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## PROBLEM WITH SV MODELS WITH CORRELATIONS

Stochastic volatility models often have the form

$dS(t) = S V(t) dw_S(t)$

$dV(t) = A(V,t) dt + B(V,t) dw_V(t)$

with the assumption that $\langle dw_S, dw_V \rangle = \rho dt$.  So this is a bad setup from the perspective of empirical facts.  The reason is that empirically, the correlation between contemporaneous volatility with returns is zero using 1900 stocks.  The Fisher Black 1976 ‘leverage correlation’ does not have any sort of decay to zero — it is a lag 1 effect.  If you define volatility as $v_t \log(r_t^2)$ then $Corr(v_{t+1},r_t) = \rho < 0$ and on the other hand

$Corr(\Delta v_{t+1}, r_t ) > 0$

There is no noticeable decay or other structure in the empirical data for individual stocks that is noticeable beyond noise.  Here’s a graph of the average of these lagged return-volatility correlations over around 1900 stocks.

Now this leverage effect does not seem to have any continuous decay in daily data.  But stochastic volatility models typically do not capture this because they put a correlation between contemperaneous brownian motions.  One could instead consider alternative correlation requirements for stochastic volatility models such as lagged correlation — i.e.  \langle dw_S(t) dw_V(t+1) \rangle = \rho dt\$.  In particular at least I am not compelled by a continuous decay for these from this chart above.

The solution idea here is to apply Novikov’s theorem following p. 14 http://arxiv.org/pdf/cond-mat/0302095v1.pdf with the same argument with (A6) replaced by

$\frac{\delta \xi_2(t+t)}{\delta \xi_1(r)} = \rho$

$\frac{\delta \xi_2(t)}{\delta \xi_1(t)} = 0$

$\frac{\delta [\sigma(t) \sigma(t + \tau)^2]}{\delta \xi_1(t)} = \rho [ \sigma(t+\tau)^2 \frac{\delta \sigma(t)}{\delta \xi_2(t+\tau)} + 2 \sigma(t) \sigma(t+\tau) \frac{\delta \sigma(t+\tau)}{\delta \xi_2(t+\tau)} ]$