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## WISHART PROCESS FOR EQUITY VOLATILITY

Volatility is modeled as a mean-reverting process employing the argument that if not and there is a law that it had followed for the past century then it would be much larger.  This is the same argument used to justify interest rates as mean-reverting.  The ‘classical’ models of volatility and interest rates had been the Cox-Ingersoll-Ross (1985)/Feller (1951) process and the Ornstein-Uhlenbeck process, i.e. the models $dv = \kappa(\theta-v)dt + \sigma\sqrt{v}dw$ and $dv=-\kappa v dt + \sigma dw$.  The former has become much more popular for volatility models because of the closed-form option pricing by Heston (1993) model.  Eraker (2003) determined that jumps in volatility are necessary in addition to jumps in the price process, so volatility models must have jumps in addition to the mean reverting diffusion but this had not become standardized and in fact I was totally confused about this aspect simply because Heston and Bates models had been well-known.  An affine class closed form model with jumps in volatility was published by Duffie-Pan-Singleton in 2000 that I have used to test which has excellent fits to the volatility surface and whose implementation is a minor variation of the Heston/Bates models.  I had shown recently that in addition to  jumps time-change by the inverse stable subordinator are necessary to fit volatility.   Now we take the next step, which is to consider the problem of multivariate volatility models with jumps and inverse stable time change.  The method I had used to fit the univariate volatility models is by minimizing the error of fit of the characteristic function of the theoretical model with the empirical characteristic function.  Recall that for a series $X_1,X_2,\dots$ the empirical characteristic function is simply $\phi_e(t,\xi) = 1/n \sum_j e^{it\xi X_j}$.  This definition is valid for $X_j$ being vector quantities so we have to work with the theoretical multivariate characteristic function.  The multidimensional analogue of the square root process is the Wishart process.  It is the process governed by

$dX_t = \sqrt{X_t} dB_t Q + Q^T dB_t^T \sqrt{X_t} + (MX_t + X_t M + \kappa Q^T Q) dt$

A formula for the Laplace transform (and therefore the characteristic function) was derived for this matrix process by Bru (1991).  More details for this process can be found here:  AlessandroGnoatto and an option pricing model where the volatility follows the Wishart process can be found here: FonsecaWishartVolatilityModel.  Under time change by an absolutely continuous time change $T_t =\int_0^s \dot{T}_s ds$ of volatility the process satisfies

$dX_t = S_{\alpha}(t) (\sqrt{X_t} dB_t Q + Q^T dB_t^T \sqrt{X_t}) + S_{\alpha}(t) (MX_t + X_tM) dt + T_{\alpha}(t) \kappa Q^T Q dt$

Now the Bru (1991) Laplace transform formula is given in laplacewishart

Now for the time-change we just insert the stopping time $S_\alpha(t)$ and integrate against the density of the inverse stable subordinator.