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# Stochastic volatility¶

We apply the sequential inference algorithm outlined here to the exchange rate of different assets.

We model the log-volatility $${\bf Z}_{\Lambda}$$ of the return of a financial asset at times $$\Lambda=\{0,1,\ldots,n\}$$ with the autoregressive process

$\begin{split}{\bf Z}_{k+1} = \mu + \phi ({\bf Z}_k - \mu) + \varepsilon_k \;, \qquad \varepsilon_k \sim \mathcal{N}(0,\sigma^2) \;, \quad \varepsilon_k {\perp\!\!\!\perp} {\bf Z}_k \\ \qquad \left.{\bf Z}_0 \right\vert \mu,\sigma,\phi \sim \mathcal{N}\left(\mu, \frac{\sigma^2}{1-\phi^2}\right) \;, \qquad \mu \sim \mathcal{N}(0,1) \;\, \\ \qquad \phi = 2 \frac{\exp(\phi^\star)}{1+\exp(\phi^\star)} - 1 \;, \qquad \phi^\star \sim \mathcal{N}(3,1) \;, \\ \sigma^2 \sim \text{InvGamma}(\alpha=1, \beta=0.1) \;.\end{split}$

For $$k \in \Xi \subset \Lambda$$, estimate parameters $$\Theta = (\mu,\phi)$$ and states $$\left\{ {\bf Z}_k \right\}$$, given observations

${\bf Y}_k = \xi_k \exp\left(\frac{1}{2}{\bf Z}_k\right) \;, \qquad \xi_k \sim \mathcal{N}(0,1) \;, \quad \xi_k {\perp\!\!\!\perp} {\bf Z}_k \;.$

## Exchange rate GBP - USD¶

We consider here the exchange rates between British Pound (GBP) and US Dollar (USD). These data and results are part of the paper TM4.

### State and parameters estimation 10/01/81 - 06/27/85¶

First we consider the problem of estimating the parameters $$\mu, \phi$$ and states $${\bf Z}_{1:945}$$ of the stochastic volatility model using the 945 observations of the daily returns associated to the GBP-USD exchange rates from 10/01/81 till 06/27/85. We fix the variance of the dynamics to $$\sigma=1/4$$. The same problem has been analyzed also in OR13 and OR14. We provide a number of files which can be used to reproduce the results in TM4.

• DurbinData.csv [md5sum: a8a223904ded9d3f19d4a3c5946541ed]: daily returns
• Distribution.dill [md5sum: ad8fd058693939a81207ea812fb44fca]: SequentialHiddenMarkovChainDistribution $$\pi\left( \left. \Theta, {\bf Z}_\Lambda \right\vert {\bf y}_\Xi \right) \propto \mathcal{L}\left({\bf y}_\Xi \left\vert \Theta, {\bf Z}_\Lambda\right.\right) \pi\left( \Theta, {\bf Z}_\Lambda \right)$$
• runner.sh [md5sum: 1471af8891c113e9851c996fbbab374b]: script used to construct the sequential map and obatin all the results. The script was run in parallel on 8 machine for a total of 128 cores.
• Sequential-map.dill [md5sum: eb7b4d90cd020a2dd237671e61a0f80e]: this contains the output of the script tmap-sequential-tm. It includes the base distribution $$\rho=\mathcal{N}(0,{\bf I})$$, the target distribution $$\pi\left( \left. \Theta, {\bf Z}_\Lambda \right\vert {\bf y}_\Xi \right) \propto \mathcal{L}\left({\bf y}_\Xi \left\vert \Theta, {\bf Z}_\Lambda\right.\right) \pi\left( \Theta, {\bf Z}_\Lambda \right)$$, the map $$T$$ such that $$T_\sharp \rho \approx \pi\left( \left. \Theta, {\bf Z}_\Lambda \right\vert {\bf y}_\Xi \right)$$, and the TransportMapSmoother used for the construction.
• Sequential-map-POST.dill [md5sum: 50c66da9e5b74792db931ac53459e906]: data structure used as output of the script tmap-sequential-postprocess.
• Sequential-map-POST.dill.hdf5 [md5sum: 1d0725ad889fe86f3e2f2c02fe7169b9]: dataset containing the output of tmap-sequential-postprocess. The data is structured as follows:
• filtering: list of samples from the approximate filtering distributions $$\pi\left(\Theta, {\bf Z}_k \middle\vert {\bf y}_{1:k}\right)$$ for $$k\in\Lambda$$.
• metropolis-independent-proposal-samples/skip-10: Monte Carlo Markov Chain $$10^5$$ long, obtained with MetropolisHastingsIndependentProposalsSampler, by subsampling every 10 samples.
• x: Monte Carlo Markov Chain with invariant $$\pi\left( \Theta, {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$.
• s: Monte Carlo Markov Chain with invariant $$T^\sharp \pi\left( \Theta, {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$.
• quadrature: Monte Carlo samples from $$T_\sharp\rho \approx \pi\left( \Theta, {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$.
• vals_var_diag: values $$\{\log\rho({\bf x}_i)\}$$ and $$\{\log T^\sharp\pi({\bf x}_i)\}$$ used to compute the variance diagnostic $$\mathbb{V}\left[\log\frac{\rho}{T^\sharp\pi}\right]$$.
• trim-%i: postprocessing of the approximation of the trimmed distribution $$\pi\left( \Theta, {\bf Z}_{\Lambda<i}\, \middle\vert {\bf y}_{\Xi<i}\, \right)$$.
• metropolis-independent-proposal-samples/skip-10: Monte Carlo Markov Chain $$10^5$$ long, obtained with MetropolisHastingsIndependentProposalsSampler, by subsampling every 10 samples.
• vals_var_diag: values used to compute the variance diagnostic.

