquiringcompany. Especially when a shareholder of the companybeing acquired is rather large, there is usually a periodwhere this shareholder is not allowed to dispose of thereceived shares of the acquiring company. This means1In practice, this strategy could only have been executed on a small part ofDCX’s stake in Mitsubishi since the equity derivatives markets are too smallto cope with such a big stake.
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that, even if this shareholder just wants to convert theshares of the acquiring company into cash, he cannot doso for a certain period. So, this shareholder finds himselfin a situation where he owns shares in a company hecannot materialize for a certain period and, therefore, isfully exposed to the share price of this acquiring companyover the specified period. Since this shareholder is onlyinterested in receiving the cash for his holding in theacquiring company, he could hedge his exposure by buying downside put options on his shareholdings whilefunding this by selling upside call options in this company and, therefore, giving up some of the upside of hisshareholdings, which he was not interested in, anyway.This strategy, where the investor buys downside putsand sells upside calls to fund it for no initial premium,is called a ‘zero cost collar’. This is clearly a very attractive strategy to the investor. As an example, suppose BPtakes over a Russian oil company with one big shareholder. The agreed takeover price is £250 million to bepaid out in BP shares. Since it would disrupt BP’s shareprice too much if the shares were to be sold in the marketstraight away, the investor can only start selling theBP shares after 6 months’ time. To protect his BPholding, the Russian shareholder decides to enter intoa zero cost collar with an investment bank for 50million underlying shares (the price of one BP share is£5) with a maturity of 6 months. For this collar to beworth 0, the strikes need to be 90% for the put and 107%for the call. In this case the Russian shareholder hasprotected himself against a drop in BP’s share price ofmore than 10% by giving up everything over 7% of theupside in BP’s share price.
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Just like the above example, the break-evens of the zerocost collar never look particularly attractive due to theeffect of skew, which causes an investor to pay relativelymore for a downside put than an upside call. However,zero cost collars on companies with low dividend yieldscan look attractive since the forwards on such companiesare relatively high, making the puts less expensive andthe calls more expensive.
11.4
RISK REVERSALS FORINSURANCE COMPANIES
Insurance companies tend to execute the same strategyas described in Section 11.3, buying downside puts andselling upside calls, the so-called ‘collar’ or ‘risk reversal’.The reason is that insurance companies have large exposures to stock markets, while at the same time havingrisk limits as to how much they can lose, were a stockmarket to go down by more than 10%. Whenever aninsurance company gets close to these limits it tendsto buy downside puts on an index and funds this partiallyor in full by selling upside calls on this same index. Thisstrategy is called a ‘risk reversal’ or a ‘collar’. For example,if the French insurance company AXA gets close to exceeding its exposure to the UK stock market, the obviousstrategy is to buy a 90% put on the FTSE2 index whileselling a 107% call on the FTSE index for zero cost.2
The FTSE index is a weighted index of the largest UK stocks.
DIFFERENT OPTION STRATEGIES
11.5
123
PRE-PAID FORWARDS
A strategy that is particularly popular in the UnitedStates among high net worth individuals and small tomidcap companies that own stakes in other companies isthe pre-paid forward. The principle is fairly straightforward and involves nothing more than a loan, a put optionand a call option. Again, the easiest way to explain a prepaid forward is by means of an example.Consider an entrepreneurial farmer who has set up hisown fizzy drink brand and is starting to market it throughout the United States. Because this fizzy drink is startingto get really popular in the United States, Coca-Cola feelsthreatened by this success and decides to offer the entrepreneurial farmer $100 million. However, the farmer willget paid in Coca-Cola shares and is not allowed to sellthese shares for the coming year, because Coca-Colawants to make sure that he will still be involved inthe management and marketing of his fizzy drink forthis 1-year period. Since the farmer is very entrepreneurial he already has a new idea of setting up his owndairy company. However, in order to do this he estimatesthat he needs $85 million as an investment. Since he isnot allowed to sell his $100 million worth of Coca-Colashares he basically wants to protect the value of his CocaCola shareholdings and he wants to get $85 million upfront. To protect his Coca-Cola holdings he can buy a putand sell a call, either with the same strike price (e.g., atthe money) or he could buy a lower strike put and a higherstrike call in which case it is nothing more than a collar.Because he owns the shares he can get a cheap loan if he
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gives the underlying shares as collateral. A cheap loanmeans that the investor can borrow at, say, Libor3 þ 15basis points4 rather than Libor þ 2%. If the investor putsthe call, the put and the loan together as a package, it iscalled a ‘pre-paid forward’. The question now becomes:How much is an investment bank prepared to give as acheap loan? The answer lies in how the shares changehands at maturity and if the investment bank is still ableto get repaid on the loan were the investor to default on it.Suppose the investor buys a 90% put and sells a 112% callon Coca-Cola for zero cost. Rather than giving the investor $85 million and charging the investor an interestrate of Libor þ 15 bps a year, the investment bank willgive the present value of $90 million where the discount1factor isand will get back $90 million atLibor þ 15bpsthe end. Economically, this is exactly the same. But, moreimportantly, the reason that the investment bank is onlycomfortable with lending the discounted value of $90million is to protect itself against a possible default ofthe loan. To see this, consider the situation where after1 year the farmer defaults on the $90 million loan andCoca-Cola’s share price has decreased by 20% to 80% ofthe current share price. At first sight, this seems reallybad for the investment bank since it holds the shares ascollateral, the value of which has decreased to $80 millionand for that reason the bank will lose $10 million. How-
3Libor stands for ‘London interbank offered rate’ and is established every dayby a group of banks and gives the rate at which banks are prepared to lendeach other money overnight.411 basis point is 100of 1%.
