2.13)
From the above equation it is clear that dividends lower the price ofthe forward and interest rates increase it. As shown in the previous subsection, one can calculate the price of an option by substituting a stockprice equal to St × e{−d(T −t)} into the Black–Scholes formula. By doingthis one can rearrange the Black–Scholes formula to express the priceof an option in terms of the forward. The price of the call can then beexpressed asct = St e−d(T −t) N (d1 ) − K e−r (T −t) N (d2 )= e−r (T −t) Ft N (d1 ) − K e−r (T −t) N (d2 ).
(2.14)
In the same way the price of the put can be expressed aspt = K e−r (T −t) N (−d2 ) − e−r (T −t) Ft N (−d1 ),where d1 and d2 are


+ r + 12 σ 2 (T − t)d1 =√σ T −t −r (T −t)  ln e K Ft + 12 σ 2 (T − t)=√σ T −t√d2 = d1 − σ T − t.ln
St e−d(T −t)K
(2.15)
(2.16)(2.17)
Conventional Options, Forwards and Greeks
11
2.7 PUT–CALL PARITYThe put–call parity is a very important formula and gives the relationbetween the European call and the put price, where the call and the puthave the same strike and maturity, in terms of the share price and thestrike price. The formula is as follows:ct − pt = St e−d(T −t) − K e−r (T −t) ,
(2.18)
where d is the dividend yield of the share and r is the risk free interestrate. One can prove this put–call parity by assuming it does not hold andshow that one can then build a portfolio that leads to a riskless profit atmaturity. For example, whenct − pt > St e−d(T −t) − K e−r (T −t) ,
(2.19)
one can prove that the portfolio
r sell a call option with strike price K and maturity T − t (income is ct );r buy a put option with strike price K and maturity T − t (incomeis − pt );
r buy a share (income is −St e−d(T −t) )leads to a riskless profit at maturity for any level of share price at maturity.Indeed, if ST > K the call will be exercised and therefore the sharewithin the portfolio is sold at € K and the put expires worthless. Thenet position at maturity is therefore zero, but since the generated incomeincluding the financing plus € K is larger than zero, the strategy makesa riskless profit. Indeed, rewriting equation 2.19 givesct − pt − St e−d(T −t) er (T −t) + K > 0.(2.20)In the same way one can prove that the above portfolio leads to a risklessprofit if ST < K .If one assumes thatct − pt < St e−d(T −t) − K e−r (T −t) ,
(2.21)
one can show that the following portfolio leads to a riskless profit forany level of share price at maturity:
r buy a call option with strike price K and maturity T − t (incomeis −ct );
r sell a put option with strike price K and maturity T − t (income is pt );r sell a share (income is St e−d(T −t) ).
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Exotic Options Trading
The put–call parity also signals for which strike the call and the put areworth the same. As one would expect, the strike for which the put andthe call are worth the same is equal to the forward value of the share.Equation 2.13 established that the forward is equal toFt = St × e{r (T −t)} × e{−d(T −t)} .
(2.22)
Therefore, if K = Ft , equation 2.18 shows that the call is worth as muchas the put.
2.8 DELTADelta is one of the most important Greeks and instrumental to the Black–Scholes derivation of the price of an option (see Section 2.2). Delta measures the sensitivity of an option price to the stock price. Mathematically,delta, δ, is the derivative of the option price with respect to the stockprice. By taking the actual derivative of equations 2.3 and 2.4, the deltaof European call and put options on a non-dividend paying stock are asgiven below. This assumes a call option price of ct , a put option price ofpt and a stock price of St .∂ct= N (d1 ) > 0,∂ St∂ ptδput, European == −N (−d1 ) < 0.∂ St
δcall, European =
(2.23)(2.24)
Equation 2.23 shows that the delta of a call option is between 0 and 1and of a put option between −1 and 0. Table 2.1 shows for which stockprice the delta reaches its extremes. The deltas of call and put optionswith strike K versus the stock price are shown graphically in Figures2.4 and 2.5 respectively.
