How to connect stochastic volatility in SDEs with its empirical version?

Question

The standard definition of stochastic volatility model is:

$dS_t = \mu S_tdt + \sqrt{v_t}S_tdW_t$

$dv_t = \alpha_{v,t}dt + \beta_{v,t}dB_t$

$\langle d W_t, d B_t \rangle = \rho$

(From wikipedia: https://en.wikipedia.org/wiki/Stochastic_volatility)

However, in Kim's paper (http://finance.martinsewell.com/stylized-facts/volatility/KimShephardChib1998.pdf), they define an empirical version:

$y_t = \beta \exp(h_t/2)\epsilon_t$

$h_{t+1} = \mu + \phi(h_t - \mu) + \sigma_t \eta_t$

$h_t \sim N(\mu, \frac{\sigma^2}{1-\phi^2})$

I have trouble in understanding how to get the empirical version from the SDE. Does it mean that other stochastic volatility models, i.e., SABR, have an empirical version?

Answer 1

"Empirical" just means they discretized it.

The original Hull-White (1987) paper has $\alpha_{v,t} = \mu v_t$, but I'm not sure that is correct, because the log-variance doesn't have mean-reverting behavior.

If you let $Y_t = \log S_t$ (log price) and $h_t = \log( V_t)$ (log-volatility), $\alpha_{v,t} = (\theta - \alpha \log v_t)v_t$ ($\alpha > 0$), $\beta_{v,t} = \sigma v_t$ and apply Ito's lemma, you get

\begin{align*} dY_t &= \left(\mu - \frac{v_t}{2}\right)dt + \sqrt{v_t} dB_{1t}\\ dh_t &= \left(\theta - \alpha\log v_t - \sigma^2/2\right)dt + \sigma dB_{2,t}\\ &= \alpha\left(\tilde{\mu} - h_t \right)dt + \sigma dB_{2t}. \end{align*} According to this, this was a model investigated by a guy named Wiggins. The log-variance follows an Ornstein-Uhlenbeck process, which is like a continuous time version of an AR(1). If you use the Euler-Maruyama method to discretize this, setting $\Delta t = 1$, you should see this \begin{align*} h_{t+1} &= \alpha \tilde{\mu} + h_t(1-\alpha) + \sigma \eta_t \\ &= \tilde{\mu}(1 - \phi) + \phi h_t + \sigma \eta_t \\ &= \tilde{\mu} + \phi(h_t - \tilde{\mu}) + \sigma \eta_t \end{align*} letting $\phi = 1-\alpha$. Note that $\tilde{\mu} \neq \mu$.

As far as the observation process goes:

\begin{align*} y_{t+1} = Y_{t+1} - Y_t &= (\mu - \frac{v_t}{2}) + \sqrt{v_t}\epsilon_{t+1} \\ &= \left(\mu - \frac{\exp h_t}{2} \right) + \exp[ \log \sqrt{v_t}] \epsilon_{t+1} \\ &= \left(\mu - \frac{\exp h_t}{2}\right) + \exp\left[ \frac{h_t}{2}\right] \epsilon_{t+1} \end{align*}

• I can't quite get the drift term to cancel out. The mean of these returns should be $0$. Maybe because that's because of some arbitrage argument I'm not privy to, or maybe it has to do with how the errors are correlated in the SDE.

NB: In the discrete time observation equation you wrote, you should set $\beta = 1$, otherwise the model isn't identifiable (because we have a non-zero mean for the log-variance process). This is mentioned in the paper you linked to, but it is also discussed on a thread on this site here.

Answer 2

If you read carefully both Kim's paper and Hull (1987) to which they refer, you'd notice that in Hull (1987) the volatility process is defined differently than in what you called "standard" model: they use log vol process, not levels like in yours. Here's equation (2) from their paper: $$dV=\mu V dt +\xi V dz$$ where $V=\sigma^2$ is the variance.

Next, notice how Kim et al say that y in their Eq.1 is the mean corrected return on asset, and $h_t$ is the log vol. Using these two observations it's rather straightforward to see how the discrete (empirical) version connects to the continuous time version.