A closer look at Black Scholes option thetas

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1 J Econ Finan (2008) 32:59 74 DOI 0.007/s A closer look a Black Scholes oion heas Douglas R. Emery & Weiyu Guo & Tie Su Published online: Ocober 2007 # Sringer Science & Business Media, LLC 2007 Absrac This aer invesigaes Black Scholes call and u oion heas, and derives uer and lower bounds for heas as a funcion of underlying asse value. I is well known ha he maximum ime remium of an oion occurs when he underlying asse value equals he exercise rice. However, we show ha he maximum oion hea does no occur a ha oin, bu insead occurs when he asse value is somewha above he exercise rice. We also show ha oion hea is no monoonic in any of he arameers in he Black Scholes oion-ricing model, including ime o mauriy. We furher exlain why he imlicaions of hese findings are imoran for rading and hedging sraegies ha are affeced by he decay in an oion s ime remium. Keywords Black Scholes oion ricing model. Oion hea. Time decay JEL Classificaions G0. G2 Inroducion An oion hea is he sensiiviy of he oion s rice o changes in he oion s ime o mauriy. I measures he rae a which he oion s ime remium decays over D. R. Emery : T. Su (*) Dearmen of Finance, Universiy of Miami, P.O. Box , Coral Gables, FL , USA ie@miami.edu D. R. Emery demery@miami.edu W. Guo Dearmen of Finance, Universiy of Nebraska-Omaha, 600 Dodge Sree, Omaha, NE 6882, USA wguo@unomaha.edu

2 60 J Econ Finan (2008) 32:59 74 ime, as he oion s remaining life dissiaes. Oion heas are used in risk managemen, and can be used o measure he cos and benefi of an oion hedge ha deends on he naural decline in ime o mauriy. This aer akes a closer look a Black Scholes oion heas of Euroean-syle call and u oions. The resuls answer wo quesions abou hese oion heas ha have imoran imlicaions for rading and hedging. Firs, a wha underlying asse value (relaive o he exercise rice) does he oion hea achieve is maximum? Or, equivalenly, a wha underlying asse value does an oion s ime remium have he fases rae of dissiaion over ime? Second, are oion heas monoonic funcions of he oher arameer values, he underlying asse s reurn volailiy, he oion s ime o mauriy, and he risk-free ineres rae? The answers o hese quesions can hel oion raders, hedgers, and seculaors o beer osiion heir oion rading sraegies and offer hem he ossibiliy of managing he decay of oion ime remium o heir benefi. In he U.S., index oions are exremely acively raded. Trading volume of U. S. index oions is much higher han ha of oions on individual socks. All index oions, wih he exceion of oions on he S&P 00 OEX Index, are Euroean syle. While all oions on individual socks are American-syle oions, oion heory ells us ha if he underlying sock does no ay dividends before he exiry of he oion, hen he value of an American-syle call oion is he same as he value of an oherwise idenical Euroean-syle call oion. Given he fac ha fewer han 22% of all socks ay any dividend, and he fac he average dividend-aying socks roduce a dividend yield ha is less han %, we conclude ha here is significan amoun of rading in Euroean-syle and Euroean-syle like oions, where he Black Scholes oion ricing model can be used o roduce aroximaion of oion rices. The Black and Scholes (973) oion-ricing model was a seminal breakhrough in ricing derivaives. Numerous sudies have examined he model s erformance wih resec o ricing. Black (975), Emanuel and MacBeh (982), MacBeh and Merville (979), and Rubinsein (985) all reor ha he Black Scholes model ends o sysemaically misrice in-he-money and ou-of-he-money oions. However, he model is accurae enough ha finance rofessionals rouinely use exended and modified versions of he Black Scholes model o value many yes of oions, including equiy oions. Oion heas have araced significan aenion from boh academia and raciioners. Discussions of oion heas can be found in almos all exbooks on derivaives and a number of academic aers. However, roeries of oion heas have no been closely examined. Pelsser and Vors (994) and Chung and Shackleon (2002) use efficien numerical differeniaion mehods o comue oion heas. Chance (994), focusing rimarily on call oions, ariions a call oion ino a margin value and an insurance olicy, and discusses inerreaions of oion Greeks. Naib (996) sudies he oion effec, underlying effec, and curve effec in an oion hea. Alexander and Suzer (996) resen a grahical descriion of Black Scholes u oion heas and ime remiums.

