theforward. If the forward goes down put options get moreexpensive and calls less expensive. If the forward goes upcall options get more expensive and put options lessexpensive.
3.5
SYNTHETICS
It is possible to synthetically create a forward by buying acall option and selling a put option with the forward priceas the strike. Intuitively, this can easily be seen by addingup the respective payoff graphs as done in Figure 3.1. Thisfigure also shows that a synthetic has the same dynamicsas a normal share, which is what one would expect. Sinceforwards do not cost anything it should also be the casethat a call minus a put with the forward price as the strikeis worth 0. Bringing the put–call parity from Section 3.6
AN INTRODUCTION TO OPTIONS TRADING
Payoff
40
Payoff call
F
0
|
Stock price
Payoff minus Put
Figure 3.1
Synthetic payoff.
back into memory will make this clear. The put–callparity for non-dividend paying stocks states:ct  pt ¼ St  KerðTtÞ
ð3:12Þ
The forward of a non-dividend paying stock isF ¼ St erðTtÞ , which shows that if the strike price forboth the call and the put is the forward price the callminus the put, the synthetic, is worth 0. In the same waythis can be proved for options on dividend paying stocks.However, in this case the put–call parity is:ct  pt ¼ St  K eðrdÞðTtÞ
ð3:13Þ
where d is the dividend yield. In practice, one would onlysee synthetic forwards on specific strikes, not necessarilyon the forward price.
Chapter
4IMPLIEDVOLATILITY
42
AN INTRODUCTION TO OPTIONS TRADING
A
lthough implied volatility is a relatively easy conceptit is usually thought to be complicated. This isbecause things can get confusing. Implied volatility isnot just a theoretical concept, it also depends on thesimple market phenomenon of supply and demand.Before implied volatility is explained in more detail,volatility will be elaborated on.As mentioned earlier volatility is a measure for the movement of the stock price. To be precise, it is the standarddeviation of the natural logarithm of the returns of thestock price. Although this is the official definition, it ismore important to understand which time period thisvolatility relates to in the Black–Scholes formula. Theanswer is of course simple, it is the volatility of the stockprice during the term of the option. It is simple, but it isgood to be aware of this fact, since this implies thatwhenever an investor wants to buy an option he doesnot know what the volatility will be during the term ofthe option. This, in turn, makes it impossible for him tocalculate the fair price of the option. However, he can tryto estimate this volatility by looking at what the volatility has been so far, the historic volatility. Although thereare techniques to estimate the volatility accurately, hewill never be able to say with certainty what the volatility will be during the term of the option. In fact, fromall the variables the Black–Scholes formula uses to determine the price of an option on a non-dividend payingstock, volatility is the only one that is really open tointerpretation. This is obvious for the stock price, strikeprice and time to maturity, but for the interest rate this isless obvious. Because, for the same reason volatility is not
IMPLIED VOLATILITY
43
known, one would argue that an investor does not knowwhat the interest rate will be during the term of the option. However, it is possible to estimate the interest rateso accurately that this variable can effectively be treatedas known. With this knowledge it is easy to explain theconcept of implied volatility.The price of an option comes about from supply and demand. This is of course confusing. The Black–Scholesformula gives the price of an option, which would suggestthere could only be one price for an option, regardless ofsupply and demand. But, as has been pointed out, at thetime an investor has to decide whether or not to buy anoption, he does not know what the volatility of the optionwill be during the term of this option. So, different marketparticipants have different views on what the volatilitywill be, and therefore have different views on what shouldbe the fair price of an option. Now an interesting situation arises. There is the Black–Scholes formula whichgives the price of an option on a non-dividend payingstock in terms of the stock price, strike price, time tomaturity, interest rate and volatility, but at the same timethe price of this option is set by supply and demand. If aninvestor wants to know whether he should buy a specificoption he could estimate what he thinks the volatilityshould be and use this volatility to plug into the Black–Scholes formula in order to calculate the price he is prepared to pay for this option. Or he could determine whichvolatility gives the option price set by the market andcompare it with what he thinks is the right volatility.This can be done by equating the Black–Scholes formulawith the market price of the option, which is an equation
44
AN INTRODUCTION TO OPTIONS TRADING
in one unknown, namely the volatility. An equation inone unknown can always be solved1 and the volatilityobtained in this way is called the ‘implied volatility’.This is a logical name because, in calculating it, the market price of an option is used. Which means that thecalculated volatility is implied by the market and, therefore, it tells what the market thinks the volatility of theunderlying stock will be during the term of the option. Inconclusion, the implied volatility is nothing more thanthat volatility, such that when it is substituted into theBlack–Scholes formula the output is equal to the marketprice of the option. The idea of implied volatility is illustrated by the following example, which is based on theexample on p. 229 of Hull (1993).
