1.(15 points). Suppose we have the following risk-neutral probability and up and down movements for a binomial lattice:

Show that the discounted asset value is a martingale. Hint, you can substitute and use basic algebra to show that under the risk-neutral measure the expectation (i.e. that is, under the risk-neutral measure , the value one-period prior is equal to the discounted value one-period hence). Hint, set , substitute for and algebraically manipulate.

With a binomial lattice where we wish to model asset prices such that, in the limit, our Random Walk (RW) becomes Brownian Motion (BM) and our process is consistent with the Black-Scholes Geometric Brownian Motion, we basically solve a system of two equations with three unknowns (i.e. one degree-of-freedom) such that we match the mean and variance. Mathematically, we have that:

For the following specification, show that we match the mean and standard deviation and therefore satisfy the system of equations:

By matching the mean and the variance, and based on our returns being i.i.d, the returns converge to which is consistent with Black-Scholes.