Problem Description

You are presented with two distinct statistical estimation challenges.

Part 1: Gaussian Volatility Estimation

Given 100 independent samples, $X_1, \dots, X_{100}$, drawn from a normal distribution $N(0, \sigma^2)$, where the mean is known to be 0 and the variance $\sigma^2$ (and thus $\sigma$) is unknown.

1. Propose an estimator for $\sigma$.
2. Determine if this estimator is unbiased.
3. If the estimator is biased, describe how it can be debiased.

Part 2: Brownian Motion Volatility Estimation

Consider a Brownian motion process $X_t = \sigma W_t$ with $X_0 = 0$, observed at integer time steps $t = 1, 2, \dots, 100$.

Define the following statistics based on these observations:

• $H = \max(X_1, \dots, X_{100})$ (the maximum observed value)
• $L = \min(X_1, \dots, X_{100})$ (the minimum observed value)
• $C = X_{100}$ (the final observed value)
• $R = H - L$ (the range of observed values)

You are asked to evaluate calibrated versions of the following three statistics for estimating volatility:

1. $|C|$
2. $R - |C|/2$
3. $R^2 + C^2$

For these three estimators:

1. Compare them.
2. Indicate which one you would prefer under a mean-squared-error (MSE) criterion.
3. State explicitly whether your proposed estimators are for $\sigma$ or $\sigma^2$.