Table 2.1 Extreme delta valuesType of OptionFar in the money call optionFar out of the money call optionFar in the money put optionFar out of the money put option
Delta (δ)10−10
1
Delta
0
K
Stock price
Figure 2.4 Variation of delta with stock price for a call option
0
K
Delta–1
Figure 2.5 Variation of delta with stock price for a put option
Stock price
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Exotic Options Trading
2.9 DYNAMIC HEDGINGBlack and Scholes showed that a portfolio consisting, at any time, ofone option and minus delta shares is risk neutral and should thereforeyield the risk free interest rate. For a long call option, delta hedgingmeans selling shares and for a long put put option, minus delta sharesmeans effectively buying shares as the delta of a put option is negative.To buy minus delta shares against one option is called delta hedging. Inpractice it is not possible to have minus delta shares against one optionat any time. This would mean that one would have to continuouslyadjust the number of shares, which is not possible. Therefore, inpractice an option is delta hedged at discrete points in time and is calleddynamic hedging. Although this might seem inconvenient, it is exactlythe reason that traders make money on volatility. This is discussed inChapter 3.
2.10 GAMMASince delta changes whenever the stock price changes, it is useful tohave a measure that captures this relationship. This measure is calledgamma and gives the sensitivity of delta to a small change in stockprice. Mathematically, gamma is the derivative of delta with respect tothe stock price. Since delta is the derivative of the option price withrespect to the stock price, gamma is the second order derivative of theoption price with respect to the stock price. For European call and putoptions, gamma is given by the following formula.∂δ∂ 2 ctN  (d1 )==> 0,√∂ St∂(St )2St σ T − t∂ 2 ptN  (d1 )∂δ==> 0.γput, European =√∂ St∂(St )2St σ T − t
γcall, European =
(2.25)(2.26)
where d1 is defined as in equation 2.5 and1x2(2.27)N  (x) = √ e− 2 .2πThe above formulae show that the gamma of a European call option isequal to the gamma of a European put option. Respectively, Figures 2.6and 2.7 indicate the way in which gamma varies with the stock priceand the time to maturity. It is helpful to interpret gamma in terms ofhow the delta hedge of an option changes for a change in stock price.
Conventional Options, Forwards and Greeks
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Gamma
K
Stock price
Figure 2.6 Variation of gamma with stock price
If gamma is positive, one needs to sell shares if the stock price goes upin order to be delta hedged and buy shares if the stock price goes down.If gamma is negative the reverse holds. Namely, one needs to buy sharesif the stock price goes up in order to be delta hedged and sell shares ifthe stock price goes down.
Gamma
Stock price is equal to strike price
Stock price is less than strike price
Stock price is bigger than strike price
0
Figure 2.7 Variation of gamma with time to maturity
Time to maturity
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Exotic Options Trading
It is worth noticing that gamma becomes very large if an at the moneyoption is close to expiring. This is caused by the fact that small stockprice changes heavily affect the probability that this option will expirein the money. Since, for options close to expiration, |δ| is approximatelyequal to this probability,3 small stock price changes heavily affect δ.This explains why ‘at the money’ options close to maturity have largegammas.Gamma is a very important Greek as it enables traders to derive theprofit on an option for any given stock move. The precise way this isdone is discussed in Chapter 3. For exotic options, gamma is usually oneof the distinguishing features as it typically changes sign for differentlevels of stock price. Since gamma prescribes how to adjust one’s deltahedge, it is closely related to the actual profit of an option (see Chapter3). Hence, for any exotic option, it is extremely important to identify thelevels of stock price where the gamma changes sign.
2.11 VEGAVega measures the option price’s sensitivity to changes in volatility, orrather implied volatility. Mathematically, it is therefore the derivative ofthe option price with respect to the implied volatility, σ . The vegas, ν, ofa European call and put option are the same. Therefore, when trading thevolatility of an underlying stock by trading options, it does not matterwhether one trades a call or a put option. In formula form, the vegas, ν,of a European call and put are expressed as:d12√∂ct1= √ e− 2 St T − t > 0,∂σ2πd12√1∂ pt= √ e− 2 St T − t > 0.νput, European =∂σ2π
νcall, European =
(2.28)(2.29)
The vega of a regular option is always positive, meaning that, if theimplied volatility goes up, the option becomes more valuable. For exotic options, it is not always the case that the price goes up wheneverthe implied volatility does. An example is an up-and-out call, which isdiscussed in Chapter 10 in Section 10.8.3The probability that a call option expires in the money is equal to N (d2 ) and the probability that a putoption expires in the money is equal to N (−d2 ). This explains why, for call and put options close to maturity,this probability is very close to |δ|.
