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## Donoho-Johnstone denoising for long memory in stochastic volatility

If you take log return-squared of a typical asset return you will find that there is long memory by estimating the fractional differencing parameter $d>0$ which corresponds to Hurst exponent $H > 1/2$.  Now the stochastic volatility model is

$r_t = \exp( h_t/2 ) \epsilon_t$

where $\epsilon_t \sim N(0,1)$.  The problem is that the stochastic volatility series $h_t$ is latent but we have used $x_t = \log(r_t^2+c_0)$ that we observe.  So what we have is $x_t = h_t + Q_t$ where $Q_t \sim \log(\chi^2)$ which could be approximated by $N(-1.27, \pi^2/2)$.  It would be excellent if we could use a denoising transformation on the observed $x_t$ to obtain the stochastic volatility $h_t$.  In the short-memory stochastic volatility model, this is done via the Kalman filter but in the long memory case no state space representation exists.  However, we can still apply the Donoho-Johnstone optimal denoising by shrinking the wavelet transform.  An excellent account of this is in Donoho’s 1992 report:  DonohoDenoiseSoftThreshold.  This allows us to filter out the stochastic volatility which can then be used to do things like calculate option prices etc.

I am rather pleased with this direction because I have solved here a very useful general problem: how does one go about filtering a long memory stochastic volatility model when Kalman filter is not available.

So practically how to implement the Donoho-Johnstone approach?  They use a specific version of wavelet transform due to Cohen-Daubechies-Jawerth-Vial for intervals.  The basic result is that for $y_t = x_t + \sigma \epsilon_t$ with standard white noise $\epsilon_t \sim N(0,1)$ the soft threshold for wavelet coefficients is $\sqrt{2 \log(n)} \sigma \gamma_1 /\sqrt{n}$ where $\gamma_1$ is determined from the maximum singular value of a matrix that is determined by the pyramid wavelet scheme.  Unfortunately I don’t know of a packaged R function to do this.  In order to proceed, we can use the ‘wavelets’ package functions ‘dwt’ and ‘idwt’.  Let $x_t$ be log return-squared which we transform as $(x_t + 1.27)/\sqrt{n}$ and take dwt say with 3 levels.  The wavelet parameters can be accessed with z@W$W1, z@W$W2, z@W\$W3 each of which we can soft threshold appropriately and then apply ‘idwt’ to reconstruct the denoised signal.  One can check that the Hurst parameter increases which is reasonable because the Hurst parameter controls the Holder-continuity of sample paths for the Brownian motion.  Now by the Donoho-Johnstone theory should approximately do the right denoising so the denoised series $y_t$ is an estimate of the latent stochastic volatility without the iid return noise.