gamma is small,stock price movements only cause small changes in delta.However, if gamma is large, delta is highly sensitive tostock price changes. So, an investor who owns an optionwith a large gamma has to adjust the number of stocks in1
Here ‘small’ means really small. Even $0:01 is too much.
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his portfolio frequently to keep this portfolio deltaneutral. Both call and put options have a positive gammaand, as will be shown shortly, these gammas appear to bethe same for both European call and put options. The factthat gamma is positive for both the call and the put optionis logical. After all, the delta of a call option is an increasing function in the stock price, and increases from 0 to 1as the stock price increases from 0 to infinity; the delta ofa put option increases from 1 to 0 as the stock priceincreases from 0 to infinity. Before the formula of gammais given for the European call and put option, note thatgamma is the second derivative of the option price withrespect to the stock price. Indeed, delta is the first derivative of the option price with respect to the stock price,and because gamma is the derivative of delta with respectto the stock price it is clear that this is true. For a European call and put option, gamma is given by:call;European ¼
@@ 2 ctN 0 ðd1 Þpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 0¼¼@St @ðSt Þ2 St  T  t
ð6:1Þ
put;European ¼
@@ 2 ctN 0 ðd1 Þpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ > 0¼¼@St @ðSt Þ2 St  T  t
ð6:2Þ
where d1 is defined as in equation (2.3) and:pﬃﬃﬃﬃﬃﬃ2N 0 ðxÞ ¼ 1 2 ex 2
ð6:3Þ
The above formulae again show that the gamma of aEuropean call option is equal to the gamma of a Europeanput option. Respectively, Figures 6.1 and 6.2 indicate theway in which gamma varies with the stock price and thetime to maturity.
AN INTRODUCTION TO OPTIONS TRADING
Gamma
64
K
Gamma
Figure 6.1
Stock price
Variation of gamma with stock price.
Stock price is equal to strike price
Stock price is less than strike price
Stock price is bigger than strike price
0
Figure 6.2
Time to maturity
Variation of gamma with time to maturity.
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It is worth noting that gamma becomes very big if an atthe-money option is close to expiring. This is caused bythe fact that small stock price changes heavily affect theprobability that this option will expire in the money.Since, for options close to expiration, jj is approximatelyequal to this probability, small stock price changesheavily affect . This explains why at-the-money optionsclose to maturity have big gammas.Lastly, from a practical point of a view it is good to remember that the gamma of any option is largest when itis close to ‘at the money’. This has the direct implicationthat prices of options close to ‘at the money’ are verysensitive to stock price movements and, therefore, needto be re-hedged frequently. Also, when the stock is closeto the strike, options with a short time to maturity havehigher gammas than options with a long time to maturity. So, the prices of at-the-money options with a shorttime to maturity are more sensitive to changes in theunderlying stock and, therefore, need to be re-hedgedmore frequently than at-the-money options with a longtime to maturity.
6.2
THETA
Theta measures the option price’s sensitivity to the passage of time while all other variables remain unchanged.So, it is the rate of change of the option price with respectto time, and is usually indicated by the Greek letter . It isgood to be aware of the impact of theta. Even if variableslike stock price, interest rate and volatility remain
66
AN INTRODUCTION TO OPTIONS TRADING
K
Stock price
Theta
0
Figure 6.3 Variation of theta with stock price for a European calloption, when the interest rate is strictly positive.
unchanged, the option price will still change. Mathematically, theta is the derivative of the option price withrespect to time. The theta of a European call option isalways negative, which means that as time passes theoption price will decrease. The variation of theta withstock price for a European call option is plotted in Figure6.3. The variation of theta with time to maturity for aEuropean call option is plotted in Figure 6.4. The theta ofa European put option is almost always negative. Anexample of a European put option with a positive thetacould be an in-the-money European put option on a nondividend paying stock, provided the interest rate isstrictly positive. For a far in-the-money put option thiscan be seen intuitively. After all, the fact that the upwardrisk is bounded is not worth anything for a far in-the-
THREE OTHER GREEKS
67Time to maturity
0
Out of the money
In the money
Theta
At the money
Figure 6.4 Variation of theta with time to maturity for a Europeancall option.
money put option. This together with an interest rateadvantage makes – as time passes – a far in-the-moneyEuropean put option become more valuable. In Figure 6.5the variation of theta with stock price is plotted for aEuropean put option when the interest rate is strictlypositive, and in Figure 6.6 when the interest rate is 0,which is the same as for the European call option.Respectively, the next two formulae give the thetas of theEuropean call and put option.d2
call;European ¼
 21
@ctSt e ¼  pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ  rK erðTtÞ Nðd2 Þ@t2 2 T  td21
ð6:4Þ
@ptSt e 2 pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ rK erðTtÞ Nðd2 Þ ð6:5Þ¼¼put;European@t2 2 T  t
AN INTRODUCTION TO OPTIONS TRADINGTheta
68
Stock price0
K
Figure 6.5 Variation of theta with stock price for a European putoption, when the interest rate is strictly positive.K
Stock price
Theta
0
Figure 6.6 Variation of theta with stock price for a European calland put option, when the interest rate is zero.