Conventional Options, Forwards and Greeks
17
Vega
0
K
Stock price
Figure 2.8 Variation of vega with stock price for a European call and put option
Respectively, Figures 2.8 and 2.9 give the variation of vega with respect to the stock price and the variation with respect to the time tomaturity for a European option on a non-dividend paying stock. Fromthese figures it is clear that vega is largest when the share is trading closeto the strike price and the longer the term of the option the larger the
Vega
0
Time to Maturity
Figure 2.9 Variation of vega with time to maturity for a European call and put option
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Exotic Options Trading
vega. So, in comparing gamma and vega it is good to remember thatboth these variables are largest when the share is trading close to thestrike price, but when considering the time to maturity, gamma is largewhen the term of the option is short whereas vega is largest when theterm of the option is long.
2.12 THETATheta measures the option price’s sensitivity to the passage of timewhile all other variables remain unchanged. So, it is the rate of changeof the option price with respect to time, and is usually indicated withthe Greek letter θ. It is good to be aware of the impact of theta. Even ifvariables like stock price, interest rate and volatility remain unchanged,the option price will still change. Mathematically, theta is the derivativeof the option price with respect to time. The theta of a European calloption is always negative, which means that as time passes the optionprice decreases. The variation of theta with stock price for a Europeancall option is plotted in Figure 2.10. The variation of theta with time tomaturity for a European call option is plotted in Figure 2.11. The thetaof a European put option is almost always negative. An example of a
0
K
Stock price
Theta
Figure 2.10 Variation of theta with stock price for a European call option, when theinterest rate is strictly positive
Conventional Options, Forwards and Greeks
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Time to maturity
0
Out of the money
In the money
ThetaAt the money
Figure 2.11 Variation of theta with time to maturity for a European call option
European put option with a positive theta could be an ‘in the money’European put option on a non-dividend paying stock, provided the interest rate is strictly positive. For a far ‘in the money’ put option this canbe seen intuitively. After all, the fact that the upward risk is boundedis not worth anything for a far in the money put option. This togetherwith an interest rate advantage ensures that as time passes, a far in themoney European put option becomes more valuable. In Figure 2.12, thevariation of theta with stock price is plotted for a European put optionwhen the interest rate is strictly positive, and in Figure 2.13 when theinterest rate is zero, which is the same as for the European call option.
2.13 HIGHER ORDER DERIVATIVES LIKE VANNAAND VOMMAVanna is the sensitivity of vega with respect to changes in the underlyingand vomma is the sensitivity of vega with respect to changes in impliedvolatility. However, this book will limit the use of these two terms andother higher order derivatives but will rather specifically describe thefeature at hand. In fact, higher order derivatives become quite meaningless for options where the Greeks themselves are already not smooth. Inparticular, exotic options have certain trigger points where the Greeks
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Exotic Options Trading
Theta
Stock price0
K
Figure 2.12 Variation of theta with stock price for a European put option, when theinterest rate is strictly positive
0
K
Stock price
Theta
Figure 2.13 Variation of theta with stock price for a European call and put option,when the interest rate is zero
Conventional Options, Forwards and Greeks
21
‘blow up’ and therefore are not smooth at all. Only for variance swaps(Chapter 23) is it worth mentioning vanna and vomma, but for most exotic options these higher order derivatives show such erratic behaviourthat they will only blur the picture.
2.14 OPTIONS’ INTEREST RATE EXPOSURE INTERMS OF FINANCING THE DELTA HEDGEIt is important to remember that an option is exposed to changes ininterest rate. Mathematically this is nothing more than the derivative ofthe option price with respect to the interest rate. However, it is much moreimportant to understand why an option is exposed to the interest rate.As usual in option theory, the answer can be found in how an optionis hedged. For example, consider a call option. A call option is deltahedged by selling |δ| shares. By selling short these shares, the traderreceives money which he can put in a savings account and subsequentlyearn interest on it. This means that the higher the interest rate the higherthe call price. For the put option it is exactly the opposite. Indeed, a putoption is delta hedged by buying |δ| shares. In order to do this the traderneeds to borrow the funds which buy him |δ| shares. On these funds thetrader incurs financing and therefore benefits from a lower interest rate.Typically, it is very useful to turn to the financing on the delta hedge inorder to derive the interest rate exposure of an option.4 Doing this alsoprovides a better understanding of the pricing of options in terms of realcash flows instead of abstract mathematical formulae.
