Method of moments for a lognormal on the raw scale

Question

Normalized prices follow a lognormal distribution: $\log X$ is normal with mean $\mu$ and variance $\sigma^2$ . Its raw moments are $\mathbb{E}[X] = e^{\mu + \sigma^2/2}$ and $\mathbb{E}[X^2] = e^{2\mu + 2\sigma^2}$ . From your data the sample mean is $1$ and the sample average of the squares is $e \approx 2.718$ .

Use the method of moments on the raw scale to estimate $\mu$ and $\sigma^2$ .

Accepted Answer

Why raw-scale moment matching. The clean approach would be to take logs and match the mean and variance of \log X. But suppose you only have the raw moments in hand. Two parameters call for two equations, so match \mathbb{E}[X] and \mathbb{E}[X^2] directly. Set up the two equations. With sample values m1 = 1 and m2 = e: e^{\mu + \sigma^2/2} = 1, \qquad e^{2\mu + 2\sigma^2} = e. Solve for \sigma^2 first. Take the ratio \mathbb{E}[X^2]/(\mathbb{E}[X])^2, which conveniently isolates \sigma^2: \frac{\mathbb{E}[X^2]}{(\mathbb{E}[X])^2} = \frac{e^{2\mu + 2\sigma^2}}{e^{2\mu + \sigma^2}} = e^{\sigma…

Method of moments for a lognormal on the raw scale

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