ANALYTICAL PRICING AMERICAN CALL OPTION WITH KNOWN DIVIDEND

Transcription

1 NLYIL PRIING MRIN LL OPION WIH KNOWN DIVIDND hie-bein hen Dep. of Inernaional Business Naional Dong Hwa Universiy Haulien, aiwan, R. O.. Wen-Hai hih Dep. of Business minisraion Naional Dong Hwa Universiy Haulien, aiwan, R. O.. Hsin-Yuan hang Dep. of Business minisraion Naional Dong Hwa Universiy Haulien, aiwan, R. O heng-lun Ko Dep. of Inernaional Business Naional Dong Hwa Universiy Haulien, aiwan, R. O.. BSR he valuaion of merican opions on ivien-paying asse is an imporan problem in financial economics. merican opions provie early exercise opporuniies o pose he aiional ifficuly o obain he close-form soluion. In his suy, a recursive formula is evelope for eermining he opimal exercise price of merican call opions wih known ivien base on he backwar ynamic programming recursions an he Black-Scholes moel combine wih he Maringale Pricing an he Girsanov heorem uner he risk-neural measure. lose-form soluion for merican call opion is obaine by aking he risk-neural expecaion of is payoff an iscouning i o he curren. Previous aemps a pricing hese opions have been accurae bu compuaionally expensive. his paper provies a simple, an an inexpensive soluion for pricing merican call opion. KYWORD: merican opion pricing, Maringale Pricing, Girsanov heorem, backwar ynamic programming recursions. INRODUION n opion is a securiy which gives is owner he righ o rae in a fixe number of shares of a specifie common sock a a fixe price a any ime on or before a given ae (Black an Scholes, 973; ox e al., 979). uropean opion can be exercise only a he en of is life; an merican opion

2 can be exercise a any ime uring is life (Hull, 003). his opion wrien on a wie variey of commoiies an commoiy fuures conrac now rae in he U.S. an anaa. n merican opion has early exercise premiums implicily embee in heir prices (Barone-esi an Whaley, 987). Since he holer of merican opion can choose he exercise ime, i is more complicae o analyze his siuaion han is uropean counerpar (Yang, 00). he soluion o he merican opion is more challenging an o ae here is non-exising close-form moel available for praciioners in he fiel of finance (Hon, 00; Yang, 00; Unerwoo an Wang, 00). I is well known ha he merican opion pricing can be reae as a free bounary problem in which no analyical formula is available. Unil recenly, here were a number of ifferen numerical mehos for valuaion of he merican opion. For insance, he finie ifference meho (Brennan an Schwarz, 978) was inrouce o price merican opion, an ox e al. (979) propose he binomial meho for valuaion merican opion. hese mehos iscreize boh ime an he sae spaces in orer o approximae he opion price. he fron-fixing finie ifference meho (Wu an Kwok, 997), he Mone arlo simulaion (Gran e al., 996), an he inegral equaion meho (Huang e al., 996) are use o approximae he merican opion. comparison of hese numerical mehos can be foun in review papers (Geske an Shasri, 985; Broaie an Deemple, 996). Some kins of quasi-analyical formulas for valuaion of he merican opion have been propose (Johnson, 983; Geske an Johnson, 984; MacMillan, 986; Barone-esi an Whaley, 987). he Geske an Johnson (984) gave an exac analyical soluion for he merican opion pricing problem, bu heir formula is an infinie series ha can only be evaluae approximaely by numerical mehos. he quaraic meho (Barone-esi an Whaley, 987; MacMillan, 986) is base on exac soluion o approximaions of he opion parial ifferenial equaion. he numerical soluions are expensive an o no offer he inuiion which he comparaive saics of an analyic soluion provie. n analyic approximaion has been evelope by Johnson (983), bu i oes no hanle iviens or hege raio an here is no way o make his approximaion arbirarily accurae. he aim of his paper is o eermine he opimal criical price an o price he value of merican call opion uner risk-neural measure, analyically. In his research, backwar ynamic programming recursions (Winson, 994; Dixi, 990), he Maringale Pricing (Nefci, 000), an he Girsanov heorem (Nefci, 000) will be applie o erive recursive formula. By using back forwar meho, he value of merican call opion on ime ( < ) is obaine an hen he expece value of merican call opion on is solve by Maringale pricing echnique uner risk neural measure o ge recursive formula. Nex secion presens he analyical soluion of non-ivien uropean call an ivien uropean call. In Secion 3, a wice exercisable an riple exercisable call opion an he fair value of calls an criical prices are erive. Furhermore, a muli-exercisable call opion is evaluae an heir criical price are obaine by mahemaical inucion.. BLK-SHOLS MODL

