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Annual Volatility for Standard & Poor 500

My undergraduate student Angel Piotrowski computed annual volatility for Standard & Poor 500 (and its predecessor, Standard & Poor 90). For each year 1928 — 2023, she took daily index values S(t) with day t in this year, and computed log returns \ln(S(t)/S(t-1)). Then she computed standard deviation of these log returns for day t in any given year. Let V(s) be this standard deviation, usually called volatility, for year s. The data is available on my web page.

Next, Angel created a time series model for this volatility: Autoregression of order 1 on the log scale. The motivation comes from plotting the autocorrelation function for X(t) = \ln V(t). It is defined as k \mapsto \rho(X(t), X(t-k)). This looks like an autocorrelation function for an autoregression of order 1.

Here is the equation for this autoregression: \ln V(s) = \alpha + \beta \ln V(s-1) + W(s) with \alpha = 0.62 and \beta = -1.776. Let us test whether the innovations W(s) are Gaussian. Apply the quantile-quantile plot versus the normal distribution. This looks like pretty close to a Gaussian law!

Next, plot the autocorrelation function for innovations W(s) and another plot of an autocorrelation function for their absolute values |W(s)| to see that they correspond to white noise.

Thus the model fits well: \ln V(s) = \alpha + \beta \ln V(s-1) + W(s). We have numerical estimates \beta = 0.620, \alpha = -1.775. Therefore, this autoregression has a stationary distribution, or an invariant probability measure, \Pi such that \ln V(t) \sim \Pi \Rightarrow \ln V(t+1) \sim \Pi. Angel has computed that mean and variance of this stationary distribution.

See updates here.

Published by

Andrey Sarantsev

Responses

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