Modeling Asymmetric Tail Dependence in Stochastic Volatility
Summary
The document examines whether a conventional stochastic volatility model captures the observed relationship between a return and the magnitude of the following return. Using historical daily returns for a single stock, it describes a binned dependence plot and reports nonlinear tail behavior: large moves in either direction are associated with higher subsequent volatility, with a stronger effect after negative returns. The plot is qualitative evidence from the described sample, not a broad empirical study.
It considers several possible extensions: a t-copula or sign-dependent copula mixture, stochastic volatility with jumps, and an EGARCH-like leverage term. The stated objective is to forecast multi-month to one-year return distributions for VaR, including both upside and downside risk. The author notes uncertainty about whether jump models reproduce regularly varying tails and distinguishes historical forecasting from fitting implied-volatility surfaces. No model is implemented or compared, so the proposals remain hypotheses requiring estimation and out-of-sample validation.
Key ideas
- The described return data show nonlinear dependence between a return and the next period's absolute return, concentrated in the tails.
- Both positive and negative extreme returns are associated with higher subsequent volatility, with stronger effects after negative returns.
- Copula mixtures, stochastic volatility with jumps, and EGARCH-like dynamics are suggested as candidate model structures.
- The intended use is forecasting longer-horizon return distributions for VaR in both tails.
- The document presents qualitative evidence and model ideas, not comparative validation or forecasting results.
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Full text
# Realistic Correlation for SV Model
# Realistic Correlation for SV Model
Classical SV model use unrealistic correlation structure $\operatorname{Corr}(\varepsilon_t,\eta_t)=\rho$:
$$r_t = \exp(h_t/2)\,\varepsilon_t,\quad \varepsilon_t \sim t_\nu(0,1) $$ $$h_t = \mu + \phi (h_{t-1}-\mu) + \eta_t,\quad \eta_t \sim \mathcal N(0,\sigma_\eta^2)$$ $$\operatorname{Corr}(\varepsilon_t,\eta_t)=\rho,\quad \rho<0$$
The historical correlation looks different. Below plot of correlation for KO stock (50y daily log returns), other stocks looks similar.
$$\frac{P(r_t, |r_{t+1}|)}{P(r_t) P(|r_{t+1}|)}$$
The $r_t$ and $|r_{t+1}|$ split into bins and the histogram built, bins are non linear to magnify tails better. The second chart is the same with center region zoomed. Click on chart to enlarge.
Observations:
- Correlation is non linear, mostly in tails.
- Both right and left tail increase volatility.
- Left tail is more extreme than right.
How to model it?
T-Copula, maybe mixture of 2 t-copulas depending on $r_t$ sign. Numerically slow, requires quantile-inv-quantile for TDistr.
SV with Jump. Seems the best option, tail correlation could be modelled naturally. As far as I know - the SVJ with normal distribution for log returns don't provide regularly varying distribution for returns - but I assume it should be close to TDist, especially for N step simulation? Especially if event clustering added? (I don't have experience with jump models).
Heuristics - is there any known? Maybe encode it like E-GARCH?
$$h_t = \mu + \phi (h_{t-1}-\mu) +\;\alpha\left(\left|\frac{r_{t-1}}{\sigma_{t-1}}\right| -\mathbb{E}|\varepsilon|\right) +\;\gamma\,\frac{r_{t-1}}{\sigma_{t-1}} +\;\xi_t,\qquad \xi_t\sim \mathcal N(0,\sigma_\xi^2) $$
Goal:
Predicting N step distribution of returns (3months-1year) from historical data, VaR simulation for both down and up.
P.S.
It seems most SV models used for IV Surface matching (fitting to moments + interpolation), which is a bit different from Historical Simulations (current state filtering + extrapolation). So maybe SV should have different structure for these two cases.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.