DIFFERENT OPTION STRATEGIES
125
ever, under the agreement of the 90% put, the investmentbank owes the farmer $10 million on which the investment bank will obviously default, making up for its $10million loss. So, even if the farmer defaults on the loanthe investment bank will still be able to collect themoney for the loan if the share price drops below thestrike of the put. If the share price does not drop belowthe strike of the put the collateral will be enough to coverfor the money of the loan in case of default by the farmer.For this reason the investment bank can never be owedmore at maturity than that percentage of collateral whichis equal to the percentage strike of the put. This meansthat the investment bank is comfortable with giving anup-front amount of money that is equal to the presentvalue of the percentage strike of the put times the value ofthe shares held as collateral. Assuming that USD Libor isequal to 2:5% the farmer will get an amount equal to90 90 ¼ $87:677 million up front.1:0265For the pre-paid forward type of structures, banks witha high credit rating have a significant advantage overbanks with a lower rating. This is because banks with ahigher rating can borrow at Libor flat whereas banks witha lower rating can only borrow at, say, Libor þ 15 bps.Especially for longer dated structures, higher rated bankswill be able to offer a much more attractive up-frontpayment. Typically, investment banks have lower ratingsthan retail banks. So, in this area retail banks have asignificant advantage over investment banks, althoughinvestment banks are typically more aggressive in optionstructures.
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11.6
AN INTRODUCTION TO OPTIONS TRADING
EMPLOYEE INCENTIVESCHEMES
Employee incentive schemes are getting more and morecommon nowadays. The idea behind these schemes is tostimulate employees by letting them benefit from theupside of the share price. This can be achieved by payingout a part of the employee’s salary or bonus in call optionson the share price of the company. This way of stimulating employees is particularly common for seniormanagement.
11.7
SHARE BUY-BACKS
Large companies usually own their own stock, which isbeing held on their balance sheets as treasury stock.Within a certain range they will also have a mandateto buy their own stock. When a company has a sharebuy-back programme in place it will typically sell putson its own stock with a strike at which it is comfortablebuying. This means that the company receives an initialpremium for the put. At maturity the company will eitherhave to buy its own shares at the put option’s strike priceif the share price has dropped below the strike or thecompany gets to keep the put premium if the share priceis above the strike. Obviously, this is a very attractivestrategy to the company. Rather than just buying theshares in the market outright, it exploits the fact thatoption traders look at options in terms of volatility instead of break-evens and are therefore willing to pay a
DIFFERENT OPTION STRATEGIES
127
premium for this strategy. Typically, this has a dampening effect on the implied volatility of this specific stocksince investment banks will get long put options on thisstock and will try to sell it out in the market place.
Chapter
12TWO EXOTICOPTIONS
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A
nother class of options that is becoming increasinglyimportant in the investment banking industry isthe so-called ‘exotic option’. The reason that this type ofoption is called an ‘exotic option’ is because the variablesthat determine the price of this option are not only theusual variables like stock price, strike price, interest rate,dividend and stock volatility but also variables like foreign exchange (FX) volatility or correlation between thestock price and a specific foreign exchange rate. In otherwords, when an investment bank executes a deal thatinvolves an exotic option both the client as well as theinvestment bank expose themselves to more variablesthan when executing a deal involving only a plain vanillaoption.This chapter will only discuss two types of exotic options:the quanto option and the composite option.