4In this approach the premium of the option is not taken into account. For the put option this premiumincreases the short interest rate exposure. However, for a call option the premium decreases the long interest rateexposure. Nonetheless, the net effect for the call option is a long interest rate exposure.
3Profit on Gamma andRelation to ThetaThe easiest way to show how the profit of an option can be determinedby the gamma and the movements of the underlying is by means of asimple example. Suppose Philips is worth € 25 and a three month calloption with a strike of € 25 has a delta of 0.50. Assuming that a traderbuys 1000 of these call options he will need to sell 500 shares to be deltahedged. If the next day Philips’ stock price goes to € 27 the delta willincrease to, let’s say, 0.70. This means that in order to be delta hedgedthe trader will sell an additional 200 shares at € 27. The next day Philips’stock price goes back down to € 25 and the corresponding delta of theoption goes back to 0.50, which means that the trader will buy back200 shares at € 25. Just from rebalancing the delta, the trader has made2 × 200 = € 400. To put this in a general formula; if a stock moves byy after which the option is delta hedged and the stock then moves backto its original level, the profit on one option is:y × (change in δ).This means that the profit from a move y without moving back to itsoriginal level is1· y × (change in δ).2
(3.1)
The change in δ can easily be rewritten in terms of gamma and is nothingmore than y × γ . This means that the profit on a long option after a movein the underlying of y is equal to1· γ · y2.2
(3.2)
Obviously, the fact that the holder of an option makes money wheneverthe stock price moves does not come for free. The way the option holderpays for the right of buying low and selling high is by means of the theta,the time decay of an option. In other words, the holder of an option needsto earn back the daily loss in value of the option by taking advantage of
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Exotic Options Trading
the moves of the underlying. Naturally, the reverse holds for the sellerof an option. In this case the option seller makes money on the theta andloses it by rebalancing the delta, which effectively means buying highand selling low.The relationship between gamma and theta is easy to understand now.The expected profit on gamma as a result of a move in a small time periodshould equal the loss in value of the option of that specific time period.The expected profit is easy to calculate as the expected move, y, of thestock in a small period of time, I , equalsy = σ St ,
(3.3)
where σ has I as the unit of time and is the implied volatility of theoption. This means that the expected profit over a small period of time,I , is1 2 2γ σ St .(3.4)2As this expected profit should be equal to theta over the small timeinterval I , the above analysis has arrived at the main formula of optiontheory, namely:1 2 2γ σ St + θ = 0.(3.5)2When plugging in the formulae for theta and gamma this can also beshown mathematically. A more detailed proof of the above is given inChapter 7 of An Introduction to Options Trading, F. de Weert.
4Delta Cash and Gamma CashBefore discussing any exotic options it is important to understand theterminology of delta cash and gamma cash. Luckily these two definitions are easy to understand and very intuitive. Instead of delta andgamma, traders tend to use delta cash and gamma cash. This is for thesimple reason that traders think in percentage movements rather thanabsolute movements. Indeed, both delta and gamma are absolute unitsand quantify the change for an absolute movement.Delta cash, δc , is defined as delta, δ, times the share price St . Mathematically,δc = δ · St .
(4.1)
The profit on delta cash for a percentage move, y %, is now nothingmore thanδc · y %.Gamma cash, γc , is defined as the change in delta cash for a 1 % movein the underlying. Now one can derive thatγ · St2.100Indeed, the change in delta for a 1 % move isγc =
(4.2)
St,100which means that the delta cash after a 1 % move isStδ+γ· St .100γ·
Therefore the change in delta cash for a 1 % move, i.e. the gamma cashisγ · St2St· St − δSt =.δ+γ100100
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Exotic Options Trading
4.1 EXAMPLE: DELTA AND GAMMA CASHTo make the use of delta and gamma cash more intuitive, consider thefollowing example. A trader buys 100 thousand 1 year € 25 Philips calloptions. The delta of each € 25 Philips call option is 0.5. This means thatthe trader delta hedges himself by selling 50 thousand Philips shares.Assume also that the gamma cash of the 100 thousand Philips optionsamounts to 50 thousand euros per 1 %. This means that, if the trader isfully delta hedged initially and Philips’ shares move down by 1 %, thetrader needs to buy € 50 thousand worth of Philips shares to be fullydelta hedged again. Indeed, when the share price moves down by 1 %,the delta cash of the portfolio comprising of 100 thousand € 25 calloptions plus a 50 thousand short share position goes from a delta cashequal to zero to a delta cash of minus € 50 thousand because of thegamma cash.
