Option Put-Call Parity Relations When the Underlying Security Pays Dividends

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1 Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, Opion Pu-all Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha, U.S.. ie Su * Deparmen of Finance, Universiy of Miami, U.S.. bsrac he original pu-call pariy relaions hold under he premise ha he underlying securiy does no pay dividends before he expiraion of he opions. Similar o Hull (23), his paper relaxes he non-dividend-paying assumpion. he underlying securiy price in he original uropean-syle pu-call pariy relaion is adjused downwards by he presen value of expeced dividends before he opion expires. he upper bound of he merican-syle pu-call pariy relaion is adjused upwards by he amoun of he presen value of expeced dividends. he resuls provide heoreical boundaries of opions prices and expand applicaion of pu-call pariy relaions o all opions on currencies and dividend-paying socks and sock indices, boh uropean-syle and merican-syle. Key words: opions, dividends, and pu-call pariy JL classificaion: G13: coningency pricing 1. Inroducion he opion pu-call pariy condiion quanifies he relaions among he price of a call opion, he price of an oherwise idenical pu opion, he price of he underlying securiy of he call and pu opions, and he presen value of he exercise price of he call and pu opions. he pariy relaions can be applied o boh uropean-syle and merican-syle opions. hey help o explain he inricae relaions among prices of call and pu opions, prices of heir underlying securiy, and he price of risk-free reasury Bills. Pu-call pariy relaions for sandard uropean-syle and merican-syle opions have been well acceped by he finance profession. arly sudies in opions, e.g., Meron (1973), Smih (1976), ox and Ross (1976), and ox and Rubinsein (1985), and essenially all opions exbooks, e.g., ucker (1991), Hull (22, 23), Jarrow Received February 2, 26, revised December 25, 26, acceped March 8, 27. * orrespondence o: Deparmen of Finance, Universiy of Miami, P.O. Box 24894, oral Gables, FL mail:

2 226 Inernaional Journal of Business and conomics and urnbull (2), and hance (23), among ohers, cover discussions on he pu-call pariy relaions. he original pu-call pariy relaions are derived under he premise ha he underlying securiy does no pay dividends before he expiraion of he opions. However, a large number of socks and almos all sock indices pay dividends. Furhermore, all foreign currencies bear foreign risk-free raes. he foreign risk-free raes of ineres can be viewed as dividends (or leakages) paid on he foreign currencies. onsequenly, he non-dividend-paying assumpion severely resrics he usage of he original pu-call pariy relaions. he original pu-call pariy relaions may no be applied o hese heavily raded opions on dividend-paying securiies. Similar o Hull (23), his paper relaxes he non-dividend-paying assumpion on he underlying securiy. I presens a variaion of he relaions when he underlying securiies pay dividends. he underlying securiy price in he original pu-call pariy relaion for uropean-syle opions is adjused downwards by he presen value of expeced dividends before he opion expires. he upper bound of he pu-call pariy relaion for merican-syle opions is adjused upwards by he presen value of expeced dividends before he opion expires. his paper shows he heoreical boundary condiions o call and pu opion prices when he underlying securiy pays dividends. Our resuls expand he applicaion of he powerful ool of pu-call pariy relaion o a much wider range of opions. I is paricularly imporan o opion raders who rade merican-syle opions on socks and sock indices (e.g., S&P 1 Index OX), currencies, and over-he-couner non-sandard opions because mos of he underlying securiies generae eiher discree cash dividends or a coninuous dividend sream. Inheried in he pu-call pariy relaion, anoher imporan propery of our resuls is ha hey are purely arbirage-driven and oally model-free. he resuls are independen of any paricular opion pricing models, e.g., he Black-Scholes (1973) opion pricing model. he only assumpion needed for he conclusions o hold is ha he financial markes are efficien, which is widely believed o be rue. his paper is organized as follows. he nex secion derives and proves he pu-call pariy relaion for uropean-syle opions. he hird secion deals wih he pu-call pariy relaion for merican-syle opions. he final secion concludes. 2. Pu-all Pariy for uropean-syle Opions If he underlying securiy does no pay dividends before he opion expires, he original pu-call pariy relaion for uropean-syle opions can be given by he following simple equaion: S + P =, (1) where and P are uropean-syle call and pu opion premiums, respecively, S is he curren price of he underlying securiy, X is he opions common exercise price, r is he annualized coninuously compounded risk-free rae of ineres, and