THREE OTHER GREEKS
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Again, in these formulae it is really important that , rand T  t are expressed in the same unit of time.
6.3
VEGA
Vega measures how sensitive the option price is tochanges in volatility, while all other variables remainunchanged. So, it specifies how much the price of theoption would change if there is a 1% change in volatility.In practice, vega is one of the most important measures ofoption risk, since option traders trade volatility. Optiontraders buy options at a certain implied volatility level inthe expectation that this implied volatility will pick up,after which they can sell it at a higher implied volatilitylevel. Or, if they think the implied volatility will decrease, they sell options at a certain implied volatilitylevel hoping to buy it back at a lower implied volatilitylevel. Note that – in interpreting vega – it is associatedwith implied volatility rather than a theoretical volatility, and this point will be elaborated on in Section 8.1.Also, for purposes of vega it does not matter whether thestrategy involves call options or put options, because callsand puts have the same vega, see formulae (2.6) and (2.7)re-stated below:@ct1 d21 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ¼ pﬃﬃﬃﬃﬃﬃ e 2 St T  t > 0call;European ¼@2
ð6:6Þ
d2pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ@pt11¼ pﬃﬃﬃﬃﬃﬃ e 2 St T  t > 0@2
ð6:7Þ
put;European ¼
The vega of any option will always be positive, whichmeans that if (implied) volatility goes up every option
AN INTRODUCTION TO OPTIONS TRADING
Vega
70
0
K
Stock price
Vega
Figure 6.7 Variation of vega with stock price for a European call andput option.
0
Time to Maturity
Figure 6.8 Variation of vega with time to maturity for a Europeancall and put option.
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will become more valuable. Respectively, Figures 6.7 and6.8 give the variation of vega with respect to the stockprice and the variation with respect to the time to maturity for a European option on a non-dividend paying stock.From these figures it is clear that vega is largest when theshare is trading close to the strike price and the longer theterm of the option the larger the vega. So, in comparinggamma and vega it is good to remember that both thesevariables are largest when the share is trading close to thestrike price, but when considering the time to maturity,gamma is large when the term of the option is short,whereas vega is largest when the term of the option islong.
Chapter
7THE PROFIT OFOPTION TRADERS
74
AN INTRODUCTION TO OPTIONS TRADING
A
s was shown in Section 5.1, an option can be hedgedby selling 1 underlying stocks. That is, if an investor has a long position in an option, and during theterm of this option he has a constant short position of underlying stocks, the return on this portfolio is the interest rate. Since  changes as the price of the underlyingstock changes and with the passage of time, an investorwho wants to execute this strategy has to adjust thenumber of shorted stocks continuously. In practice,this is not possible. The number of shorted stocks canonly be adjusted at discrete points in time. This is called‘dynamic hedging’, and whenever the number of shortedstocks is adjusted according to the value of , the portfoliois made delta-neutral. In this chapter the implications ofdynamic hedging will be discussed, and it will be shownthat dynamic hedging is exactly the reason option tradersmake a profit.Throughout this section there might be examples wherea non-integer number of stocks is bought or sold. Thisis practically impossible and is just for the sake ofargument.
7.1
DYNAMIC HEDGING OF ALONG CALL OPTION
Consider a stock which has a price of $10. The price of acall option on this stock with a strike price of $12 is $4.1
Since the  of a put option is negative, this effectively means buying jjunderlying stocks.
THE PROFIT OF OPTION TRADERS
75
The delta of this particular call option is 0:40, the gammais 0:02 and the theta is 0:01 per day. The followingportfolio is put together:. long 1 call option for $4;. short 0:40 stocks for $10 each.