5SkewSkew is the principle that lower strike options on the same underlyinghave higher implied volatilities than options with higher strikes. Thereason for skew can be explained by the observation that if markets godown they tend to become more volatile and, on very few occasions, themarket actually crashes, in which case it will be incredibly volatile. Thisalone does not explain skew as this realised volatility is the same regardless of any strike price. The existence of skew is apparently saying thatthis increase in volatility has a bigger impact on lower strike options thanon higher strike options. The reason behind this becomes apparent whenthinking in terms of realised gamma losses as a result of rebalancing thedelta of the option in order to be delta hedged. In a downward spiralingmarket the gamma on lower strike options increases, which combinedwith a higher realised volatility causes the option seller to rebalance hisdelta more frequently, resulting in higher losses for the option seller.Naturally, the option seller of lower strike options wants to get compensated for this and charges the option buyer by assigning a higher impliedvolatility to lower strikes. In other words; for any specific maturity, thelower the strike price of an option on any single underlying the higherthe implied volatility of this option. This principle applies regardless ofthe in or out of the moneyness of an option. Whether it is a lower strike‘in the money’ call option or a lower strike ‘out of the money’ put optionmakes no difference from a skew perspective. Indeed, if there were adifference there would be an arbitrage opportunity.
5.1 REASONS FOR HIGHER REALISED VOLATILITYIN FALLING MARKETSAn easy way to see why falling markets tend to become more volatile is tocompare volatility to uncertainty. Markets go down because there is moreuncertainty about the future and hence investors are more uncertain abouthow to value stocks. A price earnings ratio of 12 can be considered cheapone day and expensive the next day, just because of a different assumptionon future interest rates. For example, at the end of a rate hike cycle there
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Exotic Options Trading
is a lot of uncertainty about how to value equity markets as central banksneed to control inflation by increasing rates but can easily overshoot, inwhich case the economy can end up in a recession. The reason a centralbank could easily increase interest rates too much is because the effectof any former rate hikes can only be observed several months later andtherefore results in uncertainty about the right course of action; keepincreasing rates or acknowledge the end of the rate hike cycle.An added feature of falling markets is that the number of short players will increase to take advantage of the declining market. This willsignificantly increase volatility as short players tend to work with tightstop losses, resulting in the so-called short squeezes, which cause bigmoves on the upside, after which the market typically falls again.A natural question is of course why cheap valuations of stocks donot result in volatile markets on the upside. The answer to this questionis that investors are by nature cautious and therefore any adjustmentof too cheap a market will come slowly and is therefore unvolatile. Aninvestor first wants to see confirmation of his view before he increases hispositions, i.e. buys more stocks. However, for individual stocks wherethere is very much uncertainty about their future earnings, the stockmight be volatile on the downside as well as on the upside. Therefore,individual stocks with a bullish sentiment can have very shallow skewsor even reverse skews. However, the market as a whole will always haveskew, i.e. options on indices will have skew built into them and willtypically have steeper skews than options on stocks.Yet another argument why stocks are more volatile on the downsidethan on the upside is that volatility goes hand in hand with fear ofthe investor. Fear in turn comes from investors being scared of losingmoney and therefore cutting positions. Since the market as a whole islong stocks, the fear of the total investor community will be largest whenstock markets go down as this is when the investor community as a wholeloses money and therefore will act in fear and irrationally, which resultsin higher volatilities.
