alsotakes the price of the option into account. It is easy toactually characterize the payoff of an option at maturity.Suppose that ST is the price of the underlying stock atmaturity, and K is the strike price of the option. Thepayoff from a long position in a European call option is:maxðST  K; 0ÞThis formula is in compliance with the fact that theoption will be exercised if ST > K, and will not be exercised if ST  K. The payoff from a short position in aEuropean call is: maxðST  K; 0ÞThis is again logical, since whatever the holder of a longposition wins by exercising, the holder of a short positionloses. So, the payoff for the short position is minus thepayoff of the long position. The payoff to a holder of a longposition in a European put option is:maxðK  ST ; 0ÞThis formula shows that the put option is exercised ifST < K, and is not exercised if ST  K. The short position
16
AN INTRODUCTION TO OPTIONS TRADING
in a European put option has a payoff of: maxðK  ST ; 0Þ
1.6
PUT–CALL PARITY
The put–call parity is a relation between the call optionprice (ct ), the put option price (pt ), the stock price (St ) andthe strike price (K) of the call and put option, providedthe strike price and the time to maturity are the same forboth the call and the put option. The proof of this put–call parity uses the assumption of absence of arbitrage.Arbitrage means that it is possible to make a profit without running any risk. So, the profit of an arbitrage strategycan never be less than 0 and there is a chance that itmakes a positive profit. In practice, the absence of arbitrage assumption is a very reasonable assumption. Withthis knowledge it is easy to state and prove the put–callparity. When there is absence of arbitrage there exists aput–call parity, given by:ct  pt ¼ St  K erðTtÞwhere
ð1:1Þ
ct ¼ Price of a European call option with strikeprice K and a time to maturity of T  t;pt ¼ Price of a European put option with strikeprice K and a time to maturity of T  t;St ¼ Price of the underlying stock of both thecall and the put option at time t;r ¼ Interest rate expressed as a percentage andin the same unit of time as T  t;K ¼ Strike price.
OPTIONS
17
The put–call parity will be proved by assuming it does nothold and showing that this leads to a contradiction. So, itwill be proved in two steps. First, it will be assumedthat ct  pt > St  K erðTtÞ , and showing this leads toan arbitrage possibility. Then, it will be assumed thatct  pt < St  K erðTtÞ , and showing this also leads toan arbitrage possibility. The fact that both assumptionslead to an arbitrage possibility is a contradiction, becausefor the put–call parity it was assumed there was absenceof arbitrage:1. Suppose ct  pt > St  K erðTtÞ , that is:K þ erðTtÞ ðct  pt  St Þ > 0
ð1:2Þ
By entering in the following portfolio at time t, aprofit will always be made at maturity, T.e
At time t– long 1 underlying stock– long 1 put option– short 1 call option
(yield -St )(yield -pt )(yield ct )
The total yield of this portfolio is:ct  pt  Ste
ð1:3Þ
At maturity, T– K > STIn this case the call option will not be exercised,but the put will. By exercising the put, the underlying stock in the portfolio will be sold for
18
AN INTRODUCTION TO OPTIONS TRADING
$K, closing down the portfolio’s position. Theprofit of this portfolio is:K þ erðTtÞ ðct  pt  St Þ > 0
ð1:4Þ
– K < STThe put will not be exercised, but the call will.Because the call is being exercised, the underlying stock in the portfolio will be sold for $K tothe holder of the call option, closing down theportfolio’s position. The profit of the portfoliois:K þ erðTtÞ ðct  pt  St Þ > 0ð1:5Þ2. Suppose ct  pt < St  KerðTtÞ , that is:erðTtÞ ðSt þ pt  ct Þ  K > 0
ð1:6Þ
By entering in the following portfolio at time t, aprofit will always be made at maturity, T.e
At time t– short 1 underlying stock– short 1 put option– long 1 call option
(yield St )(yield pt )(yield -ct )
The total yield of this portfolio is:St þ pt  cte
ð1:7Þ
At maturity, T– K > STIn this case the call option will not be exercised,
OPTIONS
19
but the put will. Because the put is being exercised, the underlying stock will be bought for$K from the holder of the put option, closingdown the short position in this underlyingstock. The profit of this portfolio is:erðTtÞ ðSt þ pt  ct Þ  K > 0
ð1:8Þ
– K < STThe put option will not be exercised, but thecall will be exercised. By exercising the call,the underlying stock will be bought for $K,closing down the short position in this underlying stock. The profit of this portfolio is:erðTtÞ ðSt þ pt  ct Þ  K > 0
ð1:9Þ
Since both the assumption ct  pt > St  KerðTtÞ as wellas ct  pt < St  K erðTtÞ lead to an arbitrage possibility,the put–call parity, ct  pt ¼ St  K erðTtÞ , has to hold.