4.1
EXAMPLE2
Suppose that the value of a European call option on a nondividend paying stock is $3:67 when St ¼ 33, K ¼ 30,r ¼ 0:05 and T  t ¼ 0:25 (In this example T  t is expressed in years, so T  t ¼ 0:25 is a time to maturityof 3 months.) The implied volatility is that value of which, when substituted into formula (2.1) together withSt ¼ 33, K ¼ 30, r ¼ 0:05 and T  t ¼ 0:25, gives ct ¼ 3:67.Although this is really only an equation in one unknown,1Because the Black–Scholes formula contains the Normal distribution, thisparticular equation in the unknown  cannot be solved analytically.However, there are methods, like the Newton–Raphson method, to solveit to any required accuracy. An example of such a method will be treatedshortly.2This example is almost identical to the example on p. 229 of Hull (1993),only the numbers are different.
IMPLIED VOLATILITY
45
it cannot be reduced to an expression in which  is expressed as a function of St , K, r, T  t and ct . However, aniterative search procedure can be used to find the implied. To start with, the value  ¼ 0:15 could be used. Thisgives a value of ct equal to 3:4518, which is too low. Sincect is an increasing function of , a higher value of  isrequired. Next, a value of  ¼ 0:25 could be tried. Thisgives a value of ct equal to 3:7925, which is too high andmeans that  lies between 0:15 and 0:25. Next, a value of0:20 can be tried for . This will prove to be too low,deriving that  lies between 0:20 and 0:25. Proceeding inthis way the range for  can be halved at each iterationand the correct value of  can be calculated to any required accuracy. In this example, the implied volatility is0:22 or 22% per annum. Note that because T  t wasexpressed in years, the implied volatility is also expressedin years.
4.2
STRATEGY AND IMPLIEDVOLATILITY
Consider an option trader who wants to decide upon theimplied volatility if he bought the option in the previousexample or not. As was shown in this example, the implied volatility of this option is 22% per annum. Supposealso that he thinks that the underlying stock will have avolatility of 1:5% per trading day during the term of theoption. Now he can use formula (2.5) to determine if heshould buy the option. If there are 256 trading days ineach year, this would mean thatpﬃﬃﬃﬃﬃﬃﬃﬃhe thinks that thevolatility per annum is 1:5  256 ¼ 24%. As stated
46
AN INTRODUCTION TO OPTIONS TRADING
before, the higher the volatility the higher the price of theoption. Because the option trader thinks the volatilityshould be 24% per annum and the market thinks it willbe 22%, it means that the option trader thinks thatthe option is worth more than he has to pay for it onthe market. In this case, the option trader should buy theoption. But, if he thinks the volatility will be 1% pertrading day he should not buy the option.Because 1%pﬃﬃﬃﬃﬃﬃﬃﬃper trading day means a volatility of 1  256 ¼ 16% perannum, and this would imply that he thinks the option isworth less than he has to pay for it on the market. Ingeneral, if an investor thinks the implied volatility of anoption is too high he should not buy the option, whereasif he thinks it is too low he should buy the option. In thisanalysis it is important that volatilities are comparedover the same period of time. So, a volatility per tradingday can only be compared with another volatiltity if thisone is also expressed in trading days, and of course thesame holds for volatilities expressed in years or any otherunit of time.