12.1
THE QUANTO OPTION
The quanto option is designed for investors who want toexecute an option strategy on a foreign stock but are onlyinterested in the percentage return of that strategy andwant to get paid this return in their own currency. Thebasic principle of a quanto option is that the exchangerate will be fixed to the prevailing exchange rate at inception of the option transaction and the payout of thequanto option will be this exchange rate times the payoutof the regular option.Consider a U.S. investor who is very bullish on the shareprice of BP (British Petroleum). For that reason this U.S.
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131
investor wants to buy an at-the-money call option on BPexpiring in 1 year. However, he does not want to get hisreturn in British pounds but in US dollars. Assume thatBP’s share price is £5, the exchange rate is currently $2 perpound and in 1 year’s time the share price is worth £5.50.The quanto at-the-money call option will give the U.S.investor a payout at maturity of $1 regardless of thechange in exchange rate. In other words, the investorwould expect a 10% return in U.S. dollars on the USDnotional1 amount he bought calls on, since BP’s shareprice has increased by 10%.From the above example it is clear that it is relatively easyto structure a quanto option; however, it is much harderto see what effect it has on how this option should bepriced and what variables it depends on.To see how this option should be priced, two newvariables have to be introduced. The first variable is thecorrelation between BP’s stock price and the foreign exchange, FX, rate. To get a better understanding of howthis correlation affects the price of a quanto call option,an interesting question is whether the US investor who isbuying an ATM USD quanto call is short or long correlation. Here, ‘short correlation’ means that the investorwill benefit (quanto option will increase in value) ifthe correlation goes down and will lose in case the correlation goes up. The reverse holds for being long correlation. To answer this question, assume the correlation ispositive, which means that if the British pound becomes1
‘Notional’ is defined as the number of options times the current share price.
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more valuable against the U.S. dollar, BP’s share will goup. Now, one can easily see that the investor is shortcorrelation (selling correlation), because if the Britishpound goes up the correlation aspect will causethe share price to go up and, therefore, the dollar increaseof a plain vanilla call option is more than the dollar increase of a USD quanto call option, which has a fixedexchange rate. Similar analysis shows that if the Britishpound goes down, the correlation aspect will cause theshare price to go down and the combination of a lowershare price and a less valuable pound causes the dollarloss of a plain vanilla at-the-money call option to be lessthan the dollar loss of an at-the-money USD quanto calloption. To put it differently, if the correlation goes downfrom, say, 1 to 1 the holder of a USD quanto call optionwill benefit, since he is better off holding a plain vanillacall option if the correlation is 1 whereas if the correlation is 1 he is better off holding a USD quanto call option. The above example shows that the holder of a USDquanto call is short correlation; however, when extending this analysis to a USD quanto put one will find thatthe holder of such an option is long correlation.The second variable is FX volatility. For this variable it isless obvious whether the US investor is long or short thisFX volatility. At this point it is good to introduce a modelthat describes the stock price difference in U.S. dollarsfor a small time interval. This model shows under whichcircumstances a quanto option is long or short FX volatility and further shows once more why the holder of aquanto call is short correlation and the holder of a quantoput is long correlation.
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The model most commonly used for modelling a shareprice on a non-dividend paying stock is the Black–Scholesmodel which describes the difference in stock price for asmall time interval. The formula is as follows2:dSt¼ r dt þ S dWtStwhere
ð12:1Þ
St ¼ Stock price at time t;r ¼ Risk-free interest rate;dSt ¼ Change in stock price over time intervaldt;dt ¼ Small time interval;S ¼ Volatility of the stock price;Wt ¼ Brownian motion, which is a stochasticprocess characterised by normally distributed intervals dt with a mean of 0 and avariance equal to the length of the intervaldt. Mathematically stated, the intervals dthave a distribution equal to Nð0; dtÞ;dWt ¼ A stochastic process with distributionNð0; dtÞ.
For the purpose of understanding a quanto option asimilar model to equation (12.1) is required which includes two additional variables – namely the correlation between the logarithm of the stock price and the logarithm of the exchange rate and the FX volatility FX .However, it is important to realize that this equation2Assuming a share price movement according to equation (12.1) one canprove that the price of a call option on such a share is given by formula (2.1)and of a put option by formula (2.2).
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AN INTRODUCTION TO OPTIONS TRADING
defines a new share Ft quoted in the currency the option isquantoed into rather than the share’s own currency. Ft isdefined in such a way that a regular dollar option on Ft isin fact a quanto option on St :
dFt¼ ðrlocal  S FX Þ dt þ S dWtFt
where
ð12:2Þ
 ¼ Correlation between the logarithm of thestock price and the logarithm of the exchange rate, where the exchange rate isquoted as number of dollars per pound.A positive correlation means that if thepound increases in value the stock pricegoes up;rlocal ¼ Risk-free interest rate of the stock’s owncurrency. In BP’s case rlocal is the risk-freeinterest rate of the pound.