3 J Econ Finan (2008) 32: This aer is mos closely relaed o Alexander and Suzer (996). We examine he roeries of boh call and u oion heas in he Black Scholes model. We use he Black Scholes oion-ricing framework because of is wide acceance, is simliciy and elegance, and is mahemaical racabiliy. All conclusions and inferences derived in his aer can be used as benchmarks for oher more sohisicaed oion-ricing models. Our aer conribues o he oions lieraure by focusing on five issues ha have no been addressed by revious sudies: () deermining he uer and lower bounds on oion heas, (2) deriving he relaion beween he value of he underlying asse and oion heas, (3) deriving he relaion beween he underlying asse s reurn volailiy and oion heas, (4) deriving he relaion beween he oion s remaining ime o mauriy and oion heas, and (5) deriving he relaion beween he risk-free ineres rae and oion heas. In addiion, we examine he imlicaions of our findings for rading and hedging wih oions. The aer is organized as follows. The nex secion resens he Black and Scholes (973) oion-ricing model and he Black Scholes oion heas for Euroean-syle call and u oions. The subsequen five secions address each of he five issues abou oion heas deailed above. The eighh secion discusses how our resuls can be alied in hedging and rading oion combinaions. The final secion concludes he aer. There are four aached aendices. 2 Black Scholes oion heas The Black Scholes oion-ricing model for Euroean-syle call and u oions is given by he following sandard noaion: C ¼ S 0 Nd ð Þ Xe r Nd ð 2 Þ; P ¼ Xe r Nð d 2 Þ S 0 Nð d Þ where d ¼ ln ð S 0=X Þþðrþs ffi 2 =2Þ s, and d 2 ¼ d s ffiffi ; C and P are he values of Euroean-syle call and u oions, resecively; S 0 is he curren value of he underlying non-dividend aying asse; X is he oion s exercise rice; r is he annualized coninuously comounded risk-free rae of ineres; is he oion s ime o mauriy; A is he sandard deviaion of he rae of reurn on he underlying asse; N(d) is he cumulaive disribuion funcion of he sandard normal disribuion, whose robabiliy densiy funcion is n(d). Throughou he aer, we use hree imoran equaions in ¼ nd ð Þ ¼ ffiffiffiffi ex d2 2 ¼ d nd ð Þ 3. S 0 nd ð Þ ¼ Xe r nd ð 2 Þ

4 62 J Econ Finan (2008) 32:59 74 Algebraically, he oion hea is he arial derivaive of he oion s value wih resec o ime o mauriy. The follow equaions rovide close-form exressions of Black Scholes call and u oion heas, θ C and θ P : C ¼ S 0 Nd ð Þ Xe r Nd ð 2 Þ θ ¼ S 0 nd ð Xe r nd ð 2 2 ¼ S 0 nd ð ð d d 2 ¼ S 0 nd ð σ þ rxe r Nd ð 2 Þ þ rxe r Nd ð 2 Þ θ C ¼ 2 ffiffi S 0 nðd Þσ þ rxe r Nðd 2 Þ > 0 þ rxe r Nd ð 2 Þ The u oion hea can be derived from Eq., using u-call ariy for Euroean-syle oions S 0 +P=C+Xe r : () P ¼ C þ Xe r S 0 q ð C þ Xe r S 0 Þ ¼ q C rxe ð2þ From Eq. 2 we have: q P ¼ 2 ffiffi S 0 nd ð Þs rxe r ð Nd ð 2 ÞÞ ð3þ q C q P ¼ rxe r Noe ha Eqs. and 3 deend on he Black Scholes oion ricing formula. However, Eq. 4 holds in general regardless validiy of he Black Scholes model assumions. I can be easily derived from he u-call ariy relaion of Euroeansyle oions, which is based on no-arbirage condiions. ð4þ 3 Uer and lower bounds on heas In his secion, we derive he uer and lower bounds on call and u oion heas as funcions of underlying asse value, S 0. The boundary condiions rovide he maximum and minimum raes a which he oion s ime remium dissiaes over ime. Equaion shows ha he call oion hea is a sum of wo sricly osiive erms. As a resul, he call oion hea is always osiive. The ime remium of a Euroean-syle call oion is osiively relaed o is ime o mauriy. As ime o mauriy decreases, he call oion s ime remium always decreases. Holding all