Chapter
2THEBLACKSCHOLESFORMULA
22
AN INTRODUCTION TO OPTIONS TRADING
I
n the previous chapter the definition of an option wasgiven. Furthermore, some examples showed what thepayoff and profit of an option look like. In these examplesthe price of the option was always given. It is howeverpossible to identify what the fair price of a Europeanoption should be. In this perspective ‘fair’ means thatthe expected profit for both sides of the option contractis 0. The Black–Scholes formula is a good tool for determining the fair price of an option. From the definition ofan option it is clear that the price should depend on thestrike price, the price of the underlying stock and thetime to maturity. It appears that the price of an optionalso depends on less obvious variables. These other variables are interest rates, the volatility of the underlyingstock (the way the stock moves) and the dividends on thestock. By some simple examples it can be clarified thatthe option price should also depend on the last mentionedvariables:. Interest rate. Suppose that the interest rate given on asavings account is 5% per year. Consider a put optionwith a time to maturity of 1 year, and, given an interest rate of 5%, the price of this option is $10. Since theholder of the short position in this option gets this $10,he can put this money in a savings account, getting a5% interest rate. By doing so he will have $10:5(10  1:05) at the expiration date of the option. Sincethe price of the option is fair, the expected payoff forthe holder of the long position in the option will alsobe $10:5. Now suppose that the interest rate was not5% but 6%. Higher interest rates cause expectedgrowth rates on stocks to increase, otherwise inves-
THE BLACK–SCHOLES FORMULA
23
tors in stocks could be tempted to sell their stocks andput the money in a savings account. This means thatthe expected payoff of a put option is likely to be less.Although this is generally true, suppose that the expected payoff to the holder of the long position doesnot change and will still be $10:5. The question is:What should the price of the option be in this case?The price should be such that the holder of the shortposition has $10:5 at maturity. This implies that theprice is 10:5 discounted at a rate of 6%. In other words,the price is $9:91 (10:5=1:06). If the effect of increasingexpected growth rates on stocks when the interest rateis higher was taken into account, the option pricewould have been even less. This shows that optionprices differ as the interest rate differs. The price of aput option is less if the interest rate is higher, and for acall option the reverse holds. In Section 2.2, this effectwill be discussed in more detail. Figure 2.1 shows inwhich way the prices of an at-the-money call and putoption vary with interest rate. Note that if the interestrate is 0 the prices of an at-the-money call and putoption are the same, which is perfectly in compliancewith the put–call parity. Furthermore, it is worthnoticing that the price of an at-the-money put optionchanges faster with interest rate than an at-the-moneycall option. This means that an at-the-money putoption is more sensitive to interest rate changesthan an at-the-money call option, which will be discussed in more detail in Section 2.2.. Volatility. Volatility is a complex concept, but in thisexample a simple definition will be used to clarify thedependence of the option price on volatility. Volatility
24
AN INTRODUCTION TO OPTIONS TRADING11
tion
ll op
y caone
he-m
Price
At-t
8
At-
the
-m
on
eyp
ut
op
tio
n
4
0
0.05
0.1Interest rate
0.15
0.2
Figure 2.1 Variations of at-the-money call and put option priceswith interest rate.
