Smooth Asymmetric Stochastic Volatility for Heavy-Tailed Return Paths
Summary
The document explores how to model volatility that rises sharply and then decays more gradually, producing an asymmetric sawtooth pattern. It starts from a stochastic volatility process with centered, standardized lognormal shocks, then considers a regime-switching recursion in which occasional innovations reset volatility to a level drawn from a normal distribution. A softened version uses sigmoid weights to blend reset and ordinary dynamics, in hopes of keeping the process smooth enough for Markov chain Monte Carlo estimation.
These are proposed model formulations rather than validated results. The author reports challenges with jump magnitude control and notes that sharp switching may harm sampler behavior. The intended application is fitting historical returns for simulation and value-at-risk, with realistic extreme regimes, heavy-tailed returns induced by volatility dynamics, and few parameters. The document asks for alternatives but provides no empirical comparison or recommended final specification; latent volatility itself is secondary to matching returns.
Key ideas
- Stochastic volatility with jump-like shocks can create asymmetric volatility paths but may be difficult to estimate.
- A regime reset can model occasional upward volatility jumps followed by ordinary mean-reverting evolution.
- Sigmoid blending may smooth the transition between reset and standard dynamics, though sharpness can impair MCMC fitting.
- The target is heavy-tailed returns generated through volatility dynamics while keeping return innovations normal.
- The proposed formulations are exploratory and are not supported by reported fit comparisons.
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Full text
# Stochastic Volatility Model with Smooth Jumps in Volatility
# Stochastic Volatility Model with Smooth Jumps in Volatility
The Stochastic Volatility with Jumps can generate realistic price path dynamics with volatility like asymmetrical saw. But it has lots of params, hard for MCMC to fit, it's not smooth, params controlling jump are hard to estimate (multi modal, diverge, correlated, large uncertainty), the fit is very slow and unstable.
Is there a simple, smooth model that produces similar dynamics? I'm trying to make such model, and currently working with:
$$ \begin{aligned} r_t &= \mu + \exp(h_t)\,\varepsilon_t \\ h_t &= \omega + \phi(h_{t-1}-\omega) + \sigma J_t \\ J_t &= \frac{z_t - \mathbb{E}[z_t]}{\sqrt{\mathrm{Var}(z_t)}} \\ z_t &\sim \mathrm{LogNormal}(0,\nu) \end{aligned} $$
It produces saw like pattern:
But, the magnitude of a jump is hard to control sometimes it overshoot (maybe soft limit $h_t = h_{max}\text{sigmoid}(h_t/h_{max})$ could be used, but it may worsen MCMC convergence).
I'm thinking about a better model, but don't know how to make it smooth. The idea is to use plain recursion, but allow for innovation sometimes to bypass the recursion and set vol directly, producing asymmetrical up jumps. The jump magnitude would be correct naturally.
$$ \begin{aligned} r_t &= \mu + \exp(h_t)\epsilon_t \\ h_t &= \begin{cases} \sigma / \sqrt{1 - \phi^2} z_t, & \text{if } I_t = 1 \text{ and } z_t > h_{t-1}, \\ \omega + \phi(h_{t-1}-\omega) + \sigma z_t, & \text{otherwise} \end{cases} \\ I_t &\sim \mathrm{Bernoulli}(p) \\ z_t &\sim \mathcal{N}(0,1) \\ \end{aligned} $$
It's possible to make it soft with sigmoids. But, to make it realistic sigmoids should be sharp enough, and again - it destroy smoothness and makes it hard for MCMC.
$$ \begin{aligned} I_t &= \text{sigmoid}(\text{Normal}(\mu_p, \sigma_p))\ \text{sigmoid}(z_t - h_t) \\ h_t &= (1-I_t)[\omega + \phi(h_{t-1}-\omega) + \sigma z_t] + I_t \sigma / \sqrt{1 - \phi^2} z_t \end{aligned} $$
Question: Is there a way to make something like this smoothly and well suited for MCMC fit? Any good and simple model with asymmetrical jumps?
Notes:
Example of asymmetrical saw pattern, VIX:
Model fit to historical returns (not to IV surface) and used to simulate VaR, requirements:
- Realistic price path, especially in extreme regimes.
- Realistic volatility dynamics with asymmetric saw pattern.
- Heavy tailed returns, StudentT like (note that model must use normal for return innovation, so heavy tails must be produced by vol dynamics).
- Low param count.
Not important:
- Volatility latent process, it's not used, only the returns matter.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.