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

where ? denotes transpose.The initial strike price expressed in pounds is by definition given byK p 0 = S 0,and the corresponding dollar strike price is thusK d = K p 0 ¢ X 0 = S 0 X 0 .The dollar strike price is kept constant until the exercise date. However, expressedin pounds the strike price evolves dynamically as a result of the varyingexchange rate, so the pound strike at maturity is given byK p T = Kd ¢ X −1T= S 0 ¢ X 0 ¢ X −1T . (35)There are now two natural ways to value this option: we can work in dollars orin pounds, and initially it is not obvious which way is the easier. We will in factperform the calculations in both alternatives and compare the computationaleffort. As will be seen below it turms out to be slightly easier to work in dollarsthan in pounds.7.3 Pricing the option in dollarsIn this approach we transfer all data into dollars. The stock price, expressed indollars, is given bySt d = S t ¢ X t ,so in dollar terms the payout Ξ of the option at maturity is given by the expressionΞ d =max £ S T X T ¡ K d , 0 ¤Since the dollar strike K d is constant we can use the Black-Scholes formulaapplied to the dollar price process St d . The Itô formula applied to Std = S t X timmediately gives us the P -dynamics of Std asWe can write this asdS d t = S d t (α + α X + σ S σ ? X) dt + S d t (σ S + σ X ) dW tdS d t = S d t (α + α X + σ S σ ? X) dt + S d t δ S,d dV twhere V is a scalar Wiener process and whereqδ S,d = kσ S + σ X k = δS 2 + δ2 X +2ρδ Sδ Xis the dollar volatility of the stock price.21