From this equation it is obvious that anyone who holds aquanto call option benefits if the correlation decreasesand the holder of a quanto put benefits if the correlationincreases. Depending on the sign of the correlation thisformula also makes clear whether the holder of a quantooption is long or short FX volatility. If the correlation isnegative the holder of a quanto call is long FX volatilityand the holder of a quanto put is short FX volatility. If thecorrelation is positive the holder of a quanto call is shortFX volatility and the holder of a quanto put is long FXvolatility. Knowing that the model for the change in stock
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135
price for a USD stock with a dividend yield equal to d isdFt¼ ðr$  dÞ dt þ S dWtFt
ð12:3Þ
shows that the price of a quanto option can be derivedfrom a normal option by making an adjustment to thedividend yield of adding r$  rlocal þ S FX to the dividend yield. In equation (12.3) r$ is the USD risk-freeinterest rate.
12.2
THE COMPOSITE OPTION
The composite option is designed for investors who wantto execute an option strategy on a foreign stock but wantto fix the strike in their own currency and get the payoutof this option in their own currency. In contrast to thequanto option, where the holder gets a percentage returnregardless the exchange rate, the holder of a compositeoption has exposure to the exchange rate. One of thereasons that a composite option is traded is to protectthe value in their own currency on a foreign investment.Consider the following example. A U.S. investor ownsstock in the British pharmaceutical GlaxoSmithKline(GSK). Assume that the current value of GSK is £13.00and that the exchange rate is $2 per pound. This meansthat the dollar value of one share is $26. To protect thisholding he buys a 1-year at-the-money composite putoption on GSK. This means that the strike price of thiscomposite put option is equal to $26. Assume that after1 year the stock goes down to £11.00 and the exchangerate goes from $2 per pound to $1:5 per pound (dollar
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AN INTRODUCTION TO OPTIONS TRADING
increases in value). This means that the dollar value ofone GSK share has gone down from $26 to $16:5.However, because the strike price of the compositeoption is fixed in dollars, the dollar loss on the shareswill be offset by the payout of the composite put option,which is equal to the strike price ($26) minus the newdollar value of one GSK share ($16:5). In summary, theholder of a composite option wants to protect the sharevalue in his own currency from both exchange rate movements as well as movements in the stock price.The best way to answer the question whether the holderof a composite option is short or long correlation is tomodel the change in stock price, for a small time interval,in the currency of the composite. In the example abovethis would be the dollar. Using the same notation as inSection 12.1 the model looks as follows (see Appendix B):dFt¼ rcompo dt þ compo dWtFt
ð12:4Þ
2compo ¼ 2S þ 2S FX þ 2FX
ð12:5Þ
where
and rcompo is the risk-free interest rate of the compositecurrency.This formula shows that the holder of a composite optionis long correlation.
Chapter
13REPO
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I
n Section 1.3 it was mentioned that an investor is ableto short stock he does not own. However, this principle where the investor borrows stock to subsequentlyshort it comes at a certain cost and is often referred to as‘repo’.1 Repo stands for ‘repurchase agreement’ and meansthat the lender agrees to sell the stock with the agreement to buy it back at a specified price2 somewhere in thefuture. The lender always has the right to call the stockback.In this chapter the definition of repo will be discussed, aswill be its implications to the option market.
13.1
A REPO EXAMPLE
Consider an investor who is bearish (negative) on thestock Ahold. In this case he could short the stock Aholdat the prevailing market price, in which case he willbenefit if Ahold decreases in value. However, for thisstrategy he would have to borrow the stock which wouldcome at a certain cost. This cost is usually around 10basis points (bps)3 a year of the notional borrowed, if it isan easy borrow in which case it is referred to as ‘generalcollateral’ (gc).1Although repo is seen as the cost of borrowing a stock this is not totallycorrect. The cost of borrowing a stock is really the difference between Liborand the repo rate, since the repo rate is less than Libor.2This specified price is larger than what the lender sold the stock for initiallyand can be calculated by multiplying (1 þ Repo rate) and the initial stockprice.311 basis point is 100of 1%.
REPO
139
Suppose the investor decides to short 100,000 shares ofAhold at $10. If the lender charges him 10 bps per year, it1,000,000would cost him $1