In lieu of an abstract, here is a brief excerpt of the content:

CHAPTER 4 STOCHASTIC ASSUMPTION S AN D OPTIO N PRICIN G In Chapte r 1 we suggeste d that , sinc e optio n pricin g i s applie d eco nomics , th e choic e o f assumption s i s a matte r o f primar y importance . I n particular, w e referre d t o th e vie w o f Co x h Ros s (1976) , tha t i f a differ ent assumptio n i s introduced regardin g th e stochasti c behavio r o f an asse t price, generally a differen t formul a t o pric e options o n th e asse t woul d fol low . Th e objectiv e o f present chapte r i s to convinc e the reader o f the trut h of thi s observation . W e present a n example , i n whic h a cal l optio n o n a n equity i s priced withou t an y stochasti c assumption s abou t th e behavio r o f the equit y price . Th e reade r i s then invite d t o compar e (i n th e nex t chap ter ) thi s expressio n wit h th e classi c Black - Schole s formul a fo r pricin g th e same option , i n whic h th e equit y pric e i s assume d t o follo w a Geometri c Brownian Motion . Following Cox , Ros s & Rubinstei n (1979) , suppos e tha t ove r a singl e period o f tim e (say , on e "day" ) th e pric e o f a representativ e equit y ca n change in one of two ways. Fro m the curren t leve l 5, it ca n eithe r g o up t o hSy o r g o down t o kS. N o probabilities ar e introduced, an d n o restriction s apart fro m th e fact tha t h > 1 and 0 1 + r , where r i s the one period risk-fre e rat e o f interest). Conside r a (European ) call option on the equity in the curren t period, which will mature i n n days . What woul d b e its equilibrium pric e C(Sy n) ? Imagine a portfolio whic h is short on e call option of this type, and lon g N unit s of the underlying stock. I t is called a "hedg e portfolio", fo r reason s which wil l b e clea r later . A t curren t marke t prices , th e hedg e portfoli o i s worth: NS-C(Syn) (1 ) In on e day's time , the stock pric e becomes either hS o r kSy an d wit h n — 1 days remaining to maturity the price of the call option would be C(hSy n— 1) 19 Pricing Foreign Exchange Options or C(kS, n — 1). Th e valu e of the hedg e portfolio a t thi s tim e is then : NhS-C(hS,n-l) (2.1 ) or NkS - C(kS, n-1) (2.2)฀ Since N is lef t ope n b y construction , le t u s choos e it such tha t thes e tw o values are equal. W e then have : NhS - C(hS, n - 1 ) =NkS - C(kS, n-1) which gives: N = C(hS,n-1) - C(kS, n-1) (h - k)S The valu e of the hedg e portfolio on e perio d henc e is then : 'C(hS,n-l)-C(kS,n-l) (3)฀ NhS-C(hS,n-l) = (h - k)S kC(hS,n -1 ) - hC(kS,n -1) h-k hS-C(hS,n-l) (4)฀ No matter wha t the stock price is, hS o r kSy th e value of the portfolio woul d be th e same . Thoug h ther e i s no explici t uncertaint y i n ou r example , th e portfolio play s the par t o f a hedge agains t change s in the stoc k price . Since (4) represents a certain sum of money, to prevent arbitrage profit s the curren t valu e o f the hedg e portfoli o mus t b e equa l t o its valu e on e period hence , discounted a t th...

Share