5 J Econ Finan (2008) 32: oher arameers consan, as he underlying asse value goes o osiive infiniy, he call oion hea aroaches rxe r. As he underlying asse value goes o zero, he call oion hea aroaches zero, which is he lower bound of he call oion hea. From Eq. 3, we can see ha he Black Scholes u oion hea aroaches zero as he underlying asse value goes o osiive infiniy, and ha as he underlying asse value goes o zero, he hea aroaches rxe r, which is he lower bound of he u oion hea. Also noe ha he u oion hea is he difference beween wo osiive erms, and consequenly, he difference can be eiher osiive or negaive. Figure rovides a grahical resenaion of Black Scholes oion heas as funcions of underlying asse value. The arameer values used in he Black Scholes oion-ricing model in Fig. are X=$00, =2 monhs=0.667 years, r=5%, and σ=0.40. The figure demonsraes he following five roeries of oion heas: () as S 0 0, θ C 0, (2) as S 0 +, θ C rxe r =4.96, (3) as S 0 0, θ P rxe r = 4.96, and (4) as S 0 +, θ P 0. (5) Alhough no visually obvious from he grah because of he changes in sloe, he difference beween θ C and θ P is consan wih resec o he value of he underlying asse, i.e., for he given values of, r, and, X, θ C θ P =rxe r =4.96, S 0 R +. This resul can be seen direcly in Eq. 4, which shows ha he difference, rxe r, does no deend on he underlying asse s value. Of course, he difference does deend on, r, and X. Therefore, noe ha he difference increases wih he decline in as he oions aroach exiraion. Also, he difference would increase if here were an increase in eiher he risk-free ineres rae (as long as r>0) or he exercise rice. In Fig., noe ha here is a criical oin where he u oion hea curve inersecs he x-axis. Below ha criical value of he underlying asse, he u oion hea is negaive. Therefore, a sufficienly low underlying asse value causes he rae of ime remium decay on a Euroean-syle u oion o be negaive. In such a case, a decrease in he ime o mauriy acually causes an increase in he ime remium and value of he u oion. Above he criical value, he u oion hea is osiive, and jus as i does wih a call oion, a decrease in he ime o mauriy causes a decrease in he ime remium and value of he u oion. 4 Maximum hea and underlying asse value Traders ofen use oions o hedge or seculae. The decay of an oion s ime remium is an imoran consideraion in he cos of such rading sraegies. In his secion, we examine call and u oion heas as a funcion of underlying asse As S 0 aroaches o osiive infiniy, n(d ) aroaches o zero a a faser rae han S 0 grows. Consequenly he firs roduc erm in Eq. aroaches o zero. N(d 2 ) aroaches o as S 0 aroaches o osiive infiniy. As a resul, he whole exression in Eq. aroaches o rxe r.

6 64 J Econ Finan (2008) 32:59 74 Black-Scholes Call and Pu Oion Theas As a Funcion of Underlying Asse Value Call Thea Call and Pu Oion Theas Pu Thea S*=X*Ex(r*+variance*/2) Underlying Asse Value Fig. Black Scholes call and u oion heas as a funcion of underlying asse value. This figure los Black Scholes call and u oion heas as a funcion of underlying asse value. Parameers in he Black Scholes oion-ricing model are X=$00, =2 monhs=2/2 years, r=5%, and σ=0.40. The figure demonsraes he following six roeries of oion heas: () as S 0 0, θ C 0, (2) as S 0 +, θ C rxe r =4.96, (3) as S 0 0, θ P rxe r = 4.96, (4) as S 0 +, θ P 0, and (5) arg max(θ C )=arg max(θ P )=S*=X ex(r+σ 2 /2)=02.9. Furher, even hough i may no be visually obvious, he differences beween he call oion hea and u oion hea say a a consan across all moneyness, i.e., (6) θ C θ P =rxe r =4.96, S 0 R + value. In aricular, we deermine he underlying asse value a which oion heas reach heir maximums. Recall from Eq. 4 ha he difference beween he call and u oion heas is a consan wih resec o he value of he underlying asse, so ha he sensiiviies of call and u oion heas wih resec o underlying asse value are idenical. Therefore, call and u oion heas reach heir maximum a he same underlying asse value. This saemen holds rue indeenden from Black Scholes assumions because Eq. 4 can be imlied by Euroean-syle oion u-call ariy relaion. I is well known ha, wih all else equal, an oion s ime remium achieves is maximum value when he value of he underlying asse equals he exercise rice (Meron 973; Smih 976). From his resul, inuiion migh lead o a belief ha oion heas would also achieve heir maximum a his same oin. However, we show ha his is no he case. The maximum oion hea is reached when he arial derivaive of hea wih resec o underlying asse value equals zero. In Aendix A, we derive he underlying asse value ha maximizes Euroean-syle call and u oions heas by ¼ 0. Aendix A shows ha oion heas acually achieve heir maximum value when he underlying asse is somewha above he exercise rice: arg max ðq C Þ ¼ arg max ðq P Þ ¼ S ¼ X ex r þ s 2 2 ð5þ Therefore, oion heas for boh call and u oions are maximized when he underlying asse value exceeds he exercise rice by a facor of ex(r+σ 2 /2).