3 Le S, σ, r,,, ( ) an n ( ) be he unerlying sock price on ime, he variance of he rae of reurn on he sock, he risk-free rae, he ime o mauriy of he call, he exercise price, he uropean call price wih mauriy, an he merican call opion wih n-h muli-exercisable call opion wih mauriy, respecively. Black an Scholes (973) analysis which leas he fair value of an opion is base mainly upon he following assumpion. Firs of all, he perfec marke, () he consan ineres rae, r, an volailiy, σ, an (3) geomeric Brownian moion for he sock price. he curren ime is se o zero. he sochasic process for sock price change is assume o be: S S = µ σz () where µ is he expece reurn on he sock an z is he ifferenial of a Gauss-Wiener process. Seconly, a hege posiion is forme wih a porfolio of shor unerlying an a long posiion of a number of uropean opions. hen an arbirage argumen leas o he renowne Black-Scholes parial ifferenial equaion eermining he value of he opion. where r ( ) = e [ Max( S,0)] r = SN ) e N( ), () ( S σ ln( ) ( r ) =, σ = σ, he funcion N(x) is he cumulaive probabiliy isribuion funcion for a sanarize normal isribuion. In oher wors, i is he probabiliy ha a variable wih a sanar normal isribuion, ϕ(0,) will be less han x (Hull, 003). From q. (), he Black-Scholes moel exploiing he fac ha a fair value of an opion is given by he presen value of he expece payoff a expiry uner he risk-neural measure. hen he payoff is compue an iscoune up o he curren ime. Meron (973) value he uropean opion wih a ivien yiel a rae q. r = e [ Max( S,0)] = e q SN( ) e r N( ), (3) where S σ ln( ) ( r q) =, σ 3

4 = σ. 3. VLUION OF MRIN LL OPION For valuaion of merican call, a proceure is use o price he merican opion uner risk-neural measure an a wice exercisable call opion is also use as a basis o exen o he muli-exercisable call opion. 3. Valuaion of wice xercisable all Opion o see how o value a sanar merican call opion, we sar wih he simple siuaion- a wice-exercisable opion. ha is we can exercise a a fixe ime poin, ( 0 < < ), before mauriy or a mauriy,. Uner he assumpion of sock price following he geomeric Brownian moion, he Black-Scholes moel (973) has been use o erive he uropean opion expece payoff, r ha is ( ) = e { Max[ S, 0]}. If he merican call opion will no be exercise before expiraion ae which is jus exercising a he en of perio, he expece payoff of merican opion, ( ), is he same as ha of uropean opion. Furher problem is ha uner wha coniion, he holer will no exercise a he en of curren perio? If he sock price a ime, S, is lower han r e ( ), he holer will no exercise on. Because he payoff from exercising he call will be less han ( ), uner he risk-neural measure, he holer will no exercise on. n here will be lef in he life of he call. In conrac, if he sock price a ime is higher han r e ( ), he holer who will exercise righ now, because he payoff from exercising he call will be more han r ( ). e ( ) is he mile price, M( S,,, ), a. r herefore, he value of merica call opion, V ( ), is Max{ S, e ( )}. he fair pricing of wice exercisable merican call opion, r ( ), is r e [ Max{ S, e ( )}]. By using Maringale Pricing an Girsanov heorem (Nefci, 000), he analyical soluion of e r r [ Max{ S, e ( )}] is erive as follows. r r ( ) = e [ Max{ S, e ( )}] = r Q e S K e [ r r Q e ( ) I r ] [( ) ] { 0 e S ( )} K I r { S K e < < ( )} r = e [ e q r r N(, ( K e ( )))] e ( ) r r r SN, ( K e ( ))) e KN(, ( K e ( ))) ( 4

5 = q r r r r ( ) e SN(, ( K e ( ))) e ( K e ( )) N(, ( K e ( ))) r = ( ) ( e ( )). (4) where Q : he expece value of merican opion uner Q measure in Girsanov heorem, I : n inicaor variable, ha is,, I =, if siuion hol 0, o / w, S σ ln( ) ( r q) ( k) = k, an (5) σ, ( k) =, ( k) σ. (6) If he call opion has no alreay been exercise an he payoff from exercising he call opion r equals or excees ( ) ( e ( )), i will no be exercise an he value of call r opion is ( ) ( e ( )). Oherwise, he call opion will be exercise an he value of call opion is S0. his implies a criical price of wice exercisable call opion is r ( ) ( e ( )). Figure illusraes he fair value of wice exercisable call opion, mile price an is criical price. In Figure, if he call opion has no alreay been exercise before mauriy, he value of merican call opion is 0 if he spo price is lower han exercise price,. Oherwise, he value of merican call is S a mauriy. When ime is a, he value of r merican call opion is e ( ) if he spo price, r S, is lower han e ( ). Oherwise, he value of merican call opion is S r. Hence, e ( ) is a mile price a an r he criical price of wice exercisable opion is ( ) ( e ( )). 3. Valuaion of riple xercisable merican all Opion riple exercisable call opion is assume he holer have hree possible exercise epochs, a mauriy ( or ), an ( 0 < < < ). Similar o wice exercisable call opion, he riple exercisable call opion has wo mile prices on an. he mile price, M(S,,, ) on r is e ( ) an also he mile price, M(S,,, ), on is r r e ( ( ) ( e ( ))). herefore, he value of merica call opion, ( ), is V r r Max{ S, e ( ( ) ( e ( )))}. he fair pricing of he merican call 5