Chapter
5DELTA
48
AN INTRODUCTION TO OPTIONS TRADING
D
elta measures the sensitivity of the option price tothe stock price while all other variables remainunchanged. Mathematically, it is the derivative of theoption price with respect to the stock price. So, if thedelta of an option is 0:5, it means that – if the stock priceincreases by $1 – the option price increases by $0:5,1 and ifthe stock price decreases by a small amount the optionprice decreases by 50% of that amount. If delta is negativethe reverse holds. Suppose an option has a delta of 0:5.In this case, if the stock price increases by $1 the optionprice decreases by $0:5, and if the stock price decreases by$1 the option price increases by $0:5. The delta of a putoption is negative, since the price of a put option decreases as the stock price increases. For a call optionthe reverse holds, since the price of a call option increasesas the stock price increases. The delta of a call option isbetween 0 and 1, and for a put option it is between 1 and0. It was argued that the delta of a call option had to begreater than 0. But why does the delta of a call option haveto be less than 1? To answer this question it is importantto know what comprises the price of a call option. Thisprice has an intrinsic2 part and an option part. It has anintrinsic part since the price of a call option on a nondividend paying3 stock is always at least as much as theintrinsic value. After all, an American call option can be1
This is not totally correct because if the stock price changes,  also changes.So, in fact, it should be that if the stock price changes by a small amount theoption price changes by 50% of that amount. But, for the sake of this exampleit is expressed in a $1 stock price movement.2The intrinsic value of an option is the payoff of that option were it exercisedimmediately. See Section 1.3.3The value of a European call option on a dividend paying stock can be lessthan intrinsic, which will be shown in Section 5.3.
DELTA
49
exercised immediately to get a payoff equal to the intrinsic value of the option, and, for the European calloption, the strategy4 of going long the call option andshorting the underlying stock locks in a discounted payoffgreater than the intrinsic value. There is also an optionpart to the price of the option, since payoffs can be unlimited but are bounded from below by 0 (an investor canchoose not to do anything). This means that the investor’s discounted expected payoff is always higher than theintrinsic value. Logically, the option part is defined as thediscounted expected payoff of the option minus the intrinsic value. In conclusion, an investor who owns a calloption can guarantee himself a payoff equal to the intrinsic value by exercising immediately, and if he choosesnot to exercise the option immediately his discountedexpected payoff is higher than the intrinsic value. Now, itis easy to see why the delta of a call option has to be lessthan 1. To show this, it is convenient to distinguish between an in-the-money and an out-of-the-money calloption, respectively. Suppose the delta of an in-themoney call option is bigger than 1. This would meanthat if the stock price increased by $1, the option pricewould increase by more than $1. Since a $1 increase instock price implies a $1 increase in intrinsic value of thecall option, this implies that the option part of the pricealso increases. But this cannot be true. Since, if the stockprice has increased by $1, it is less likely that the stockprice will drop below the strike price and for that reasonthe downside risk being bounded5 becomes less valuable,45
Argument put forward on p. 158 of Hull (1993).See Section 1.3.
50
AN INTRODUCTION TO OPTIONS TRADING
which in turn implies that the option part becomes lessvaluable. The same conclusion can be drawn if the stockprice decreased by $1. So, if the delta of an in-the-moneycall option were bigger than 1, a $1 decrease in stock pricewould cause the option price to decrease by more than theintrinsic value, implying that the option part of the pricebecomes less. But, in this case this is a contradiction,because the downside risk being bounded becomes morevaluable. This shows that the delta of an in-the-moneycall option cannot be bigger than 1, so it has to be lessthan 1. In the same way, it can be argued that the delta ofan in-the-money put6 option has to lie between 1 and 0.For an out-of-the-money call or put option it is even morestraightforward. Because, in this case, the intrinsic valueof the option is 0, the price only consists of the optionpart. Recalling that the option part is defined as the discounted expected payoff minus the intrinsic value, thismeans that the price is equal to the discounted expectedpayoff. Since the change in discounted expected payoff ofan out-of-the-money option is less than the change instock price (otherwise, the probability that this changein payoff is realized would be 1, which is clearly not true),it is easy to see that the delta of an out-of-the-money calloption lies between 0 and 1 and the delta of an out-of-themoney put option lies between 1 and 0. These phenomena also become clear if the derivatives of formulae (2.1)and (2.2) are taken with respect to the stock price, St . The6
One could think that the European put option case differs slightly, sincethe price of this option is not necessarily as much as the intrinsic value. But,by noting that in this case the option part is negative and since the intrinsicpart can never change by more than the change in stock price, the samereasoning still holds.
