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Unformatted text preview: 6.6 Black-Scholes formula Many options take place over a time period of one or more months, so it is natural consider St to be the stock price after t years. We could use a binomial model in which prices change at the end of each day but it would also be natural to update prices several times during the day. Let h be the amount of time measured in years between updates of the stock price. This h will be very small e.g., 1/365 for daily updates so it is natural to let h ! 0. Knowing what will happen when we take the limit we will let p Snh = S(n 1)h exp(µh + hXn ) where P (Xn = 1) = P (Xn = 1) = 1/2. This is binomial model with p p u = exp(µh + h) d = exp(µh h) (6.23) Iterating we see that Snh n pX = S0 exp µnh + h Xm m=1 ! (6.24) If we let t = nh the first term is just µt. Writing h = t/n the second term becomes n p 1X t· p Xm n m=1 To take the limit as n ! 1, we use the Theorem 6.9. Central Limit Theorem. Let X1 , X2 , . . . be i.i.d. with EXi = 0 and var (Xi ) = 1 Then for all x we have !...
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