93 203 68 exercises example 610 put call parity allows

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Unformatted text preview: n 1X Pp Xm x ! P ( x) (6.25) n m=1 where has a standard normal distribution. That is, Zx 2 1 p e y /2 dy P ( x) = 2⇡ 1 The conclusion in (6.25) is often written as n 1X p Xm ) n m=1 where ) is read “converges in distribution to.” Recalling that if we multiply a standard normal by a constant c then the result has a normal distribution with mean 0 and variance 2 , we see that p n p 1X t· p Xm ) t n m=1 and the limit is a normal with mean 0 and variance t. This motivates the following definition: 199 6.6. BLACK-SCHOLES FORMULA Definition. B (t) is a standard Brownian motion if B (0) = 0 and it satisfies the following conditions: (a) Independent increments. Whenever 0 = t0 < t1 < . . . < tk B (t1 ) B (t0 ), . . . , B (tk ) B (tk 1) are independent. (b) Stationary increments. The distribution of Bt Bs is normal (0, t s). (c) t ! Bt is continuous. To explain (a) note that if ni = ti /h then the sums ni X Xm i = 1, . . . k 1 <mni are independent. For (b) we note that that the distribution of the sum only depends on the number of ter...
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