DELTA
51
next two formulae are the deltas of the European call andEuropean put option on a non-dividend paying stock,respectively:call;European ¼
@ct¼ Nðd1 Þ > 0@St
ð5:1Þ
put;European ¼
@pt¼ Nðd1 Þ < 0@St
ð5:2Þ
Formulae (5.1) and (5.2) not only point out that the  of aEuropean call option is positive and the  of a Europeanput option is negative, they also show that the  of aEuropean call option lies between 0 and 1 and of a European put between 1 and 0. This last mentioned observation stems from the fact that Nðd1 Þ and Nðd1 Þ liebetween 0 and 1. An interesting question is, of course:When is the  of a call option 0 and when is it 1? Again, bydividing the option price into an intrinsic part and anoption part the answer to this question is easily given.Consider a call option which is far in the money, whichmeans that the stock price is substantially higher thanthe strike price of the call option. For this option, the factthat the downside risk is bounded from below is virtuallynot worth anything. So, the price of this call option isequal to the intrinsic value. This implies that if the stockprice goes up by $1 the option price also has to go up by $1,since otherwise the option price will be less than theintrinsic value. For the same reason, if the stock pricegoes down by $1 the option price also goes down by $1.This shows that the  of a far in-the-money call optionis 1. Now consider a far out-of-the-money call option,which means that the stock price is substantially less
52
AN INTRODUCTION TO OPTIONS TRADING
Table 5.1
Extreme delta values.
Type of option
Delta ()
Far in-the-money call optionFar out-of-the-money call optionFar in-the-money put optionFar out-of-the-money put option
1010
than the strike price of the option. This option is notworth anything, since the probability that a profit willbe made is 0. Whether the price of the stock goes up ordown the call option will still not be worth anything. Thisshows that a far out-of-the-money call option has a  of 0.With a similar reasoning one would find that a far in-themoney put option (stock price is substantially less thanthe strike price of the option) has a  of 1 and the  of a farout-of-the-money put option (stock price is substantiallyhigher than the strike price of the option) is 0. Table 5.1summarizes these observations, and Figures 5.1 and 5.2show the variation of delta with the stock price for thecall and the put option, respectively.
5.1
DELTA-HEDGING
‘Hedging’ literally means reducing the risk. So, if an investor has a portfolio which exposes him to a high risk, hecould try to hedge this portfolio in order to reduce hisrisk. The question arises: How can a holder of a stockoption hedge himself against future stock movements?This question is easily answered with the knowledge ofdelta. Consider an investor with 100 call options and each
DELTA
53
Delta
1
0
Figure 5.1
Stock price
Variation of delta with stock price for a call option.
K
Stock price
Delta
0
K
1
Figure 5.2
Variation of delta with stock price for a put option.
54
AN INTRODUCTION TO OPTIONS TRADING
option has a delta of 0:5. This investor can hedge himselfagainst future stock movements by shorting (selling) 50of the underlying stocks. After all, if the stock price goesdown by $1, the call option price decreases by $0:5,causing a loss of $50 (100 options times $0:5), but atthe same time the investor makes a profit of $507 on the50 stocks he had shorted. With this hedging method theinvestor not only makes no more losses, he also no longermakes any profits. Because, if the stock price goes up by$1, he makes a profit of $50 (100 options times $0:5) onthe call option, but this is offset by the loss on the shortedunderlying stocks. The same holds for a put option, but inthis case the investor has to buy jj8 times the number ofoptions of the underlying stock. It is logical that, in thecase of a long call option, stocks have to be sold to hedgethe position and, in the case of a long put option, stockshave to be bought. Because a call option gives the right tobuy the underlying stock, the holder of a call option doesnot face a payoff at maturity if he is one stock shortwhenever the call option expires in the money, and whenever the call option expires out of the money if he doesnot hold a position in the underlying stock at all. Since,for call options very close to maturity, the probabilitythat a call option will expire in the money is approximately equal to , this particular long call option is hedgedif  stocks are shorted. Although the probability that acall option with a longer time to maturity will expire in
