We show that, for a class of mean-reverting models , the correlation function of stochastic variance (squared volatility) contains only one -- relaxation -- parameter. We generalize and simplify the expression for leverage for this class of models. We apply our results to specific examples of such models -- multiplicative, Heston , and combined multiplicative-Heston -- and use historic stock market data to obtain parameters of their steady-state distributions and cross-correlations between Weiner processes in the models for stock returns and stochastic variance .
We investigate relaxation and correlations in a class of mean-reverting models for stochastic variances. We derive closed-form expressions for the correlation functions and leverage for a general form of the stochastic term. We also discuss correlation functions and leverage for three specific models -- multiplicative, Heston (Cox-Ingersoll-Ross) and combined multiplicative-Heston -- whose steady-state probability density functions are Gamma, Inverse Gamma and Beta Prime respectively, the latter two exhibiting "fat" tails. For the Heston model, we apply the eigenvalue analysis of the Fokker-Planck equation to derive the correlation function -- in agreement with the general analysis -- and to identify a series of time scales, which are observable in relaxation of cumulants on approach to the steady state. We test our findings on a very large set of historic financial markets data .
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, the correlation function of stochastic variance (squared volatility) contains only one -- relaxation -- parameter. We generalize and simplify the expression for leverage for this class of models. We apply our results to specific examples of such
for stochastic variances. We derive closed-form expressions for the correlation functions and leverage for a general form of the stochastic term. We also discuss correlation functions and leverage for three specific
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and use historic stock market data to obtain parameters of their
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distributions and cross-correlations between Weiner processes in the models for stock returns and stochastic variance
probability density functions are Gamma, Inverse Gamma and Beta Prime respectively, the latter two exhibiting "fat" tails. For the Heston model, we apply the eigenvalue analysis of the Fokker-Planck equation to derive the correlation function -- in agreement with the general analysis -- and to identify a series of time scales, which are observable in relaxation of cumulants on approach to the steady state. We test our findings on a very large set of historic financial markets data