is a variable that measures the movement of a stock.Consider the stock Unilever with a price of $50. If thevolatility is 0:1 this means, in this example, that after1 year the stock price is either $55 or $45. If thevolatility is 0:2 this means that after 1 year it is either$60 or $40. Consider a European call option on Unilever with a time to maturity of 1 year. If the volatilityof Unilever is 0:1, the expected payoff at maturity is$2:5 (ð55  50Þ  0:5 þ 0  0:5). If the volatility ofUnilever is 0:2, the expected payoff at maturity is $5(ð60  50Þ  0:5 þ 0  0:5). Since the expected payoff isgreater if the volatility is 0:2 than if the volatility is0:1, the price of the option should be higher if thevolatility is 0:2. In general, the higher the volatility
25
Price
THE BLACK–SCHOLES FORMULA
ey
llca
n
tio
op
on
-mthe
In-
ey
llca
n
tio
op
on
-mthe
At-
S t−K
on
pti
ey
on
0
0.2
0.4
0.6
-mthe
f-
t-o
Ou
local
0.8
1
Volatility
Figure 2.2
Variation of the price of a call option with volatility.
the higher the option price. This has to do with the factthat losses are bounded from below by the optionprice, but profits can be unlimited. Figures 2.2 and2.3 show the variation of the option price with volatility for the call and the put option, respectively. Notethe similarity between variations of the call and putoption prices with volatility.. Dividends. Consider an option on the stock Unileverwith a strike price of $50 and the expiration date is in1 year. Again, the stock price is $50. If in 6 months adividend of $1 per stock is to be paid, this will causethe stock price to drop by $1. For that reason thisdividend is favourable for a put option on Unilever andunfavourable for a call option on Unilever. In general,
AN INTRODUCTION TO OPTIONS TRADINGPrice
26
ey
tpu
n
tio
op
n
on
tio
-mthe
pto
u
In-
peyon
e-m
-th
At
n
tio
pto
u
peyon
K−S t
e-m
-th
ft-o
Ou
0
Figure 2.3
0.2
0.4
Volatility
0.6
0.8
1
Variation of the price of a put option with volatility.
dividends will cause reductions in call option pricesand the prices of put options to increase.Now that it is clear which variables the price of an optionshould depend on, it is time to state the Black–Scholesformula. The Black–Scholes formula gives the price ofan option in terms of its characteristics. It is beyondthe scope of this book to actually prove this formula.Although it looks very complicated, do not be deterred.It is only for the sake of completeness that the Black–Scholes is stated here. It is perfectly possible to work withoptions without fully understanding the Black–Scholesformula. The most important thing to remember fromthe Black–Scholes formula is that it depends on the strikeprice (K), stock price (St ), time to maturity (T  t), interest
THE BLACK–SCHOLES FORMULA
27
rate (r), volatility () and dividends. Because dividendsmake the theory unnecessarily complicated, this bookwill only deal with non-dividend paying stocks. In thiscase dividends can be left out. The Black–Scholes formulauses the following notation:ct ¼ Price of a European call option at time t;pt ¼ Price of a European put option at time t;St ¼ Price of the underlying stock at time t;T  t ¼ Time to maturity;K ¼ Strike price of the option; ¼ Volatility of the underlying stock;r ¼ Interest rate:The prices, at time t, of a European call option ct and aEuropean put option pt are given by:ct ¼ St Nðd1 Þ  K erðTtÞ Nðd2 Þ;
ð2:1Þ
pt ¼ K erðTtÞ Nðd2 Þ  St Nðd1 Þ
ð2:2Þ
In these formulae, NðxÞ is the standard Normal distribution, and d1 , d2 are defined as: Stlnþ r þ 12 2 ðT  tÞKpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃd1 ¼ð2:3Þ Ttpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃð2:4Þd2 ¼ d1   T  tOf course, the above formulae reflect the fact that theprice of an option on a non-dividend paying stock dependson the strike price, price of the underlying stock, timeto maturity, volatility of the underlying stock and the
28
AN INTRODUCTION TO OPTIONS TRADING
interest rate. The next three subsections will elaborate onthe Black–Scholes formula.