7
For each stock he received $1 more than he has to pay for it now.This is the absolute value of delta, which means that if  ¼ 0:5, jj ¼ 0:5.Note that the absolute value is taken because the delta of a put option isnegative.8
DELTA
55
the money is not equal to ,9 such a long call option is stillhedged by shorting  underlying stocks. This phenomenon will be explained in Section 7.5. Because a putoption gives the right to sell the underlying stock, theholder of a put option does not face a payoff at maturity ifhe is one stock long whenever the put option expires inthe money, and if the put option expires out of the moneyif he does not hold a position in the underlying stock atall. Since, for put options very close to expiration, theprobability that a put option will expire in the money isapproximately equal to jj, this particular long put optionis hedged if jj stocks are bought. Just like call options, forput options with longer times to maturity the probabilitythat a put option expires in the money is not equal to jj,10but such a put option is still hedged by buying jj underlying stocks. Another way to look at the difference between hedging a call option and a put option is by meansof the sign of . The sign of the  of a call option is positiveand of a put option negative. So, to hedge either a long putoption as well as a long call option,  underlying stockshave to be shorted. But, since the  of a put option isnegative this hedging strategy effectively means that theunderlying stocks have to be bought.Black and Scholes showed that if at any point in time along option is hedged by shorting the  of the underlying9
The probability that a call option expires in the money is equal to Nðd2 Þ (cf.Appendix A). This explains why, for call options close to maturity, thisprobability is equal to .10The probability that a put option expires in the money is equal to Nðd2 Þ(cf. Appendix A). This explains why, for put options close to maturity, thisprobability is equal to jj.
56
AN INTRODUCTION TO OPTIONS TRADING
stocks, this portfolio is riskless, which means that thereturn on the portfolio is equal to the interest rate. Mathematically, the following portfolio is riskless (return onthe portfolio is equal to the interest rate)þ1 :
Option
 :
Underlying stock.
This means that if money is to be borrowed to set up thisportfolio, the return on the portfolio is the interest rateand the borrowed money can be repaid with interest,making neither loss nor a profit. If money is to be receivedto set up this portfolio, the return on the portfolio isnegative the interest rate, which cancels out the returnon the money received for setting up the portfolio. Important in the analysis of Black and Scholes is the boldface part, at any point in time, since  already changes ifthe stock price changes by a small amount and also withthe passage of time. This means, to really set up a risklessportfolio, the number of shorted stocks has to be adjustedcontinuously. Aside from the fact that this portfolio isonly riskless if no transaction costs have to be paid foradjusting the number of stocks in the portfolio, this isphysically impossible. In practice, the number of stocksin the portfolio is only adjusted according to the value of at discrete points in time. Such a hedging scheme isreferred to as ‘dynamic hedging’, and whenever the number of shorted stocks is adjusted according to , this isreferred to as making the portfolio delta neutral.Although, at this point, it may seem inconvenient thatin practice it is only possible to hedge dynamically, it will
DELTA
57
become apparent that this is exactly why option tradersmake a profit.
5.2
THE MOST DIVIDENDSENSITIVE OPTIONS
Since there is always a chance that a company willchange its dividend payments it is good to have a feelfor which options are most dividend-sensitive. In Section3.2 it was shown that the price of an option on a dividendpaying stock can easily be calculated by putting into theBlack–Scholes formula a stock price St equal to the current stock price minus the present value of the dividends.The delta measures the sensitivity of an option price to achange in the underlying St . From this it is clear that anoption with a large delta in absolute terms will be verysensitive to dividend changes. This means that far in-themoney options as well as synthetics have large dividendexposures.