7 J Econ Finan (2008) 32: This oin is in-he-money for he call oion, and ou-of-he-money for he u oion. For examle, wih r=5%, =2/2, and σ=0.40, his facor equals ex r þ s 2 2 ¼ ex 0:05 2=2 þ 0:40 2 2=2=2 ¼ 0:029 Therefore, wih hese arameer values, oion heas achieve heir maximum values when he value of he underlying asse is 2.9% above he exercise rice. Wih X=$00, he maximum hea values would occur a an underlying asse value of $02.9. Figure illusraes his relaion by grahing oion hea values as a funcion of underlying asse value for hese values of he risk-free ineres rae, ime o mauriy, and underlying asse reurn volailiy. From Eq. 5, we can see ha he underlying asse value a which he oion heas reach heir maximum decreases wih he decline in as he oions aroach exiraion. I also shows ha he underlying asse value ha maximizes he oion heas increases wih increases in he oher variables, he risk-free ineres rae, he volailiy of he reurns of he underlying asse, he exercise rice. 5 Non-monooniciy of hea as a funcion of sigma Oion value, and more secifically, an oion s ime remium, is known o be a monoonic funcion of he underlying asse s reurn volailiy; he higher he volailiy, he larger he ime remium, and consequenly, he higher he value of he oion. Again, inuiion migh lead one o seculae ha he same would hold for oion heas, and again, inuiion would lead us asray. In his secion, we show ha oion heas are no monoonic funcions of reurn volailiy. We firs noe ha he sensiiviy of he call oion hea wih resec o reurn volailiy is exacly he same as i is for ha of he u oion. Once again, his resul is clear from Eq. 4, and arallels he sensiiviy o underlying asse value. Aendix B derives he arial derivaive of he oion heas wih resec o underlying asse reurn volailiy. Secifically, Aendix B shows ¼ S 0nd ð Þ h i 2 :5 s 2 ½ln ðs 0 =X ÞŠ 2 r 2 2 0:25s 4 2 þ s 2 rs 2 2 ð6þ The arial derivaive can be osiive or negaive, which roves ha he oion heas are no monoonic in reurn volailiy. Figure 2 los he sensiiviy of Black Scholes call and u oion heas as a funcion of sigma, he reurn volailiy of he underlying sock. This sensiiviy is he arial derivaive of he call and u oion hea wih resec o sigma. This sensiiviy changes dramaically as sigma aroaches zero from he osiive side. Therefore, o rovide a clearer figure, we ransform he horizonal axis from sigma o he logarihm of sigma. For examle, 0.0 on he horizonal axis reresens ln(sigma)=0.0, or sigma=ex(0.0) =00% of annualized underlying asse reurn volailiy. Similarly, 2.0 on he horizonal axis reresens ln(sigma) = 2.0, or sigma=ex( 2.0) = 3.53% of annualized underlying asse reurn volailiy. Our ransformaion sreches he curve horizonally as sigma aroaches zero. Parameers in he Black Scholes oion-ricing model are S 0 =$00, X=$00, =6 monhs=0.5 years, and r=5%. As

8 66 J Econ Finan (2008) 32:59 74 Black-Scholes Oion Thea Sensiiviy wih Resec o Sigma Thea Sensiiviy Log Sigma Fig. 2 Black Scholes call and u oion hea sensiiviy wih resec o changes in sigma. This figure los he sensiiviy of Black Scholes call and u oion heas as a funcion of sigma, he reurn volailiy of he value of he underlying asse. The verical axis is he arial derivaive of boh call and u oion heas wih resec o sigma. The horizonal axis is he logarihm of sigma. For examle, 2.0 on he horizonal axis reresens ln(sigma)= 2.0, or sigma=ex( 2.0)=3.53% of annualized securiy reurn volailiy. Parameers in he Black Scholes oion-ricing model are S 0 =$00, X=$00, =6 monhs= 6/2 years, and r=5%. As shown mahemaically in Aendix B, he figure confirms ha Black Scholes oion heas are no monoonic in sigma. Thea sensiiviy becomes negaive when sigma is sufficienly close o zero or sufficienly large. Noe ha he sensiiviy of call oion hea wih resec o changes in sigma is he same as ha of a u oion hea is shown mahemaically in Aendix B, he figure confirms ha Black Scholes oion heas are no monoonic in sigma. The sensiiviy of oion heas wih resec o sigma becomes negaive when sigma is sufficienly close o zero or sufficienly large. However, for mos commonly seen reurn volailiies of 20% o 200% (.6 o 0.7 on he log volailiy scale on he horizonal axis), Fig. 2 shows ha he sensiiviy of hea o sigma is osiive. 6 Non-monooniciy of hea as a funcion of ime o mauriy We also sudy he behavior of oion heas as a funcion of an oion s remaining ime o mauriy. I is well known ha oion value generally declines as ime o mauriy dissiaes. In aricular, he value of a Euroean-syle call oion declines monoonically as ime o mauriy dissiaes. However, here are arameer values for which he value of a Euroean-syle u oion can acually increase as ime o mauriy dissiaes, so ha a u oion s ime remium is no sricly monoonic in ime o mauriy.