2.1
VOLATILITY AND THEBLACK–SCHOLES FORMULA
The volatility used in the Black–Scholes formula is actually more difficult than depicted before. In the Black–Scholes formula  is the standard deviation of the naturallogarithmic returns of the stock price, instead of just thestandard deviation of the stock price. This means thatif, over 3 days, a stock has a price trend of $50, $55,$53, the volatility per day is the standard deviation of  lnð55Þ  lnð50Þ ¼ ln 55andlnð53Þ  lnð55Þ ¼ ln 535055 ,rather than just the standard deviation of the percentagechanges in stock price. Furthermore, it is important thatthe unit of time of volatility coincides with the unit oftime of the time to maturity and with the unit of time ofthe interest rate. So, if the time to maturity and interestrate are expressed in years, volatility should also be expressed in years. Luckily, there is an easy formula toswitch from volatility per (trading) day to volatility perannum (cf. p. 231 of Hull, 1993):sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃNumber of tradingVolatility per Volatility per¼days per annumtrading dayannumð2:5ÞAs mentioned earlier, the higher the volatility the higherthe option price. This also becomes clear when looking atthe Black–Scholes formula. The derivative of the option
THE BLACK–SCHOLES FORMULA
29
price with respect to the volatility gives the price’s sensitivity to volatility. If this derivative is positive it meansthat the price increases as volatility increases. This isexactly what appears to be the case if the derivatives offormulae (2.1) and (2.2) are taken with respect to :d2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ@ct11¼ pﬃﬃﬃﬃﬃﬃ e 2 St T  t > 0@2
ð2:6Þ
@pt1 d21 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃput;European ¼¼ pﬃﬃﬃﬃﬃﬃ e 2 St T  t > 0@2
ð2:7Þ
call;European ¼
The option price’s sensitivity to volatility is often referred to as vega, and is the same for both Europeancall and put options. In this book ‘vega’ is indicatedwith the Greek letter .1
2.2
INTEREST RATE AND THEBLACK–SCHOLES FORMULA
In the previous section it was mentioned that it is important that r,  and T  t are expressed in the same unitof time. In the same section it was shown how to switchfrom volatility per annum to volatility per trading day.To switch from interest rate per annum to interest rateper trading day is very straightforward. The followingrelationship holds between interest rate per annumand interest rate per trading day.1
Vega is not actually a Greek letter, but in option theory it is still referredto as an option Greek.
30
AN INTRODUCTION TO OPTIONS TRADING
Interest rate Interest rate per Number of trading¼days per annumtrading dayper annumð2:8ÞThe same analysis as in Section 2.1 can be conducted forthe interest rate. The derivative of the option price withrespect to the interest rate gives its sensitivity to theinterest rate. Of course, since the price of a European putoption decreases as the interest rate increases, this derivative should be negative for the put option. For a calloption it is hard to deduce economically what the sign ofthis derivative should be. The next formulae give for boththe call and the put option the sign of the derivative of theoption price with respect to the interest rate. It appearsthat for the call option this derivative is positive, whichmeans that the price of the call option increases as theinterest rate increases:@ct¼ KðT  tÞ erðTtÞ Nðd2 Þ > 0ð2:9Þ@r@pt¼ KðT  tÞ erðTtÞ Nðd2 Þ < 0 ð2:10Þput;European ¼@r
call;European ¼
From an economical point of view it is logical that theprice of a European put option is less when the interestrate is higher. Because both the fact that expected growthrates on stocks increase as well as the fact that higherinterest rates are earned on savings accounts cause theprice to decrease. Since, for a European call option, thefirst effect causes the price to increase and the second todecrease, it is less obvious what happens to the price.Formula (2.9) states that the price of a European call option increases if the interest rate increases. This means
THE BLACK–SCHOLES FORMULA
31
that the first effect more than offsets the second. Because,for a put option, there are two effects causing the price todecrease, a put option is more sensitive to interest ratechanges than a call option, which has one effect causingthe price to decrease and one effect causing it to increase.The option price’s sensitivity to the interest rate is oftenreferred to as rho, and is indicated with the Greek letter .
