Question: S(t + At) = ] S(t)u with probability p S(t)d with probability 1 - p. where 1 =eat d = 1/u, p= erAt - e-ovAt

S(t + At) = ] S(t)u with probability p S(t)d with probability 1 - p. where 1 =eat d = 1/u, p= erAt - e-ovAt egvAt _ p-ovAt* over the time interval At. (a) Show that ES(t + At)|S(t)] = S(t)erAt where E[.] is the expectation. (b) Show that Var[S(t + At)|S(t)] = S(t)2 62At + 0(At?)], where Var is the variance. Hint: the Taylor expansion of e is e = 1+x+ 2! + 3! n! *-0

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