6 opion, r ( ), is r r e [ Max{ S, e ( ( ) ( e ( )))}] 3. By using Maringale Pricing an Girsanov heorem, he analyical soluion of e r r r [ Max{ S, e ( ( ) ( e ( )))}] is erive as follows: r r r = ( ) ( e ( )) ( e ( ( ) ( e ( )))) (7) 3 Sock price ime mauriy n of perio o r xercise price () riical price: ( ) ( e ( )) ) M(S,,, )= r e ( ) 0,,0 S, S, S S, ime a r ( ) e r, e ( r ),, e ( ) S S Now (ime perio 0 ) Do no exercise xercise Figure he Value of wice xercisable all, Mile Price an is riical Price If he call opion has no alreay been exercise an he payoff from exercising he call opion r r r equals or excees ( ) ( e ( )) ( e ( ( ) ( e ( )))), r i will no be exercise an he value of call opion is ( ) ( e ( )) r r ( e ( ( ) ( e ( )))). Oherwise, he call opion will be exercise an he value of call opion is S0. his implies a criical price of wice exercisable call opion is r r r ( ) ( e ( )) ( e ( ( ) ( e ( )))). Figure illusraes a fair value of riple exercisable call opion, mile prices an is criical price. In Figure, here are wo mile prices separaely a an. If he call opion has no alreay been exercise before mauriy, he value of merican call opion is 0 if he spo price is lower han exercise price,. Oherwise, he value of merican call opion is S a mauriy. When ime is a, he value 6

7 r of merican call opion is e ( ) r if he spo price, S, is lower han e ( ). Oherwise, he value of merican call opion is S. Similarly, when ime is a, he value of r merican call opion is r e ( ( ) ( e ( ))) if he spo price, S, is lower han r r e ( ( ) ( e ( ))). Oherwise, he value of merican call opion is S. r herefore, he criical price of wice exercisable opion is ( ) ( e ( )) r r ( e ( ( ) ( e ( )))). Sock price ime o mauriy n of perio ime a ime a Now (ime perio 0 ) r xercise price () M(S,,, )= r e ( ( ) ( e ( ))) M(S,,, )= r e ( ) 0,,0 S, S, S r r e ( ), e ( r ),.., e ( ) r r e [ ( ) ( e ( ))] Do no exercise S S, S S S S riical price S xercise Figure he Value of riple xercisable all, Mile Prices an is riical Price ables an summarize he fair value of wice an riple exercisable call opion an heir criical prices. From ables an, he fair values of wice an riple exercisable call opion, an heir criical prices show hree noable feaures. Firs of all, he relaionship of boh criical prices in wice an riple exercisable call opions is no linear. his means ha wo ypes of exercisable call opions have ifferen criical price an heir relaionship is no linear. Seconly, uner he same assumpion wih Black-Sholes moel (973), he criical price is he 7

8 summaion of uropean opions price wih ifferen exercise prices an ifferen mauriies a he exercise price. Finally, he relaionship of boh he fair values or close-form soluions in wice an riple exercisable call opions is no linear. he fair value of call opion riical price able he Fair Value of wice xercisable all Opion an is riical Price r ( ) ( e r ( ) ( e ( )) ( )) he fair value of call opion riical price able he Fair Value of riple xercisable all Opion an is riical Price r r r ( ) ( e ( )) ( e ( ( ) ( e r r r ( ) ( e ( )) ( e ( ( ) ( e ( )))) ( )))) 3.3 Valuaion of Muli-exercisable all Opion general siuaion, wih n possible exercise epochs a ime,, n an mauriy,, ( 0 < < <... < n < ) is consiere. Base on he iscussion in secion 3. an 3., here are (n-) mile prices an n is obaine from qs. (4) an (7). n = r n ( ) ( e ( n )) rn rn ( e [ ( ) ( e n n ( ))])... herefore, he criical price of merican call opion, S,is (...) (8) S = rn ( ) ( e n ( )) rn rn ( e [ ( ) ( e n n ( ))])... (...) (9) qs. (8) an (9) have hree feaures. Firs of all, he analyical soluion is he summaion of n uropean call opions prices wih ifferen mauriies an ifferen exercise prices. here are wo avanages of hem: () he valuaion of merican call opion can be obaine easily by he summaion of Black-Sholes formula (or uropean call opion pricing formula), an () an analyical formula is more efficien han approximaing he sock price process or he parial ifferenial equaion by binomial 8

