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Asset prices have time-varying volatility (variance of day over day `returns`). In some periods, returns are highly variable, while in others very stable. Stochastic volatility models model this with a latent volatility variable, modeled as a stochastic process. The following model is similar to the one described in the No-U-Turn Sampler paper, Hoffman (2011) p21.
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Asset prices have time-varying volatility (variance of day over day `returns`). In some periods, returns are highly variable, while in others very stable. Stochastic volatility models model this with a latent volatility variable, modeled as a stochastic process. The following model is similar to the one described in the No-U-Turn Sampler paper, {cite:p}`hoffman2014nuts`.
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$$ \sigma \sim Exponential(50) $$
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$$ \nu \sim Exponential(.1) $$
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$$ s_i \sim Normal(s_{i-1}, \sigma^{-2}) $$
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$$ log(r_i) \sim t(\nu, 0, exp(-2 s_i)) $$
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$$\log(r_i) \sim t(\nu, 0, \exp(-2 s_i)) $$
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Here, $r$ is the daily return series and $s$ is the latent log volatility process.
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