Recall that Brownian motion, which we identify with the log-return of an asset price, does not...

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Recall that Brownian motion, which we identify with the log-return of an asset price, does not have bounded variation. Here is an exercise to motivate why this is consistent with our finance models: Let T, , , So be fixed, with T,, So > 0. Consider a real-world CRR model for the price Sn of the security S. Show that in the limit as n , the variation of the log-returns of the real-world CRR trajectory is unbounded: n lim log S; - log Sj1| = . nX j=1 (This statement is true with probability 1.) Recall that Brownian motion, which we identify with the log-return of an asset price, does not have bounded variation. Here is an exercise to motivate why this is consistent with our finance models: Let T, , , So be fixed, with T,, So > 0. Consider a real-world CRR model for the price Sn of the security S. Show that in the limit as n , the variation of the log-returns of the real-world CRR trajectory is unbounded: n lim log S; - log Sj1| = . nX j=1 (This statement is true with probability 1.)