Exercise: Log-Returns as an Additive Random Walk

Prerequisites: Random Walk, Log-Normal Distribution

Problem

Consider a multiplicative random walk for a stock price: $S_n = S_{n-1} Z_n$ with

and $S_0 = 100$.

1. Show that $R_n := \ln(S_n / S_{n-1})$ is a random variable taking only two values, and compute its mean $\mathbb{E}[R_n]$ and variance $\operatorname{Var}(R_n)$.
2. Show that the log-price $\ln S_n$ is an additive random walk. Write $\ln S_n$ in terms of $\ln S_0$ and the $R_i$.
3. Compute $\mathbb{E}[\ln S_{250}]$ and $\operatorname{Var}(\ln S_{250})$ for one trading year.
4. Even though $\mathbb{E}[R_n] > 0$ (positive expected log-return), show that the expected price $\mathbb{E}[S_n]$ grows at a different rate than $e^{n\mathbb{E}[R_n]}$. Compute both rates and interpret the gap financially.

Hint

For part 4, use the fact that $\mathbb{E}[S_n] = S_0 (\mathbb{E}[Z_n])^n$ by independence, and compare to $S_0 e^{n\mathbb{E}[\ln Z_n]}$. Jensen's inequality guarantees they are different.