5.2 SKEW THROUGH TIME: ‘THE TERMSTRUCTURE OF SKEW’A natural question is how skew behaves for different maturities. In otherwords, is the skew for long term maturities lower or higher than for shortterm maturities? The answer is that the skew for any specific stock is
Skew
29
higher (steeper) for short term maturities than for long term maturities.1To understand this, one first needs to make the observation that skewis mainly there because traders are afraid to lose money on downsidestrikes in case the market goes down and becomes more volatile. Thismeans that skew is a result of traders wanting to be compensated on shortdownside options because rebalancing the delta on these options morefrequently results in higher losses. Obviously, the larger the gamma themore imminent the problem. Since short term downside options havelarger gammas when the share price moves down to the lower strikes,the effect of skew is largest for short term options. Another reason is thatfor short term maturities the trader exactly knows whether an option is adownside option or not. For long term options the trader cannot qualifywhether it is a downside strike, as the trader does not know where thestock will be trading in, for example, two years’ time. This means thata two year downside option currently, might be an upside option in twoyears’ time if the markets go down. Two years from now is exactlywhen the effects of skew for this particular two year option are mostpronounced, hence reducing the effect of skew for longer term options.Graphically, the term structure of skew looks like Figure 5.1, where theskew is quantified as the volatility difference per 10 % strike differential.For example, Figure 5.1 shows that the one year maturity has a skew of2 volatility points differential between strikes that are 10 % apart.
5.3 SKEW AND ITS EFFECT ON DELTAThe skew curve of a particular stock can have a big impact on a trader’sdelta hedge against any option positions he has. Consider a trader whoowns a 1 year 120 % call on BMW. Suppose the skew curve of BMW forthe 1 year maturity indicates that every 10 % decrease in strike translatesinto a 1.5 % increase in implied volatility. This means that, if the 1 yearATM implied volatility of BMW is 20 %, the 1 year 120 % strike has animplied volatility of 17 %. Suppose that the trader decides to mark this120 % call on an implied volatility of 21 %, i.e. higher than the ATMimplied volatility. As a result the implied volatility curve plotted againststrike takes the shape of a smile. Because the trader marks the 120 %call on a 21 % implied volatility instead of a 17 % implied volatility, his1Only in very special situations is the skew for long term maturities higher than for short term maturities.For example, when a stock is expected to go bankrupt but people know it will not be in the next few months butany time from 6 months to 2 years.
Exotic Options Trading
Skew (vol difference per 10 % strike)
30
2
1Time to maturity (years)
Figure 5.1 Skew plotted per time to maturity, clearly showing that the shorter the timeto maturity the steeper the skew. Skew is expressed as the volatility differential per 10 %difference in strike
delta is larger than it should be and therefore sells more BMW sharesagainst the 120 % call. This can be seen intuitively by comparing deltato the probability that an option expires in the money.2 By increasing theimplied volatility the probability that the 120 % call expires in the moneyincreases and hence the delta increases. This shows that skew and theway a trader marks his volatility surface have a large impact on a trader’sdelta hedge. A good question is whether one would expect a downwardsloping implied volatility curve per maturity or a smile curve, i.e. bothhigher and lower strikes than ATM have a larger implied volatility thanthe ATM implied volatility. The answer to this question is that it dependson the asset class. For options on equities one can say that the impliedvolatility curve per maturity is always downward sloping, except forvery special situations. This can be seen by looking at the way a traderdelta hedges his options for upside strikes and by comparing the deltato the probability that the option expires in the money.2The probability√ that a call option expires in the money is N (d2 ) and the delta of a call option is N (d1 ),where d2 = d1 − σ T − t. Since d1 is also a function of σ it is the case that for upside strikes a higher σ resultsin a greater d2 and therefore a higher probability that the call expires in the money. This is also what one wouldexpect intuitively.
Skew
31
Assuming that a trader marks his upside strikes on a higher impliedvolatility than the ATM implied volatility means that he sells more sharesagainst these long upside calls than when the skew curve has a downwardsloping shape. This also means that, considering that the ATM impliedvolatility for this particular maturity is a trader’s best estimate for whatthe stock will realise (volatility wise) during the term of the option,the trader assigns too high a probability to the option expiring in themoney and therefore hedges the upside call on too high a delta. To makematters worse, the trader will continue to sell too many shares with theshare price going up. However, with the share price going up the stockis likely to realise even less, making the probability and therefore theproper delta of this option even lower. In other words, by marking upsidestrikes on a higher implied volatility than the ATM implied volatility,the trader sells too many shares and, especially when