Random Walks

We develop the formal mathematics of the lognormal random walk model. We start by discussing continuous compounding for risk-free investments. We introduce a random variable to model the uncertainty of a risky investment. We apply the Central Limit Theorem to argue that under three strong assumptions, the values of risky investments at any time horizon are lognormally distributed. We model the random walk using a stochastic differential equation. We define the notion of an " Ito process " and prove that it is equivalent to our formulation. We apply the model to the S&P 500 stock market index as an example. We learn how to do parameter estimation for the model using historical time series data and how to do calculations in the model in computer programs. We discuss how uncertainty and risk increase with time horizon when investing in volatile assets like stocks, contrary to popular opinion. We conclude by asking the all-important question of how well the simple random walk model describes how financial markets actually work. We mention known failings of the model and conclude that at best it is a rough approximation to reality and should be used for real-life financial planning with caution. We assume that the reader is familiar with the normal and lognormal probability distributions as presented in reference [8].