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CHAPTER 5 THE BLACK-SCHOLE S OPTION S THEOR Y 5.1 Introductio n As we have seen in Chapter 4 , the questio n is : wha t i s the equilibriu m price o f an option , give n it s natur e (whethe r i t i s American o r European) , exercise price , exercis e date , th e curren t pric e o f the underlyin g asset , th e discount rate , an d give n a n assumptio n regardin g th e behavior o f the asse t price over time? A major breakthroug h wa s achieved when Blac k&; Scholes (1973) obtaine d a closed-for m expressio n t o pric e a n Europea n optio n o n a stoc k whic h doe s no t pa y dividends , assumin g tha t it s pric e follow s a Geometric Brownia n Motion 1 . 5.2 Th e Geometri c Brownia n Motio n Assumptio n In thei r semina l wor k (1973) , Blac k an d Schole s adopte d th e assump tion , that th e price of the representative stock follows a Geometric Brownia n Motion. A s we have seen in §3.6 above, the standard discret e approximatio n to Brownia n Motio n i s the symmetrica l rando m walk . I n th e cas e of Geo metric Brownia n Motio n i s th e symmetri c rando m wal k (se e §3. 6 above) . In th e cas e o f geometri c Brownia n motion , thi s mean s tha t ove r a smal l length o f time , proportiona l change s i n th e pric e o f a stoc k ar e normall y distributed. I t the n follows tha t a t an y future poin t o f time, the stock pric e is lognormally distribute d (se e §3.7 above). Writin g S(t)y t > 0 for the pric e of the stoc k ( a rando m variable ) an d t = 0 for curren t time , w e ca n writ e its probability densit y functio n as : (SQ)tyS) = -—7= = = ex p {logS- [lo gSp + (jt - 0 are constants . - (5-1 ) 1 Though the analysi s has been extended to take dividends into account an d a number of other assumptions have been relaxed, for ease of exposition we will centrate on th e classic simplicit y o f th e origina l formulation . 24฀ The Black-Scholes Options Theory 5.3 Th e Black-Schole s Optio n Pric e Formul a In addition to the assumption that stock prices follow geometric Brownian motion, Blac k Sz Scholes assume that : (a) tradin g i n asset market s i s continuous; (b) transactio n cost s and taxe s ar e zero; (c) asset s ar e perfectly divisible ; (d) dividend s ar e not pai d on th e stock ; (e) riskles s arbitrat e opportunitie s ar e absent ; (f) th e ris k fre e rat e o f interest r i s given an d constant . (As we have noted above , many o f these assumptions hav e been relaxe d b y later researchers. ) Black an d Schole s demonstrat e tha t i t i s possibl e t o creat e a riskles s hedge by forming a portfolio which contains units of the stock and Europea n call options on it. A t any point o f time, since the quantitie s o f assets in th e portfolio ar e fixed, th e sources of change i n its value must b e the prices . I f the cal l price is a function o f the stock pric e and it s time t o maturity , the n changes i n th e cal l pric e ca n b e expresse d a s a function o f change s i n th e stock pric e an d change s i n th e tim e t o maturity . Blac k an d Schole s the n observe tha t a t an y point o f time, the portfoli o ca n be mad e int o a riskles s hedge by choosing a n appropriat e mixtur e o f the stock an d Europea n calls : e...

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