9 J Econ Finan (2008) 32: In his secion, we show ha call and u oion heas can eiher increase or decrease as he oion s ime o mauriy dissiaes. Therefore, oion heas are no monoonic funcions of ime o mauriy. Noe ha, unlike he siuaions wih underlying asse value and sigma, he sensiiviies of he call and u oion heas wih resec o ime o mauriy are no he same. Figure 3 los Black Scholes call oion heas as a funcion of he oion s ime o mauriy. There are hree curves in he figure. Parameers in he Black Scholes oion-ricing model are X=$00, r=5%, and σ=0.40. The o curve los a-hemoney call oion heas, where S 0 =$00. The middle curve los in-he-money call oion heas, where S 0 =$20. The boom curve los ou-of-he-money call oion heas, where S 0 =$80. Noe ha near-he-money oion heas behave similarly o ahe-money oions heas. As he oion s ime o mauriy dissiaes, away-fromhe-money call oion heas decrease, bu a-he-money call oion heas increase. The direcions of he hree curves mach inuiion. We omi u oion heas from he figure because one can easily use Eq. 4 o comue u oion heas. Noe ha as a u oion s ime o mauriy aroaches zero, is hea aroaches he call oion hea minus rx, i.e., as 0, θ P θ C rx. Call Oion Theas As a Funcion of Time o Mauriy A-he-money call oion hea 30 Call Oion Thea In-he-money call oion hea Ou-of-he-money call oion hea Time o Mauriy (Years) Fig. 3 Black Scholes call oion heas as a funcion of ime o mauriy. This figure los Black Scholes call oion heas as a funcion of oion s ime o mauriy and moneyness. Parameers in he Black Scholes oion-ricing model are X=$00, r=5%, and σ=0.40. There are hree curves in he figure. The o curve los a-he-money call oion heas, where S 0 =$00. The middle curve los in-he-money call oion heas, where S 0 =$20. The boom curve los ou-of-he-money call oion heas, where S 0 =$80. As oion s ime o mauriy dissiaes, away-from-he-money call oion heas decrease, bu a-hemoney call oion heas increase

10 68 J Econ Finan (2008) 32: Non-monooniciy of hea as a funcion of he risk-free ineres rae Finally, we examine Black Scholes oion heas as a funcion of he risk-free ineres rae. In Aendix D we derive he sensiiviy of Black Scholes call and u oion heas wih resec o changes in he risk-free ineres rae. The arial derivaives of heas are no always osiive for boh call and u oions. Consequenly, he Black Scholes oion heas are no monoonic funcions of he risk-free ineres rae. Figure 4 los he arial derivaive of Black Scholes call and u oion heas wih resec o he risk-free ineres rae. Parameers in he Black Scholes oionricing model are S 0 =$80, X=$00, =6 monhs, and σ=0.70. The o curve los call oion hea sensiiviy and he boom curve is he u oion hea sensiiviy. The call oion hea sensiiviy is osiive for mos arameer values. However, for very large values of sigma (no shown in Fig. 4), he call oion curve can go negaive, showing numerically ha he Black Scholes call oion hea is no monoonic funcions of he risk-free ineres rae. Non-monoonic behavior of hea occurs only using exreme values of sigma. Black-Scholes Oion Thea Sensiiviy o Risk-Free Ineres Rae Call oion hea sensiiviy o ineres rae changes Thea Sensiiviy Pu oion hea sensiiviy o ineres rae changes Risk-Free Ineres Rae Fig. 4 Black Scholes call and u oion hea sensiiviy wih resec o changes in risk-free ineres rae. This figure los he sensiiviy of Black Scholes call and u oion heas as a funcion of risk-free ineres rae. The verical axis is he arial derivaive of boh call and u oion heas wih resec o r. The horizonal axis is he risk-free ineres rae. Parameers in he Black Scholes oion-ricing model are S 0 =$80, X=$00, =6 monhs=6/2 years, and sigma=0.70. The exac mahemaical exressions of arial derivaives are given in Aendix D. Noe ha he sensiiviy of call oion hea o ineres rae changes ends o be osiive for mos arameer seings, bu he sensiiviy can become negaive (no shown in his figure) for very large values of sigma