3 ie Su and Weiyu Guo 227 is he ime o opions mauriy. Proof of he above relaion can be found in mos exbooks on opions. he pu-call pariy relaion for uropean-syle opions saes ha he sum of he curren underlying securiy price and a pu opion premium equals o he sum of a call opion premium and he presen value (discouned a he risk-free rae) of he opions exercise price. If he underlying securiy pays a dividend (or dividends) before he opions expire, he pu-call pariy relaion can be modified as: ( Div) + P = S PV, (2) where PV (Div) is he presen value (discouned a he risk-free rae) of all expeced cash dividend paymens generaed by he underlying securiy o be paid on or before he opions expire. For example, if he underlying securiy is expeced o pay a r dividend D a ime where < <, hen PV ( Div) = De. he proof of he dividend-adjused pu-call pariy for uropean-syle opions is sraighforward. For simpliciy and wihou loss of generaliy, assume here is only one cash dividend paymen D, occurring a ime during he life of he opion, where < <. We rea coninuous dividend yields a he end of he nex secion. onsider ha an invesor holds he following wo porfolios from oday unil he r opions expire: S De + P and. Noe ha he firs porfolio conains a share of an underlying securiy ha pays dividend D a ime. he erminal value r r r ( ) of he firs porfolio a ime is S De e + P + De = S + P because he fuure value of he dividend received a ime cancels he fuure value of PV (Div). I s now easy o see ha when he opions expire, he wo porfolios have exacly he same erminal value S + P = + X. his occurs because if S X, hen S + P = S and + X = S, so ha S + P = + X. lernaively, if S X, hen S + P = X and + X = X, so ha again S + P = + X. Because he wo porfolios always have he same erminal value, hey mus have exacly he same presen value in an efficien marke. onsequenly, we mus have S PV ( Div) + P =. he pu-call pariy relaion for uropean-syle opions is hus proved. 3. Pu-all Pariy for merican-syle Opions Under he assumpion of no dividends, he original pu-call pariy relaion for merican-syle opions can be given by he following chain of inequaliies: P + S + X, (3) where and P are merican-syle call and pu opion premiums, respecively. Insead of one simple equaion for uropean-syle opions, he pu-call pariy relaion for merican-syle opions is a chain of inequaliies, where he difference beween he upper and lower bounds, i.e., he widh of he inerval, is ( + X ) ( ) = X (1 e ). For a reasonable exercise price, risk-free rae

4 228 Inernaional Journal of Business and conomics of ineres, and ime o mauriy, he original pu-call pariy relaion for merican-syle opions provides a igh inerval ha brackes he pu opion premium and underlying securiy price. If he underlying securiy pays a dividend (or dividends) before he opions expire, hen he merican-syle pu-call pariy relaion can be modified as: P ( Div) + S + X + PV. (4) he proof can be shown by proof by conradicion. For simpliciy and wihou loss of generaliy, assume here is only one cash dividend paymen D, occurring a ime during he life of he opions, where < <. he case of a coninuous dividend yield is addressed below. Noe ha, due o he early exercise feaure of merican-syle opions, an opion holder can choose o exercise her opion early when doing so is opimal. arly exercise of a call opion occurs when he underlying securiy pays a significanly large dividend, he amoun of which exceeds he remaining ime (or speculaive) value. arly exercise of a pu opion occurs when he price of he underlying securiy is sufficienly low ha ineres income earned on he inrinsic value of he pu opion is greaer han he remaining ime value. he proof of expression (4) is divided ino wo pieces: P P + S, (4a) ( Div) + S + X + PV. (4b) We prove boh pieces by proof by conradicion. Firs, suppose ha inequaliy (4a) does no hold for all securiies and heir opions. hen here exiss a leas one underlying securiy and is opions which saisfy > P + S. We show ha if his case happens, here mus be an arbirage opporuniy. n arbirageur buys he merican-syle pu opion and buys a share of he underlying securiy. he same ime, she wries he merican-syle call opion and sells a risk-free bond wih a face value of he exercise price of he opions. he iniial cash flow is posiive because ( ) ( P + S ) >. onsequenly, he posiion P + S Xe is held. Because he arbirageur wroe an merican-syle call opion, he buyer of he call opion may choose o early exercise he call opion in order o capure a significanly large cash dividend paymen from he underlying securiy. he insan before he underlying securiy goes ex-dividend, he call opion buyer may exercise he call opion early by paying X dollars in exchange for a share of he underlying securiy. In his case, he arbirageur loses he underlying securiy, and he call opion buyer capures he underlying securiy and is forhcoming dividend. he value of he arbirageur s porfolio becomes: ( ) ( ) P + S S + X Xe r = P + X Xe r >. If he cash dividend D is no large enough o rigger early exercise of he call

