2 To show that lognormal price dynamics is the continuum limit of a family of binomial...

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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 uSnow or dSnow, where u = eAt+oAt and d = eAto At. 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(NAt). (a) Suppose that over the course of the N time steps the stock goes upj times and down Nj times. Express InS(NAt) in terms j. (b) What is the mean of InS(NAt)? (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 uSnow or dSnow, where u = eAt+oAt and d = eAto At. 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(NAt). (a) Suppose that over the course of the N time steps the stock goes upj times and down Nj times. Express InS(NAt) in terms j. (b) What is the mean of InS(NAt)? (c) What is the variance of InS(NAt)?

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a If the stock goes up j times and down Nj times over the course of N time steps then the ... View the full answer