5.3
EXERCISE-READY AMERICANCALLS ON DIVIDENDPAYING STOCKS
Section 1.4 showed that it is never optimal to exerciseearly a call option on a non-dividend paying stock and as aresult prices of American and European call options arethe same for non-dividend paying stocks. However, it canbe optimal to exercise early an American call option on adividend paying stock. For that reason the price of an
58
AN INTRODUCTION TO OPTIONS TRADING
American call option on a dividend paying stock candiffer from its European counterpart. To see this consideran American call option on the French pharmaceuticalcompany Sanofi-Aventis with a strike price of $50 and amaturity of 3 months. Sanofi is currently trading at $80and the stock is due to go ex-dividend11 tomorrow with apayable dividend of $1:50. Since this option is clearly farin the money the time value12 of this option is very little,which means that this option has a very high delta. Sincethe stock will go ex-dividend tomorrow one can expectthe share price to drop by $1:50 and as a result the intrinsic value of the option to go from $30 to $28:50. Theonly way the option can regain this drop in intrinsic valueis by a sharp increase in time value. However, the optionis so far in the money that, although the time value willincrease slightly, it will never be enough to make up forthe loss in intrinsic value. Assuming that the time valueof this option goes from $0:02 to $0:10 with the stockdropping to $78:50, the option on the ex-dividend date isworth $28:60 where the day before it was worth $30:02.From the above it is clear that it is optimal to exercise the$50:00 call on Sanofi. When comparing the price of theEuropean counterpart of the above option the day beforethe stock goes ex-dividend, one would expect this priceto be equal to the price of the American option on the ex-
11A stock trading ex-dividend means that whoever purchases this stock is nolonger entitled to receive the dividend. As a result the stock will theoretically be worth less by exactly the amount of the dividend payable. If it takes2 days to settle the shares of a company, the company will trade ex-dividend2 days before the record date of the dividend.12The time value of an option is equal to the option part, as described inChapter 5.
DELTA
59
dividend date plus the time value of the option of the daybefore – that is, $28:62.When determining whether an American call option on adividend paying stock is exercisable, one has to realizefirst that one would only exercise an American call optionthe day before the stock goes ex-dividend. Because, evenif the American call option is a clear exercise, there isalways a chance that the stock will drop considerablygiving the option a large time value again, and becausethe option was delta-hedged, one would have made thesame amount of money on stocks that were shorted as onthe drop in intrinsic value (delta needs to have been veryclose to 1 otherwise it can never be an exercise). Bearingthis in mind there are basically three option characteristics that signal that an American call option on a dividend paying stock is an exercise. The first one is that thedelta of the option should be very close to 1, which meansthat the time value is very little. Second, the Americanoption should be worth almost intrinsic the day beforethe stock goes ex-dividend. This second characteristic isin fact the most reliable one because it is basically tellingone that exercising the option will not be costly. Third,the European call option should be worth less than intrinsic the day before the stock goes ex-dividend and,therefore, be less than the American call. However, justcomparing the European with the American call can betricky because, when there are several dividend payments during the term of the option, the European callwill almost always be worth less.Although the aforementioned three characteristics givegood guidelines as to whether an American call on a
60
AN INTRODUCTION TO OPTIONS TRADING
dividend paying stock is exercisable, the only way to besure is by testing the following. If on the day before thestock goes ex-dividend the price of an American call isworth more than the same American call, only then –with the dividend taken out and a share price which is thecurrent share price less the dividend – should the call beexercised.13 If this is not the case the call should not beexercised.
13Obviously, there is a 1-day difference. So, the American call should reallybe compared with the same American call and only then, with the dividendtaken out and a share price equal to the current share price less the dividend,should the start date of the call be moved 1 day forward.
Chapter
6THREE OTHERGREEKS
62
AN INTRODUCTION TO OPTIONS TRADING
I
n this chapter three other Greeks will be introduced –respectively, gamma, theta and vega. Gamma measures delta’s sensitivity to the stock price, theta measuresthe option price’s sensitivity to the passage of time, andvega measures the option price’s sensitivity to volatility.
6.1
GAMMA
Previously, in most examples concerning  it was saidthat a $1 change in stock price would cause the optionprice to change by . Although in these examples this is auseful interpretation of , it is not totally correct. This isbecause  changes even if the stock price changes by asmall amount and also with the passage of time. Thus,while the stock price changes by $1,  takes many different values. This means that it is not correct to calculatethe option price change, caused by a $1 change in stockprice, using only one -value. The right way to deal with is, if the stock price changes by a small1 amount, then theoption price changes by  times this amount. Since changes if the stock price changes, it would be nice tohave a unit that measures delta’s sensitivity to stock pricemovements. This unit is called ‘gamma’ and is indicatedby the Greek letter . Mathematically,  is the derivativeof  with respect to the stock price. If