5 ie Su and Weiyu Guo 229 opion, he arbirageur collecs he dividend on he ex-dividend dae and holds he porfolio o he opion s dae of mauriy. he value of his porfolio becomes r ( ) r ( ) P + S + De X = De >. I s herefore clear ha he assumpion > P + S induces an arbirage opporuniy. hus, in an efficien financial marke, he inequaliy P + S mus hold for all securiies and a all imes. Nex, suppose ha inequaliy (4b) does no hold for all securiies and opions. hen here exiss a leas one underlying securiy and is opions which saisfy P + S > + X + PV ( Div). gain we argue ha an arbirage opporuniy mus resul. n arbirageur buys he merican-syle call opion, buys a risk-free bond wih a face value in he amoun of he expeced cash dividend, and buys a risk-free bond priced a X dollars. he same ime, she wries an merican-syle pu opion and shor sells a share of he underlying securiy. he iniial cash flow is posiive because ( P + S ) ( + X + PV ( Div)) >. he arbirageur now holds he posiion + X + PV ( Div) P S. Because he arbirageur shor sold a share of he underlying securiy, she is responsible for paying any dividends ha he underlying securiy generaes. he componen PV (Div) in her porfolio allows her o exacly mee his obligaion. onsequenly, he remaining discussion assumes ha he componen PV (Div) neuralizes dividend obligaions from he underlying securiy when sold shor. any ime before he opion expires, if he pu opion holder exercises he opion by submiing he underlying securiy in exchange for he exercise price, he r r value of he arbirageur s porfolio is + S X S = X >. If he pu opion holder does no exercise he opion early, hen he arbirageur holds his porfolio unil he opion expires. he value of his porfolio becomes: r r P S = Xe X >. gain, he assumpion P + S > + X + PV ( Div) induces an arbirage opporuniy. hus, in an efficien financial marke, he inequaliy P + S + X + PV ( Div) mus hold for all securiies. he dividend-adjused pu-call pariy relaion for merican-syle opions implies a wider inerval widh ( + X + PV ( Div)) ( ) = X Xe + PV ( Div). he wider inerval is due o he addiional uncerainy of early exercise before he underlying securiy goes ex-dividend. In summary, he dividend-adjused pu-call pariy relaions for uropean-syle and merican-syle opions are given by expressions (2) and (4): S PV Div uropean-syle opions: ( ) + = +, merican-syle opions: P + S + X + PV ( Div) P Xe. In some cases, he underlying securiy does no pay discree cash dividends. Insead, i generaes a coninuous sream of dividends. ypical examples include all

6 23 Inernaional Journal of Business and conomics foreign currencies which bear foreign risk-free ineres raes as dividend yields, mos sock indices, and cusom-designed over-he-couner producs. In some cases, he erm leakage is used insead of dividends. If he underlying securiy generaes a coninuously compounded dividend yield d, hen he presen value of all dividends d before he opion expires is PV ( Div) = S (1 e ). Subsiuing he new PV (Div) expression ino equaions (2) and (4), we conclude ha he pu-call pariy relaions under he coninuous dividend yield case are: d uropean-syle opions: S e + P =, (5) merican-syle opions: d P + S + X + S e ). (6) 4. onclusion his paper derives pu-call pariy relaions for uropean-syle and merican-syle opions when he underlying securiy pays dividends before he opions expire. he dividend-adjused pu-call pariy relaion provides heoreical boundary condiions for call and pu opion prices when he underlying securiy pays dividends. Our resuls expand he scope and applicaion of he original pu-call pariy relaion o all uropean-syle and merican-syle opions on foreign currencies and dividend-paying socks and sock indices. References Black, F. and M. S. Scholes, (1973), he Pricing of Opions and orporae Liabiliies, Journal of Poliical conomy, 81, hance, D., (23), nalysis of Derivaives for he F Program, IMR. ox, J. and S. Ross, (1976), Survey of Some New Resuls in Financial Opion Pricing heory, Journal of Finance, 31, ox, J. and M. Rubinsein, (1985), Opions Markes, Prenice-Hall. Hull, J., (22), Fundamenals of Fuures and Opions Markes, Fourh diion, Prenice Hall. Hull, J., (23), Opions, Fuures, and Oher Derivaives, Fifh diion, Prenice Hall. Jarrow, R. and S. urnbull, (2), Derivaives Securiies, Second diion, Souh-Wesern ollege Publishing. Meron, R.., (1973), heory of Raional Opion Pricing, Bell Journal of conomics and Managemen Science, 4, Smih Jr.,. W., (1976), Opion Pricing: Review, Journal of Financial conomics, 3, ucker,., (1991), Financial Fuures, Opions, and Swaps, Wes Publishing ompany.

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