Exponentiating the average log-price underestimates the average price

Question

A log-price (or log-return) Y is normally distributed, Y \sim \mathcal{N}(\mu, \sigma^2). You want to estimate the average price \mathbb{E}[e^{Y}]. A trader estimates it by exponentiating the sample mean of the logs, reporting e^{\bar{Y}}. Does e^{\bar{Y}} correctly target \mathbb{E}[e^{Y}]? Which way does it err, and by how much?

Accepted Answer

Direction, from Jensen. The exponential g(x) = e^x is strictly convex (g'' = e^x 0), so Jensen gives \mathbb{E}[e^{Y}] \ge e^{\mathbb{E}[Y]} = e^{\mu}. Meanwhile e^{\bar{Y}} estimates e^{\mu} (since \bar Y targets \mu), so e^{\bar{Y}} systematically underestimates the true average price \mathbb{E}[e^{Y}]. Exact size. For a normal Y \sim \mathcal{N}(\mu, \sigma^2), the lognormal mean is known exactly: \mathbb{E}[e^{Y}] = e^{\mu + \sigma^2/2}. So the true average price carries an extra factor e^{\sigma^2/2} 1 that the plug-in e^{\mu} misses entirely. The multiplicative gap is e^{\sigma^2/2}; fo…

Exponentiating the average log-price underestimates the average price

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