Math for Finance question

2 To show that lognormal price dynamics is the continuum limit of a family of binomial trees, we considered the following multiperiod multiplicative tree: the stock price at the initial node is S(0); if at a given node the stock price is Snow, then at the next time it can be either uS now or dSnow, where u = eMAt+oAt and d = At-oAt, at each time, the stock goes up or down with probability 1/2 for each choice. Fixing N, consider the stock price after N time steps, s(N At). (a) Suppose that over the course of the N time steps the stock goes upj times and down N-; times. Express InS(NAT) in terms j. (b) What is the mean of InS(N At)? (c) What is the variance of InS(NAt)? 2 To show that lognormal price dynamics is the continuum limit of a family of binomial trees, we considered the following multiperiod multiplicative tree: the stock price at the initial node is S(0); if at a given node the stock price is Snow, then at the next time it can be either uS now or dSnow, where u = eMAt+oAt and d = At-oAt, at each time, the stock goes up or down with probability 1/2 for each choice. Fixing N, consider the stock price after N time steps, s(N At). (a) Suppose that over the course of the N time steps the stock goes upj times and down N-; times. Express InS(NAT) in terms j. (b) What is the mean of InS(N At)? (c) What is the variance of InS(NAt)