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CHAPTER 7 NON RANDO M WAL K EFFECT S AN D A NE W STOCHASTI C SPECIFICATIO N 7.1 Intr o duction As we have seen in Chapter 5 , a serious problem arises when geometri c Brownian motion i s used to model asset prices. I n addition, recent researc h is beginnin g t o "questio n th e rando m wal k dogma " associate d wit h geo metric Brownia n motio n (Samuelso n 1991) . On e factor whic h may accoun t for "runs " i n th e price s o f asset s lik e stock s i s provide d b y standar d eco nomic theory, which tells us that i n long period equihbrium, the value of the firm's balanc e shee t i s determine d b y exogenou s variable s lik e technolog y and tastes. I f this value changes in response to changes in any one exogenous variable, th e 'intrinsi c value ' o f a firm's share s wil l change i n a systemati c way. However , b y virtu e o f th e assumptio n o f geometric Brownia n motio n in the Black-Schole s theory , i t i s not possibl e t o tak e such standar d result s into account . The abov e observations suggest i t is in order to consider a n alternativ e system o f stochastic asse t pric e dynamics . Followin g th e approac h o f Co x k Ros s (1976), we propose a candidate for additio n t o the list of "plausibl e alternative [t o geometric Brownia n motion ] form s o f stochastic processes" , which ar e useful i n financial analysis . Th e remainder o f this Chapte r draw s on Cheun g & Yeung (1994) , an d introduce s a tw o dimensiona l stochasti c process t o mode l th e dynami c behavio r o f th e representativ e equity . T o characterise th e solutio n o f th e resultin g syste m o f stochasti c differentia l equations i n th e mos t complet e for m known , w e obtai n a close d for m ex pression fo r it s transitio n densit y function . I t i s show n tha t th e solutio n (stochastic) proces s o f th e stoc k pric e exclude s th e possibilit y o f almos t certain ruin. Sinc e "th e option valuation proble m is equivalent t o the problem o f determining th e distributio n o f the stoc k price " (Co x k Ros s 1976 , p. 154), we procee d t o tak e mathematica l expectation s directl y i n term s o f the transitio n densit y function . A n exac t formul a t o pric e option s o n th e stock i s then obtained . 57฀ Pricing Foreign Exchange Options One important feature of this option pricing formula is that non random walk effect s ca n b e incorporated . I n particular , i t i s show n tha t a stock' s intrinsic value enters i n an essential way into the valuation o f options on it . 7.2 A Ne w Stochasti c Specificatio n As we have seen, the concept o f geometric Brownia n motio n was introduced int o financial modellin g b y Samuelso n (1965) . Th e marke t pric e o f a stoc k S(t) i s assume d t o behav e accordin g t o th e stochasti c differentia l equation dS(t) = aS(t) + aS(t)dz(t) y (7.1 ) where a an d a are constant parameter s an d {dz(t)} i s a Wiener process . I n their classi c work on option pricing , Blac k an d Schole s (1973 ) als o adopte d the assumptio n tha t stoc k price s follow geometri c Brownia n motion . A fundamenta l resul t followin g fro m thi s assumptio n i s that th e stoc k price would displa y th e characteristic s o f a random walk . I n particular, th e price o f th e stoc k a t an y tim e i n...

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