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

where Q f denotes the risk neutral martingale measure for the foreign market.At this point we have to make some probabilistic assumptions, and in fact weassume that we have a Garman-Kohlhagen model for Y . Standard theory thengives us the Q f dynamics of Y asdY t = Y t (r f ¡ r d )dt + Y t σ Y dW t . (15)For simplicity we assume that also the inflation follows a geometric Brownianmotion, with Q f -dynamics given bydI t = I t α I dt + I t σ I dW t . (16)Note that W is assumed to be two-dimensional, thus allowing for correlationbetween Y and I. Also note that economic theory does not say anything aboutthe mean inflation rate α I under Q f .When computing the expectation in (14) we cannot use a standard change ofnumeraire technique, the reason being that none of the processes Y , I or Y ¢I areprice processes of traded assets without dividends. Instead we have to attackthe expectation directly.To that end we define the process Z as Z t = Y t ¢ I t and obtain the followingQ f -dynamics.dZ t = Z t (r f ¡ r d + α I + σ Y σ ? I ) dt + Z t (σ Y + σ I ) dW t .¿Fromthisitiseasytoseethatifwedefine S t bythen we will have the Q f -dynamicsS t = e −(r f −r d +α I +σ Y σ ? I )t Z t ,dS t = S t (σ Y + σ I ) dW t ,thepointbeingthatwecaninterperetS t as a stock price in a Black-Scholesworld with zero short rate and Q f as the risk neutral measure. With thisnotation we obtain easilyΠ (0; Ξ f )=e cT E Q £ £f max ST ¡ e −cT Y 0 , 0 ¤¤ + Y 0 ,wherec = α I + σ Y σ I ? .The expectation above can now be expressed by the Black-Scholes formula fora call option with strike price e −cT Y 0 , zero short rate and a volatility given byqσ = kσ Y k 2 + kσ I k 2 +2σ Y σI?12