In the following we report some of the results obtained. For a complete treatment we refer to TM4.

Mean and $$\{5,95\}$$ percentiles of the approximate filtering marginals $$\pi\left({\bf Z}_k \middle\vert {\bf y}_{1:k}\right)$$ (blue) along with one realization (black)

Mean and $$\{5,95\}$$ percentiles of the approximate smoothing marginals $$[T_\sharp\rho]_k \approx \pi\left({\bf Z}_k \middle\vert {\bf y}_\Xi\right)$$ (red) along with one realization (black)

Mean and $$\{5,95\}$$ percentiles of the approximate (red) and exact (black) smoothing marginals obtained with the map $$T$$ and Markov Chain Monte Carlo respectively

(xy)-axis: mean and $$\{5,25,40,60,75,95\}$$ percentiles of the approximate filtering marginal $$[T_\sharp\rho]_{\mu} \approx \pi\left(\mu\middle\vert {\bf y}_{1:k}\right)$$ of the hyper-parameter $$\mu$$. (xyz)-axis: for a subset steps $$k$$, we show the density of the approximate (solid lines) and the exact (dashed lines) filtering marginal obtained with Markov Chain Monte Carlo

(xy)-axis: mean and $$\{5,25,40,60,75,95\}$$ percentiles of the approximate filtering marginal $$[T_\sharp\rho]_{\phi} \approx \pi\left(\phi\middle\vert {\bf y}_{1:k}\right)$$ of the hyper-parameter $$\phi$$. (xyz)-axis: for a subset steps $$k$$, we show the density of the approximate (solid lines) and the exact (dashed lines) filtering marginal obtained with Markov Chain Monte Carlo

(shaded) $$\{5,25,40,60,75,95\}$$ percentiles of the posterior predictive (conditioned on all the data). (dots) data.

### Filtering and smoothing 10/01/1981 - 08/24/2017¶

Here we fix the hyper-parameters $$\mu,\phi$$ of the stochastic volatility model to the medians $$\mu=0.667$$ and $$\phi=0.879$$ found through the preceding analysis of the first 945 steps, and apply the algorithm for filtering and smoothing on an extended dataset of 9009 observations from 10/01/1981 till 08/24/2017. This means that we will sequentially construct 9008 two dimensional maps in order to approximate the full posterior $$\pi\left({\bf Z}_{1:9009}\middle\vert {\bf y}_{1:9009}\right)$$ and the filtering distributions $$\pi\left({\bf Z}_{k}\middle\vert {\bf y}_{1:k}\right)$$ for $$k=1,\ldots,9009$$. This setting is also described in TM4. Here we provide the dataset used and the results obtained.

• GBP-USD.csv [md5sum: 195a260b45b113051756d1297f082714]: daily returns
• Distribution.dill [md5sum: 65c4cc50ff8eb6200cfc373523dad46a]: SequentialHiddenMarkovChainDistribution $$\pi\left({\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right) \propto \mathcal{L}\left({\bf y}_\Xi \middle\vert {\bf Z}_\Lambda\right) \pi\left( {\bf Z}_\Lambda \right)$$
• runner.sh [md5sum: 51de28a3588809bbe8965646b7a4d0a4]: script used to construct the sequential map and obatin all the results. The script was run in parallel on one machine with 10 cores.
• Sequential-map.dill [md5sum: ecff7757ea414f045259e8e5caca903b]: this contains the output of the script tmap-sequential-tm. It includes the base distribution $$\rho=\mathcal{N}(0,{\bf I})$$, the target distribution $$\pi\left( {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right) \propto \mathcal{L}\left({\bf y}_\Xi \middle\vert {\bf Z}_\Lambda\right) \pi\left( {\bf Z}_\Lambda \right)$$, the map $$T$$ such that $$T_\sharp \rho \approx \pi\left( {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$, and the TransportMapSmoother used for the construction.
• Sequential-map-POST.dill [md5sum: 72a755383fba437e4dead6ff3e3d81e3]: data structure used as output of the script tmap-sequential-postprocess.
• Sequential-map-POST.dill.hdf5 [md5sum: d1b5686c3680f623b8cba2764c92eb0c]: dataset containing the output of tmap-sequential-postprocess. The data is structured as follows:
• filtering: list of samples from the approximate filtering distributions $$\pi\left({\bf Z}_k \middle\vert {\bf y}_{1:k}\right)$$ for $$k\in\Lambda$$.
• metropolis-independent-proposal-samples/skip-10: Monte Carlo Markov Chain $$10^5$$ long, obtained with MetropolisHastingsIndependentProposalsSampler, by subsampling every 10 samples.
• x: Monte Carlo Markov Chain with invariant $$\pi\left( {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$.
• s: Monte Carlo Markov Chain with invariant $$T^\sharp \pi\left( {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$.
• quadrature: Monte Carlo samples from $$T_\sharp\rho \approx \pi\left( {\bf Z}_\Lambda \middle\vert {\bf y}_\Xi \right)$$.

The following images show the smoothing marginals at different timesteps. We makred some historical events to put this results into context. If you, by any chance, have a better historical insight on the evolution of the volatlity for certain periods, we would be happy to know it.

Mean and $$\{5,95\}$$ percentiles of the approximate (red) and exact (black) smoothing marginals obtained with the map $$T$$ and Markov Chain Monte Carlo respectively