11 J Econ Finan (2008) 32: Trading imlicaions Hedgers and seculaors who ake oion osiions can make use of our resuls. Figure and Eq. 5 show ha somewha in-he-money call oions and somewha ou-of-he-money u oions have he highes hea. All else equal, a rader aking a long osiion in such oions will end o suffer he mos loss from ime remium decay. Consequenly, in aking a long osiion, a rader may wan o consider using farher away-from-he-money oions o minimize he amoun los o ime remium decay. In he mirror-image case, a rader aking a shor osiion will end o rea he larges gain from ime remium decay if he value of he underlying asse is somewha above he exercise rice. Consequenly, in aking a shor osiion, a rader may wan o consider using oions wih an exercise rice ha is somewha below he value of he underlying asse o maximize he amoun gained from ime remium decay. A rader engaging in a bull call oion sread buys a call oion wih a low exercise rice and sells a call oion wih a higher exercise rice o aniciae oenial uward movemen in he underlying asse s value rior o he oion s exiraion. If he rader uses call oions ha are boh in-he-money, hen Fig. imlies ha he call oion wih he higher exercise rice has a higher hea han he call oion wih a lower exercise rice. As a resul, he rader gains more from ime remium decay on he shor call oion wih he higher exercise rice han he rader loses from ime remium decay on he long call oion wih he lower exercise rice. The rader caures a ne gain from ime remium decay in he bull sread. However, aking a bull call oion sread using ou-of-he-money call oions reverses he resul. Figure imlies ha a bull sread using ou-of-he-money call oions resul in a ne loss from ime remium decay. A calendar sread (also known as a ime, or horizonal, sread) involves buying an oion and selling an oherwise idenical oion wih a differen ime o mauriy. Mos rading sraegies relaed o calendar sreads are ime-remium lays. Figure 3 offers insighs ino how o rofi from a call oion calendar sread. If he oions are a-he-money, he rader should sell he shorer-ime-o-mauriy oion and buy he longer-ime-o-mauriy oion because he a-he-money oion hea line in Fig. 3 shows ha shorer-ime-o-mauriy oions lose ime remium a a higher rae han longer-ime-o-mauriy oions. If he call oions are away-fromhe-money, hen he boom wo lines in Fig. 3 sugges an oosie sraegy, i.e., buy he shorer-ime-o-mauriy oion and sell he longer-ime-o-mauriy oion, because longer-ime-o-mauriy oions lose ime remium a a higher rae han shorer-ime-o-mauriy oions. Consequenly, a rader should buy an oion wih lile ime o mauriy and sell a longer-ime-o-mauriy oion o consruc a rofiable calendar sread. We have rovided wo secific examles of rading sraegies where ime remium decay migh be managed o an advanage. There are, of course, oher sraegies o which he resuls here can also be generalized, such as sraddles, sras, sris, buerfly sreads, raio sreads, among ohers. A more comlee undersanding of oion heas can hel marke aricians o formulae beer oion rading sraegies.

12 70 J Econ Finan (2008) 32: Conclusion This aer examines call and u oion heas wihin he Black Scholes oionricing framework. Secifically, we address five issues. Firs, Black Scholes call and u oion heas are bounded. We rovide heir uer and lower bounds and discuss heir relaion. Second, we derive he criical underlying asse value ha maximizes call and u oion heas. Third, we demonsrae ha Black Scholes oion heas are no monoonic funcions of he reurn volailiy of he underlying asse. Fourh, we show how remaining ime o mauriy affecs oion heas. Fifh, we derive he relaion beween oion heas and he risk-free ineres rae. Alying he resuls from his aer, we discuss imoran rading and hedging imlicaions associaed wih oion heas. We show ha a bull oion sread wih in-he-money call oions is more favorable wih he shor (long) osiion in he oion ha has he higher (lower) exercise rice. We also show ha a call oion calendar sread is more favorable wih he shor (long) osiion in he oion ha has he shorer (longer) ime o mauriy. Parallel conclusions hold for many oher sread sraegies, such as buerfly sreads and raio sreads. Our findings have aricular relevance for rading and hedging sraegies in risk managemen ha are affeced by he decay in an oion s ime remium. Acknowledgmen We hank Gordon Alexander for helful commens. Aendix A This aendix derives he underlying asse value ha maximizes a Euroean-syle call and u oions heas. Secifically, we rove he following equaion based on he Black Scholes oion ricing framework: arg max ðq C Þ ¼ arg max ðq P Þ ¼ S ¼ X ex r þ s 2 2 : The derivaion is based on he firs-order condiion. We comue he arial derivaive of he oion s hea, se i o zero, and solve for he criical value of he underlying P ð q C rxe r ¼ ffi 2 i S 0 nd ð Þs þ rxe r Nd ð 2 0 ¼ 2 ffiffi nd ð Þs þ 2 ffiffi S 0 nd ð Þsð d Þ S 0 s ffiffi þ rxe r nd ð 2 Þ S 0 s ffiffi ¼ 2 ffiffi nd ð Þs 2 nd ð Þd þ rs 0 nd ð Þ S 0 s ffiffi ¼ 2 ffiffi nd ð Þs 2 nd ð Þd þ rnðd ¼ nd ð Þ 2 ffiffi s 2 d þ r s ffiffi Þ s ffiffi