9 (ox e al., 979) or finie ifference mehos (Brennan an Schwarz, 978). Seconly, from qs. (8), he same conclusion is compare wih Barone-esi an Whaley s (987) conclusion; he value of merican call opion is he value of uropean call opion a early exercise premium. hus, he early exercise premium, ε ( S, ), can be obaine easily. n ε ( S, ) can be expresse mahemaically as ε ( S, ) = ( ) ( ) (0) n hirly, from qs. (8) an (0), he feaures of he early exercise premium, ε ( S, ), can be easily capure by he Greek Leers propery (Hull, 003). 4. ONLUSIONS he valuaion of merican opions on ivien-paying asse is an imporan problem in financial economics. he heoreical values of uropean opion can be evaluae by a simple formula. However, he heoreical values of merican opions are very ifficul o eermine since he possibiliy of merican opions can be early exercise. In his suy, analyical soluion of pricing merican call opion an he criical price are obaine. n from analyical soluion an he criical price can ge wo feaures. Firs of all, he analyical soluion is he summaion of n uropean call opions prices wih ifferen mauriies an ifferen exercise prices. here are wo avanages of hem: () he valuaion of merican call opion can be obaine easily by he summaion of Black-Sholes formula (or uropean call opion pricing formula), an () an analyical formula is more efficien han approximaing he sock price process or he parial ifferenial equaion by binomial (ox e al., 979) or finie ifference mehos (Brennan an Schwarz, 978). Seconly, he same conclusion is compare wih Barone-esi an Whaley s (987) conclusion; he value of merican call opion is he value of uropean call opion a early exercise premium. Previous aemps a pricing hese opions have been accurae bu compuaionally expensive. his suy provies a simple, an inexpensive soluion for pricing merican call opion. RFRN. Broaie M. an Deemple J., merican Opion Valuaion: New Bouns, pproximaion, an a omparison of xising Mehos, Review of Financial Suies, 9, 996, pp Barone-esi G. an Whaley R., fficien nalyic pproximaion of merican Opion Values, Journal of Finance, 4, No., 987 (June). 3. Black. F. an Scholes. M., he Price of Opions an orporae Liabiliies, Journal of Poliical conomy, 8, 973, pp Brennan M. an Schwarz., Finie Difference Mehos an Jump Process rising in he Pricing of oningen laims: Synhesis, Financial an Quaniaive nalysis, 3 (3), 978, pp

10 5. ox J.., Ross S.. an Rubinsein M., Opion Pricing: Simple pproach, Journal of Financial conomics, 7, 979, pp Geske R. an Johnson H.., he merican Pu Opions Value nalyically, Journal of Finance, 39, 984, pp Geske R. an Shasri K., Valuaion by pproximaion: omparison of Opion Valuaion echniques, Financial an Qualiaive analysis, 0, (), 985, pp Gran D., Vora G. an Weeks D., Simulaion an he arly-exercise Opion Problem, Financial ngineering, 5, 996, pp Hon, Y.., Quasi-Raial Basis Funcions Meho for merican Opions Pricing, ompuers an Mahemaics wih pplicaions, 43, 00, pp Huang H. Z., Subrahmanyam M. G. an Yu G. G., Pricing an Heging merican Opions: Recursive Inegraion Meho, Review of Financial Suies, 9, 996, pp Hull, J., Opions, Fuures & Oher Derivaives,4 h e., Upper Sale River New Jersey: Prenice Hall, Johnson H., n nalyic pproximaion for he merican Pu Price, Financial an Qualiaive nalysis, 8, No., 983, pp MacMillan L., nalyic pproximaion for he merican Pu Opion, vances in Fuures an Opions Research,, 986, pp Meron, R.. heory of Raional Opion Pricing, Bell Journal of conomics an Managemen Science, 4, Spring, Nefci, S. N., n Inroucion o he Mahemaics of Financial Derivaives, alifornia: caemic Press, Unerwoo, R. an Wang, J., n Inegral Represenaion an ompuaion for he Soluion of merican Opions, Nonlinear nalysis: Real Worl pplicaions, 3, 00, pp Winson, W. L. Operaion Research, Duxbury, Wu L. an Kwok Y. K., Fron-fixing Finie Difference Meho for he Valuaion of merican Opions, Financial ngineering, 6, 997, pp Yang H., Numerical nalysis of merican Opions. Disseraion, Dep. Mah., Universiy of lbera, Spring, 00. 0