ON THE USE OF NUMERAIRES IN OPTION PRICING by Simon ...

The dollar price (expressed in dollar data) at t of the option is now obtaineddirectly from the Black-Scholes formula asCt d = St d N[d 1 ] ¡ e −r d(T −t) K d N[d 2 ], (36)ln ¡ St d /K d¢ ³ ´+ r d + 1 2 δ2 S,d(T ¡ t)d 1 =p ,δ S,d T ¡ tpd 2 = d 1 ¡ δ S,d T ¡ t.The corresponding price in pound terms is finally obtained asso we have the final pricing formulaC p t = C d t ¢1X t,C p t = S t N[d 1 ] ¡ e −r d(T −t) S 0X 0N[d 2 ], (37)X³ ´ ³t´lnSt X tS 0 X 0+ r d + 1 2 δ2 S,d(T ¡ t)d 1 =p ,δ S,d T ¡ tpd 2 = d 1 ¡ δ S,d T ¡ t,qδ S,d = δS 2 + δ2 X +2ρδ Sδ X7.4 Pricing the option directly in poundsAlthough this is not immediately obvious, pricing the option directly in poundsis a bit more complicated than pricing the option in dollars. The pricing problem,expressed in pound terms, is that of pricing the T -claim Ξ p defined byΞ p =max[S T ¡ K p T , 0] .Using (35) and denoting the pound/dollar exchange rate by Y (where of courseY =1/X) weobtain∙¸Ξ P Y T=max S T ¡ S 0 , 0 .Y 0It is now tempting to use the punds/dollar exchange rate Y as the numeraire butthis is not allowed. The reason is that although Y is the price of a traded asset(dollar bills) it is not the price of a traded asset without dividends, the obviousreason being that dollars are put into an American (or perhaps Eurodollar)account where they will command the interest rate r d . Thus the role of Y israther that of the price of an asset with the continuous dividend yield r d . Inorder to convert the present situation into the standard case covered by Theorem2.1 we therefore do as follows.22