13 J Econ Finan (2008) 32: ffiffi ¼ nd ð Þ s2 d s þ 2r 2s ¼ nd ð Þ s2 ln S 0 =X ffiffi ð Þþ r þ s 2 2 þ 2r ffiffi 2s ¼ nd ð Þ ln ðs 0=X Þ r s2 ffiffi 2 2s ¼ 0, we obain S0 ¼ Xe ð rþs2 =2Þ ¼ X ex r þ s 2 2. Aendix B This aendix shows ha oion heas are no monoonic funcions of sigma, he reurn volailiy of he underlying asse. We derive he arial derivaive of heas wih resec o sigma and show ha he arial derivaive can be eiher osiive or negaive, which roves ha he oion heas are no monoonic in sigma. The following derivaion uses ¼ d 2 s ¼ d s. ð θ C rxe r ffi ¼ 2 S 0 nd ð Þσ þ rxe r Nd ð 2 ¼ 2 ffiffi S 0 nd ð Þþ 2 ffiffi σs 0 nd ð Þð þ rxe r nd ð 2 ¼ 2 ffiffi S 0 nd ð Þþ 2 ffiffi σs 0 nd ð Þð þ 2 0nd ð ¼ 2 ffiffi S 0 nd ð Þþ 2 ffiffi σs 0 nd ð Þð d Þ d 2 σ þ rs d 0nd ð Þ σ ¼ S 0nd ð Þ 2 ffiffi σ þ d d 2 σ 2 ffiffi rd σ ¼ S 0nd ð Þ 2 ffiffi σ þ ln ½ S 0= ðxe r ÞŠþ 2 σ2 σ σ ffiffi ln ½S 0 = ðxe r ÞŠ 2 σ2 σ ffiffi σ 2 ffiffi ln ½S 0 = ðxe r ÞŠþ r 2 σ2 σ ffiffi " ¼ S 0nd ð Þ 2 ffiffi f σ þ ln ½ S 0= ðxe r ÞŠg 2 0:25σ 4 2 2r σ σ σ ln ½ S 0= ðxe r ÞŠþ # 2 σ2 ¼ S 0nd ð Þ h i ln S 2 :5 σ 2 f ½ 0 = ðxe r ÞŠg 2 2rfln ½S 0 = ðxe r ÞŠg 0:25σ 4 2 þ σ 2 rσ 2 2 ¼ S 0nd ð Þ h i ½ln ðs 2 :5 σ 2 0 =X ÞŠ 2 r 2 2 0:25σ 4 2 þ σ 2 rσ 2 2 The erm, r σ 4 2 +σ 2 rσ 2 2 can be negaive, as σ aroaches zero or osiive infiniy. I imlies ha he whole can be eiher osiive

