1. Assume the following (idealized) annual return estimates (, ²) from a dataset. Arithmetic: (0.08, 0.20²), Continuous: (0.06, 0.20²) note the perfect mean shift for continuous returns. In the lectures we saw two binomial models, the Hull model with simplified u/d formulas and more complicated p*, and the general u/d formulas with p*=1/2. In theory, both binomial models converge to the same lognormal model. With S₀=100 and T=1, confirm one example of convergence, comparing Pr(S₁>110) from these binomials (Hull and General) as t0 to the theoretical lognormal.

Hint: Given n, let t=1/n and note that in either binomial model

S₁= 100u^kd^n-k=100dⁿ(u/d)^k where k is between 0 and n. Algebraically translate Pr(S₁>110) into a statement about k (which has a binomial distribution), and then evaluate with Excels BINOM function since you have p* (note that these models have different u/d definitions, different p* values, and these all change with n). Compare these probabilities as n increases to exact Pr(S₁>110) calculated with the limiting lognormal distribution. Try n=50, 500, 1000, 100000.