Returns should not be modeled by non-Markovian processes directly; the Delbaen-Schachermayer 1994 fundamental theorem of asset pricing states that the existence of a martingale measure is equivalent to ‘no free lunch with vanishing risk’ and many people have shown that fractional Brownian motion models for returns have explicit arbitrage, for instance Rogers (2013). On the other hand, Hurst exponents for a big basket of 1900 stocks estimated from daily returns are spread widely around leading to a quandary. Now non-Markovian models for volatility are known not to have a direct arbitrage problem: in the 2006 ICM David Nualart (nualart-icm2006-fbm-apps) presented the results of Hu (2005) where a stochastic volatility model driven by fractional Brownian motion was shown to have a non-unique equivalent martingale measure where and where and he found that the call option prices are expectations with respect to of with , while stochastic volatility models with long memory of this type were initiated by Comte and Renault (1998). Thus stochastic volatility models have no arbitrage problems when volatility models are non-Markovian; these problems arise when the return process are non-Markovian.

We now consider the empirical features of volatility from daily data for 1419 stocks and consider whether it makes any sense to model volatility by a fractional Brownian motion or any other Gaussian process.

Gaussians have zero kurtosis, while the empirical kurtosis of volatility jumps from a sample 1900 stocks have mean kurtois and therefore volatility jumps are not modeled well by Gaussian processes such as the Wiener process or the fractional Brownian motion which have zero kurtosis. At the same time, -stable distributions have infinite variance and kurtosis so these cannot be used directly, and the well-known solution is the truncated Levy flight models where the stable distributions are modified by a cutoff function.

Recall that the characteristic function of an exponentially truncated Levy distribution is

where

(see mariani-liu-2007-truncated-levy where these were used successfully to fit index returns and cumulants-truncated-levy-distribution where the theoretical kurtosis of truncated Levy distribution was computed for various truncation schemes)

What we find is that the shocks to volatility follow approximately a fractional Poisson process with while the Markovian Poisson process would have (details on the fractional Poisson process can be found in Dexter Cahoy’s 2007 Ph.D. thesis cahoy-phd-thesis-fractional-poisson-estimation and the estimator for is from cahoy-parameter-estim-fpp)

The jumps have distribution whose fit by truncated Levy flight model with parameters where is the index (which is for Gaussian) and . The parameter estimates for 1419 stocks are here.

Note that we can safely reject Gaussian from the estimate alone.

The tweakable parameter in the three-parameter truncated Levy flight model is l the cutoff parameter. The fitting is by an ad-hoc estimation procedure. The importance of this preliminary result is that the empirical kurtosis of jumps cannot be modeled by alpha-stable distributions because these have infinite kurtosis, and this result gives hope for a model for volatility jumps that can accurately model the empirical kurtosis of volatility jumps.

This is significant because GAUSSIAN PROCESSES generally will not have high kurtosis and these are the standard models used for stochastic volatility models since 1998 (Comte-Renault ComteRenault1998). Technical history of stochastic volatility models can be found in Shepard (stochastic-volatility-shepard).

The kurtosis of the jump distributions is ; mean which can be modeled by these truncated Levy flight distributions fitting but not by Gaussian or other well-known parametric models.

Also it is reasonable to assume that the non-Markovian nature of financial markets originally brought up by Mandelbrot in the 1960s can be safely put in volatility rather than returns (which can be modeled by Markov Brownian motion models as has been done in many ways described by David Nualart in 2006 ICM). Note that Nualart in his 2006 presentation describes fbM models of volatility but this kurtosis observation would dissuade us from considering the Gaussian models on empirical grounds.

The non-Markovian nature of financial markets and its regularity seem to be in the volatility rather than in returns themselves.

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