2.3
PRICING AMERICANOPTIONS
The Black–Scholes formula is one way of deriving the fairprice of an option. This works well for European options,but for American put options and American call optionson dividend paying stocks this method does not work anylonger. For that reason there is another model, the binomial tree model. In practice, the binomial tree model is apopular tool for determining the price of American options. The binomial tree model reaches a price by simulating price trends and looking at different points in time,whether it is optimal to exercise the option or not. This isdifferent from the Black–Scholes model, but it still usesthe same variables. For that reason, to get an understanding of options it is enough or maybe better to use theBlack–Scholes model.
Chapter
3DIVIDENDS ANDTHEIR EFFECTON OPTIONS
34
AN INTRODUCTION TO OPTIONS TRADING
A
lthough dividends are not the first concern of anoption trader it is important to understand howthey affect option pricing. In this section this will beexplained and the reader will be introduced to forwards.
3.1
FORWARDS
Forwards are agreements to buy or sell shares at a futurepoint in time without having to make a payment upfront. Unlike an option, the buyer of a forward doesnot have an option at expiry. For example, the buyer ofa 6-month forward in Royal Dutch/Shell commitshimself to buying shares in Royal Dutch/Shell at a preagreed price determined by the forward contract. Thenatural question is, of course: What should this preagreed price be? Just like Black and Scholes did for thepricing of an option the price is determined by how muchit will cost to hedge the forward position. To show this,consider the following example. An investment banksells a 2-year forward on Royal Dutch/Shell to an investor. Suppose that the stock is trading at $40, the interestrate is 5% per year and after 1 year Royal Dutch/Shell willpay a dividend of $1. Because the bank sells the forward itcommits itself to selling a Royal Dutch/Shell share in 2years’ time. The bank will hedge itself by buying a RoyalDutch/Shell share today. By buying a Royal Dutch/Shellshare the bank pays $40, over which it will pay interestfor the next 2 years. However, since the bank is long aRoyal Dutch/Shell share it will receive a dividend of $1 in1 year’s time. So, over the first year the bank will pay
DIVIDENDS AND THEIR EFFECT ON OPTIONS
35
interest on $40 and over the second year interest on $39.This means the price of the forward should be:F ¼ 40 þ 40  0:05 þ 39  0:05  1 ¼ 42:95 ð3:1ÞA more general formula for the forward price is:F ¼ Price of underlying þ Cost of carry
ð3:2Þ
In the previous example the cost of carry is the interestthe bank has to pay to hold the stock minus the dividendit receives for holding the stock.
3.2
PRICING OF STOCK OPTIONSINCLUDING DIVIDENDS
When dividends are known to be paid at specific points intime it is easy to adjust the Black–Scholes formula suchthat it gives the right option price. The only change oneneeds to make is to adjust the stock price. The reason forthis is that a dividend payment will cause the stock priceto go down by exactly the amount of the dividend. So, inorder to get the right option price one needs to subtractthe present value of the dividends paid during the term ofthe option from the current stock price, which can thenbe plugged into the Black–Scholes formula – see equations (2.1) and (2.2). As an example, consider a 1-year calloption on Daimler Chrysler (DCX) with a strike price of$40. Suppose DCX is currently trading at $40, the interestrate is 5%, stock price volatility is 20% per annum andthere are two dividends in the next year, one of $1 after2 months and another of $0:5 after 8 months. It is now
36
AN INTRODUCTION TO OPTIONS TRADING
possible to calculate the present value of the dividendsand subtract it from the current stock level. For thepresent value calculation see Section 3.3:2
8
PV of dividends ¼ ef120:05g  1 þ ef120:05g  0:5¼ 1:4753
ð3:3Þ
