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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

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

## 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.

### 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.