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CHAPTER 8 PRICING FOREIG N EXCHANG E OPTION S INCORPORATING PURCHASIN G POWE R PARIT Y 8.1 Introduction 1 One o f th e mos t usefu l an d widel y use d application s o f th e theor y o f options is to price options on foreign exchange. 2 A s we have seen in Chap ter 4 , according t o a fundamental assumptio n i n Black an d Scholes ' theory , the stochasti c proces s governin g th e (spot ) pric e o f the underlyin g foreig n currency follow s geometri c Brownia n motion. 3 A s a result , th e exchang e rate display s th e characteristic s o f a random walk. 4 O n th e othe r hand , a fundamental theore m in the international trad e predicts that ove r time, th e spot (relative ) pric e o f a currenc y woul d converg e t o it s purchasin g powe r parity.5 Fluctuation s i n the exchange rate ar e anchored, so to speak, by its long period equilibriu m value . T o synthesise th e theor y o f foreign currenc y options an d th e theor y o f internationa l trade , th e presen t Chapte r intro duces the elemen t o f non rando m wal k behavior represente d b y purchasin g power parit y int o th e stochasti c specificatio n o f the exchang e rate . A n al ternative t o geometri c Brownia n motion , i n th e for m o f a two dimensiona l stochastic process , is introduced t o model the dynamic s of foreign currenc y prices. A close d for m expressio n fo r th e transitio n densit y functio n i s ob tained , t o characteris e th e solutio n o f th e resultin g syste m o f stochasti c differential equation s in the most complet e form. Sinc e "optio n valuation i s equivalent t o th e proble m o f determining th e distributio n o f th e [underly ing ] asse t price", 6 w e proceed t o tak e mathematica l expectatio n i n term s of the transition densit y function. A n exact formula t o price options on th e currency i s obtained, whic h in particular i s conditional upon it s purchasin g power parity. Thi s result is line line with recent research (see e.g. McQuee n k Thorle y 1991 , Samuelso n 1991 , Kaehle r k Kugle r eds . 1994 , Hauge n 1 Thi s chapte r is a n expanded versio n of Cheun g & Yeung (1994a) . 2 Se e e.g . Bige r & Hul l (1983) , Garma n & Kohlhage n (1983) , Grabb e (1983) , Bodurtha & Courtadon (1987) , an d Hull (1991) . 3 Se e e.g . Bodurth a & Courtadon (1987) . 4 Se e e.g. Samuelso n (1965 ) an d McKean's Appendix . 5 See . e.g . Frenke l & Mussa (1984) , Levic h (1984) , and Lothian and Taylor (1996) . 6 Se e Co x & Ross (1976) , p.154 , and Goldenberg (1991) , p.6 . 71 Pricing Foreign Exchange Options 1995, Malkie l 1996 , Campbell , L o k Mackinla y 1997) , whic h i s beginnin g to "questio n th e rando m walk dogma" (Samuelso n 1991 ) so much a part o f the geometri c Brownia n motio n assumption . A second proble m arise s whe n exchang e rate s (an d othe r asse t prices ) are assume d t o follo w geometri c Brownia n motion . I n a classi c paper , Samuelson (1965 ) pointed out tha t whe n modelling asset prices in this way, one must tak e int o accoun t a bia s in the (Brownian ) rando m walk . I n par ticular , a theore m o n "th e virtua l certaint y o f relativ e ruin " (Samuelso n 1965, p.795) was obtained, such that i f the diffusion paramete r is more tha n twice the instantaneous expected rate of return of the asset price, its typical sample values would dro p t o zero with probabilit y on e over time, a positiv e expected rat e o f return notwithstanding . A detaile d mathematica l verificatio n o f the existenc e...

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