The aim of this section is to identify the profit of thisportfolio for the following price trend of the underlyingstock, when the portfolio is made delta-neutral everydollar and every 2 dollars, respectively:.....
day 0day 1day 2day 3day 4
$10$9$8$9$10
7.1.1 Hedging dynamically every $1. After 4 days the price of the stock is $10 again, whichmeans that, apart from the effect of theta, the price ofthe call option is $4. But, because of theta, the calloption depreciates 0:01 every day. So, after 4 days thecall option is only worth $3:96:2 Hence, the incurredloss on the call option is $0:04.. Hedging dynamically every $1 gives the following cashflows. Because  is 0:02, to make the position deltaneutral 0:02 stocks have to be bought if the stock price2
The fact that theta also changes with the passage of time is not taken intoaccount here.
76
AN INTRODUCTION TO OPTIONS TRADING
goes down by $1, and 0:02 stocks have to be sold if thestock price goes up by $1:3eeee
0.02 stocks bought for $9 on day 1;0:02 stocks bought for $8 on day 2;0:02 stocks sold for $9 on day 3;0:02 stocks sold for $10 on day 4.
Hence, the profit of dynamic hedging is $0:04.For this particular hedging scheme and price trend of thestock, the total profit of this portfolio is $0.
7.1.2 Hedging dynamically every $2. After 4 days the price of the stock is $10 again, whichmeans that, apart from the effect of theta, the price ofthe call option is $4. But, because of theta, the calloption depreciates 0:01 every day. So, after 4 days thecall option is only worth $3:96. Hence, the incurredloss on the call option is $0:04.. Because the portfolio is hedged dynamically every $2,the portfolio is made delta-neutral on day 2 and day 4.Remembering that a  of 0:02 means that if the stockprice goes down by $1 delta goes down by 0:02, it isclear that on day 2, 2  0:02 ¼ 0:04 stocks have to bebought to make the portfolio delta-neutral. In the sameway, on day 4 0:04 stocks have to be sold to make theportfolio delta-neutral. This leads to the following3
The fact that delta also changes with the passage of time is not taken intoaccount here.
THE PROFIT OF OPTION TRADERS
77
cash flows:ee
0:04 stocks bought for $8 on day 2;0:04 stocks sold for $10 on day 2.
So, the profit of dynamic hedging is $0:08.For this particular hedging scheme and price trend of thestock, the total profit of this portfolio is $0:04.These two numerical examples show that, when hedginga long call option dynamically, it is more profitable tomake the portfolio delta-neutral after big stock pricechanges than after small stock price changes. However,this strategy bears a bigger risk. For example, the nextprice trend of the stock still gives a profit on dynamichedging when done every $1, but when done every $2 itdoes not give a profit at all:.....
day 0day 1day 2day 3day 4
7.2
$10$9$8:50$9$10
DYNAMIC HEDGING OF ASHORT CALL OPTION
Again, consider a stock which has a price of $10. Theprice of a call option on this stock with a strike price of$12 is $4. The delta of this particular call option is 0:40,the gamma is 0:02, and the theta is 0:01 per day. The
78
AN INTRODUCTION TO OPTIONS TRADING
following portfolio is put together:. short 1 call option for $4;. long 0:40 stocks for $10 each.
The aim of this section is to identify the profit of thisportfolio for the following price trend of the underlyingstock, when the portfolio is made delta-neutral everydollar and every 2 dollars, respectively:.....
day 0day 1day 2day 3day 4
$10$9$8$9$10
7.2.1 Hedging dynamically every $1. After 4 days the price of the stock is $10 again, whichmeans that, apart from the loss in time value, the priceof the call option is $4. But, because of theta, the calloption depreciates 0:01 every day. So, after 4 days thecall option is only worth $3:96.4 Because the call option was shorted, the profit on the call option is $0:04.. Hedging dynamically every $1 gives the following cashflows. Because  is 0:02 and the call option is shorted,to make the position delta-neutral, 0:02 stocks have tobe sold if the stock price goes down by $1, and 0:024
The fact that theta also changes with the passage of time is not taken intoaccount here.
THE PROFIT OF OPTION TRADERS
79
stocks have to be bought if the stock price goes upby $1:5eeee
0:02 stocks sold for $9 on day 1;0:02 stocks sold for $8 on day 2;0:02 stocks bought for $9 on day 3;0:02 stocks bought for $10 on day 4.
The loss because of dynamic hedging is $0:04.For this particular hedging scheme and price trend of thestock, the total profit of this portfolio is $0. Although thisprofit is equal to the profit of the long call option, now theprofit was made on the option and a loss was incurred ondynamic hedging, whereas with the long call option itwas the other way around. The next example will showthat the profit on the short call is minus the profit of thelong call option whenever the same hedging scheme isused.