14 72 J Econ Finan (2008) 32:59 74 or negaive, i.e., call and u oion heas are no monoonic funcions in he reurn volailiy of he underlying asse. Aendix C This aendix derives he exression of he sensiiviy of oion hea o changes in he oion s ime o mauriy. The exression of he arial derivaive of hea wih resec o ime o mauriy does no allow us o solve for he criical remaining life ha will maximize hea. Numerical mehods are needed o solve his C ffi i 2 S 0 nd ð Þσ þ rxe r Nd ð 2 Þ ¼ ffiffi S 0 nd ð Þσ þ 4 2 ffiffi S 0 nd ð Þσð r 2 Xe r Nd ð 2 Þ þ rxe r nd ð 2 ¼ ffiffi S 0 nd ð Þσ þ 4 2 ffiffi S 0 nd ð Þσð d Þ r þ σ2 2 σ ffiffi d 2 r 2 Xe r Nd ð 2 ÞþrXe r r nd ð 2 Þ σ ffiffi d 2 ¼ S 0nd ð Þ ffiffi σ 2 þ d σ2 rþ σ2 4σ 2 σ ffiffi d 4 ffiffi r rσ 2 σ ffiffi d 2 r 2 Xe r Nd ð 2 Þ ¼ S 0nd ð Þ ffiffi σ 2 þd σ2 ffiffi ffiffi rþd σ 3 d 2 4σ σ 2 4r 2 þ2 ffiffi rσd r 2 Xe r Nd ð 2 Þ ¼ S 0nd ð Þ ffiffi σ 2 þ4 ffiffi rσd σ 2 d d 2 4r 2 r 2 Xe r Nd ð 2 Þ 4σ ¼ S 0nd ð Þ ffiffi σ 2 d d 2 4 ffiffi rσd þ 4r 2 σ 2 r 2 Xe r Nd ð 2 Þ 4σ ð q C rxe r þ r 2 Xe r ¼ S 0nd ð Þ ffiffi s 2 d d 2 4 ffiffi rsd þ 4r 2 s 2 þ r 2 Xe r ð Nd ð 2 ÞÞ 4s The above exressions can be eiher osiive or negaive in value for various arameer inus. I shows ha call and u oion heas are no monoonic funcions in oions remaining ime o mauriy.

15 J Econ Finan (2008) 32: Aendix D This aendix derives he exression of he sensiiviy of oion hea o changes in he risk-free ineres rae. The exression of he arial derivaive of hea wih resec o he risk-free ineres rae does no allow us o sign he exression easily. Numerical mehods are needed o resolve his ffi i ¼ 2 S 0 nd ð Þs þ rxe r Nd ð 2 ¼ 2 ffiffi S 0 nd ð Þsð þ Xe r Nd ð 2 Þ rxe r Nd ð 2 ÞþrXe r nd ð 2 ¼ 2 ffiffi S 0 nd ð Þsð d Þ s ffiffi þ Xe r Nd ð 2 Þ rxe r Nd ð 2 ÞþrXe r nd ð 2 Þ s ffiffi ¼ ffiffi 2 S 0d nd ð ÞþXe r Nd ð 2 Þ rxe r Nd ð 2 ÞþrXe r nd ð 2 Þ s ffiffi ¼ Xe r Nd ð 2 Þð rþþxe r r nd ð 2 Þ s d ð q C rxe r Þ C Xe r þ rxe r ffiffi ¼ Xe r Nd ð 2 Þð rþþxe r r nd ð 2 Þ s d Xe r ð rþ 2 ffiffi ¼ Xe r ð rþðnd ð 2 Þ ÞþXe r r nd ð 2 Þ s d 2 ffiffi ¼ Xe r Nð d 2 Þð rþþxe r r nd ð 2 Þ s d 2 The above exressions can be eiher osiive or negaive for various arameer inus, and demonsraes ha call and u oion heas are no monoonic funcions of he risk-free ineres rae. References Alexander G, Suzer M (996) A grahical noe on Euroean u heas. J Fuures Mark 6: Black F (975) Fac and fanasy in he use of oions. Financ Anal J 3:36 72 Black F, Scholes MS (973) The ricing of oions and cororae liabiliies. J Poli Econ 8: Chance DM (994) Translaing he Greek: he real meaning of call oion derivaives. Financ Anal J 50(4):43 49 Chung S, Shackleon M (2002) The binomial Black Scholes model and he Greeks. J Fuures Mark 22(2):43 53 Emanuel DC, MacBeh JD (982) Furher resuls on he consan elasiciy of variance call oion ricing model. J Financ Quan Anal 8:

16 74 J Econ Finan (2008) 32:59 74 MacBeh JD, Merville LJ (979) An emirical examinaion of he Black Scholes call oion ricing model. J Finance 34:73 86 Meron RC (973) Theory of raional oion ricing. Bell J Econ Manage Sci 4:4 83 Naib FA (996) FX oion hea and he joker effec. Deriv Q 2(4):56 58 Pelsser A, Vors T (994) The binomial model and he Greeks. J Deriv (3):45 49 Rubinsein M (985) Nonarameric ess of alernaive oion ricing models using all reored rades and quoes on he 30 mos acive CBOE oion classes from Augus 23, 976 hrough Augus 3, 978. J Finance 40: Smih CW Jr (976) Oion ricing: a review. J Financ Econ 3:3 5

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