Now the option price can be calculated by plugging intothe Black–Scholes formula – see equation (2.1) – a stockprice of St ¼ 40  1:4753 ¼ 38:5247 and using K ¼ 40,r ¼ 0:05,  ¼ 0:2 and T  t ¼ 1:38:527þ 0:05 þ 12  0:22  1ln40pﬃﬃﬃd1 ¼¼ 0:1621 ð3:4Þ0:2  1pﬃﬃﬃð3:5Þd2 ¼ d1  0:2  1 ¼ 0:0379So, the price of the call option will be:ct ¼ 38:5247  Nð0:1621Þ  40  e0:05 Nð0:0379Þ¼ 3:2934
3.3
ð3:6Þ
PRICING OPTIONS IN TERMSOF THE FORWARD
Instead of expressing the option price in terms of thecurrent stock price, interest rate and expected dividendsit makes it more intuitive to price an option in termsof the forward, which comprises all these three components. The easiest way to rewrite the Black–Scholes
DIVIDENDS AND THEIR EFFECT ON OPTIONS
37
formula in terms of the forward is to assume a dividendyield rather than dividends paid out at discrete points intime. This means that a continuous dividend payout isassumed. Although this is not what happens in practiceone can calculate the dividend yield in such a way thatthe present value of the dividend payments is equal toSt 1  edðTtÞ , where d is the dividend yield. So, if thedividend yield is assumed to be d and the interest rate isr the forward at time t can be expressed as:Ft ¼ St  efrðTtÞg  efdðTtÞg¼ St  efðrdÞðTtÞg
ð3:7Þ
From the above equation it is clear that dividends lowerthe price of the forward and interest rates increase it. Asshown in the previous subsection one can calculate theprice of an option by putting a stock price equal toSt  efdðTtÞg into the Black–Scholes formula. By doingthis one can easily rearrange the Black–Scholes formulato express the price of an option in terms of the forward.The price of the call will then be expressed as:ct ¼ St edðTtÞ Nðd1 Þ  K erðTtÞ Nðd2 Þ¼ erðTtÞ Ft Nðd1 Þ  K erðTtÞ Nðd2 Þ
ð3:8Þ
In the same way the price of the put can be expressedas:pt ¼ K erðTtÞ Nðd2 Þ  erðTtÞ Ft Nðd1 Þ
ð3:9Þ
38
AN INTRODUCTION TO OPTIONS TRADING
where d1 and d2 look like: dðTtÞ St elnþ r þ 12 2 ðT  tÞKpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃd1 ¼ Tt rðTtÞ eFtlnþ 12 2 ðT  tÞKpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ¼ Ttpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃd2 ¼ d1   T  t
3.4
ð3:10Þð3:11Þ
DIVIDEND RISK FOROPTIONS
The problem option traders face when trading options isthat they usually do not know the exact dividendsa company is due to pay out. So, the only way to pricean option is to estimate the future dividends a company isexpected to pay. Usually, it is relatively easy to estimatethe dividends for the near future; however, the furtherone goes into the future the harder it gets. This is becausedividends are very much related to the profitability of acompany. In this perspective it is also useful to think ofan option price in terms of the forward. If the actualdividends a company has paid out are higher than thetrader had expected, the trader’s forward should havebeen lower which causes the prices of call options to godown because it is less likely that the call option willexpire in the money. With the same argument the pricesof put options will go up because they are more likely toexpire in the money. So, the risk an option trader faces
DIVIDENDS AND THEIR EFFECT ON OPTIONS
39
when buying a call option is that the dividends appear tobe higher than he had expected, and when buying a putoption that they appear to be lower than he had expected.Of course, the reverse holds for selling either a call or aput. When it comes to assessing the risks of dividends it isbest to think of the option price in terms of the forward. Ifthe dividends go up the forward will go down whichmakes it less likely that a call option will expire in themoney, so the price will go down. A put will more likelyexpire in the money, so the price will go up. If dividendsgo down the forward will go up, so the price of a call goesup and the price of a put goes down. In conclusion, toassess how dividend or interest rate changes affect theprice of an option just think about what happens to