7.2.2 Hedging dynamically every $2. After 4 days the price of the stock is $10 again, whichmeans that, apart from the loss in time value, the priceof the call option is $4. But, because of theta, the calloption depreciates 0:01 every day. So, after 4 days thecall option is only worth $3:96. Because the call optionwas shorted, the profit on the call option is $0:04.. Because the portfolio is hedged dynamically every $2,the portfolio is made delta-neutral on day 2 and day 4.5
The fact that delta also changes with the passage of time is not taken intoaccount here.
80
AN INTRODUCTION TO OPTIONS TRADING
Remembering that a  of 0:02 means that if the stockprice goes down by $1 delta goes down by 0:02, it isclear that for the shorted call on day 2, 2  0:02 ¼ 0:04stocks have to be sold to make the portfolio deltaneutral. In the same way, on day 4 0:04 stocks haveto be bought to make the portfolio delta-neutral. Thisleads to the following cash flows:ee
0:04 stocks sold for $8 on day 2;0:04 stocks bought for $10 on day 2.
So, the loss of dynamic hedging is $0:08.For this particular hedging scheme and price trend of thestock, the total loss of this portfolio is $0:04.The foregoing examples show that whenever the samehedging scheme is used the profit of the short call isminus the profit of the long call. Moreover, for a long calloption it is more profitable to make the portfolio deltaneutral after big stock price changes than after smallstock price changes, whereas for a short call option it ismore profitable to make the portfolio delta-neutral aftersmall stock price changes.
7.3
PROFIT FORMULA FORDYNAMIC HEDGING
In this section a formula is given that expresses the profitof dynamic hedging in terms of the change in stockprice, after which the portfolio is made delta-neutral,and gamma. This formula will be derived for both thelong call and the short call position in a situation of 0
THE PROFIT OF OPTION TRADERS
81
interest rates. Derivation for the long put and short put issimilar to the one given in this section.Suppose the value of the underlying stock is St and thevalue of the call option is ct . Suppose also that the delta ofthis call option is  and the gamma of this call option is .Furthermore, assume the interest rate is equal to 0. Thepurpose of this section is to give a general formula of theprofit on dynamic hedging, when this is done after a stockprice movement of y dollars. It should be emphasizedthat such a formula only holds for really small stockprice changes – that is, for really small y.
7.3.1 Long call option. Upward stock movement:e
Because of the long position in the call option aprofit will be made on this option, since its priceincreases. Since the new delta after an upwardstock movement of y is  þ y, the average deltaover the upward stock movement is: þ ð þ yÞ2
ð7:1Þ
The increase in option price is equal to the averagedelta over the upward stock movement times thedifference in stock price, which reads mathematically as: þ ð þ yÞyð7:2Þ2
82
AN INTRODUCTION TO OPTIONS TRADING
e
Because of the long position in the call option,there is a short position in  underlying stocks (elsethe portfolio would not be delta-neutral). Afterthe price of the stock has gone up by $y, ystocks have to be sold to make the portfoliodelta-neutral again. So, a loss is incurred on thestocks, and it is equal to:y
ð7:3Þ
The total profit of this dynamic hedge is y22
ð7:4Þ
. Downward stock movement:e
Because of the decrease in stock price, the value ofthe long call position decreases as well. The average delta over downward stock movement is: þ ð  yÞ2
ð7:5Þ
Because of the long position in the call option, theincurred loss on this option is: þ ð  yÞ y2e
ð7:6Þ
Because of the long position in the call option,there is a short position in  underlying stocks(else the portfolio would not be delta-neutral). Afterthe price of the stock has gone down by $y, ystocks have to be bought to make the portfolio
THE PROFIT OF OPTION TRADERS
83
delta-neutral again. So, a profit is made on thestocks, and it is equal to:y
ð7:7Þ
Again, the total profit of the dynamic hedge is: y22
ð7:8Þ
In conclusion, whether the stock price goes up or downthe profit from dynamic hedging a long call option after asmall stock price change y is always equal to  y 2 .2
7.3.2 Short call option. Upward stock movement:e
Because of the increase in stock price, the value ofthe short position in the call option decreases. Theaverage delta over the upward stock movement is: þ ð þ yÞ2
ð7:9Þ
Because of the short position in the call option theincurred loss on this option is equal to:e
 þ ð þ yÞy2
ð7:10Þ
Because of the short position in the call option,there is a long position in  underlying stocks (elsethe portfolio would not be delta-neutral). After theprice of the stock has gone up by $y, y stocks
84
AN INTRODUCTION TO OPTIONS TRADING
have to be bought to make the portfolio deltaneutral again. So, a profit is made on the stocks,and it is equal to:yð7:11ÞThe total loss on this dynamic hedge is:  y22
ð7:12Þ
. Downward stock movement:e
