Why the average sits above the typical, the lognormal gap

Question

Asset prices are often modeled as lognormal: $\ln X \sim N(\mu, \sigma^2)$ . A colleague assumes the mean price and the median price are basically the same.

Find the mean, median, and mode of $X$ , explain why they differ, and how the gap depends on volatility.

Accepted Answer

The three centers. With \ln X \sim N(\mu, \sigma^2): - Median: since \ln X has median \mu and the log is monotonic, the median of X is e^{\mu}. - Mean: \mathbb{E}[X] = e^{\mu + \sigma^2/2} (this is the moment generating function of the normal evaluated at 1). - Mode: e^{\mu - \sigma^2}. So they line up as mode < median < mean: e^{\mu - \sigma^2} \;<\; e^{\mu} \;<\; e^{\mu + \sigma^2/2}. Why the mean exceeds the median. The lognormal is right-skewed: it is bounded below by zero but has a long upper tail. Those rare large values pull the average upward, while the median, the middle value, is un…

Why the average sits above the typical, the lognormal gap

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