The problem of long memory in finance has been on the agenda of top econometricians like Clive Granger since the 1960s. In fact, he wrote about the typical spectral shape of economic variables in 1966 and then in 1980-81 introduced the long memory models that I have been using to show that long memory is a real phenomenon probably for the n-th time since the 1960s, the fractionally differenced ARIMA models, for which a nice open source implementation in the ‘forecast’ module in R makes this simple. Well, so Granger and coworkers were responsible for the Ding-Granger-Engle model from 1993 which fit 1900 stock volatility autocorrelations extremely well, which I was trying to understand for the last few days. The DGE autocorrelation model is:

which is log-linear and can be fit with so its very good. Attempting to analyze this equation directly produces a complicated mess. But we are fortunate because the fundamental idea behind this model is essentially to interpolate between an exponential and a power law decay, so the question is whether a similar quality fit to actual empirical volatility autocorrelations in stocks can be obtained by a more theoretically justified model, and the answer is yes. We can consider the model and obtain fits with . For now here are some pictures of fits.

The natural model to consider for these, following the breakthrough of Comte-Renault in 2003 of an analyzable stochastic volatility model with long memory, where they took a square root process as used by Cox-Ingersoll-Ross in 1985 and studied by Feller in 1951 and essentially considered the volatility to be a fractional integral of . This allowed them to use the Heston 1993-type analysis for closed form solutions in the affine class which are solvable — essentially this allows one to take a complicated second order PDE which comes out of Black-Scholes analysis via Ito formula and solve it by ordinary differential equations; the affine class is essentially a generalized view for when this is possible. Regardless of analytic tractability, there is still no science without some exact numerial model fitting data, and the fits above tell us what the process should be if we did not worry about option pricing: we’d be looking at a fractional version of the square root type process:

which is the non-rigorous way of adding a white noise scaled by to a fractional ordinary differential equation. The Mittag-Leffler functions act as analogs of exponentials and describe the autocorrelations of these processes just as exponentials occur as the autocorrelations of the standard square root process. See samorodnitsky-spectrum-autocorrelation for introduction to long memory and comte-renault-2003-affine-sv

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