Because of the decrease in stock price, the value ofthe short position in the call option increases. Theaverage delta over the downward stock movementis: þ ð  yÞð7:13Þ2Because of the short position in the call option theprofit made on this call option is:
e
 þ ð  yÞ þ ð  yÞ y ¼y22
ð7:14Þ
Because of the short position in the call option,there is a long position in  underlying stocks (elsethe portfolio would not be delta-neutral). After theprice of the stock has gone down by $y, y stockshave to be sold to make the portfolio delta-neutralagain. So, a loss is made on the stocks, and it isequal to:yð7:15ÞAgain, the total loss on the dynamic hedge is equal
THE PROFIT OF OPTION TRADERS
85
  y22
ð7:16Þ
to:
In conclusion, whether the stock price goes up or downthe profit from dynamic hedging a short call option after asmall stock price change y is always equal to   y 2 .2Similar to the derivations of the long call and the shortcall, the profit formulas of a dynamic hedge can be derivedfor the long put and short put. One will find that the profitformula for the long put is y 2 and for the short put2 y 2 . Again it should be emphasized that these formulae2only hold for small values of y. It turns out that dynamichedging of short call and put options leads to losses. Thisraises a logical question: Why would one want to hedgeshort positions in call and put options? The answer iseasy for an American option. Because of dynamic hedgingthe position is made delta-neutral again, so should theoption be exercised the loss will not be that big. For aEuropean option it is also logical. It has been shown that,when holding a short position in either a call or a put,hedging dynamically after big stock price changes leadsto bigger losses than if done after small stock pricechanges. So, to reduce the risk, the holder of a shortposition in a European option should hedge his positiondynamically.In conclusion, dynamic hedging of a long position in anoption leads to a profit and compensates for the price paidfor the option. An option trader who buys an optionspeculates on earning more by dynamic hedging thanthe price he has to pay for the option. Dynamic hedging
86
AN INTRODUCTION TO OPTIONS TRADING
of a short position in an option comes down to reducingthe risk. An option trader who writes an option speculates on losing less on dynamic hedging than the price hegets for the option.
7.4
THE RELATIONSHIPBETWEEN DYNAMICHEDGING AND h
In the previous section it was derived that the profit fromdynamic hedging of a long call option is y 2 , provided the2interest rate is 0. In this formula, y is the movement ofthe stock after which the portfolio is made delta-neutral.If  is the volatility over a small time interval I, then theexpected magnitude of movement of the stock in thistime interval is equal to:St
ð7:17Þ
So, when the portfolio is made delta-neutral after thistime interval I, the expected movement of the stock willbe St . This means that, if the portfolio is made deltaneutral after a period of time equal to I, y ¼ St in theprofit formula of dynamic hedging. Hence, the profit ofdynamic hedging every time interval I is equal to:  2  S2t2
ð7:18Þ
Substituting the expression for  – cf. equation (6.1) – intoequation (7.18) gives the following expression for the
THE PROFIT OF OPTION TRADERS
87
profit of dynamic hedging every time interval I. Since is measured over a time interval I, the unit of T  tshould also be I:d212
e Stpﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 2 T  t
ð7:19Þ
If the interest rate is 0, the expression for theta – cf.equation (6.4) – is equal to (the unit of time for  andT  t is I):d21
e 2 StI ¼  pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 2 T  t
ð7:20Þ
From equations (7.19) and (7.20) it can be concluded that,for a long call option, the profit of dynamic hedging everytime interval I is expected to be as much as the incurredloss on the option per time period I.6 It should be emphasized that this relation only holds if I is really small.The same derivation can be executed for a long put option.6Although in this derivation it is not taken into account that  also changeswith the passage of time and  changes with the stock price, this relation stillholds. This is caused by the fact that these two effects are very smallcompared with the effect of  and , respectively. After all, had this beentaken into account, then the profit of the dynamichedge would have been:001St22ddln11C@e2e2 BKy¼St Bpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ  pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃCpﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ pﬃﬃﬃﬃﬃﬃ Byþy ¼ BÞSt@@2@t2 2 T  t2 2ðT  tÞ T  t 4 T  t ApﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃBecause  is equal to some constant times 1= T  t , and T  t is very big(the unit of time is I), the second term in brackets it negligibly smallcompared with the first term in brackets. This means the profit of a dynamicd21
e 2 Sthedge of a long call option is pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ. In the same way it can be argued2 2 T  tthat the loss in time value