Euler Discretization of a Stochastic Volatility Model
Summary
The document asks how a continuous-time price and variance process relates to a discrete-time stochastic volatility model. Applying Itô’s lemma to log price and log variance gives a mean-reverting log-variance process and a log-return process whose conditional drift includes a variance adjustment. An Euler step for the variance state yields an autoregressive form for log volatility, with parameters tied to the continuous-time mean reversion and diffusion terms.
The accepted answer instead demonstrates an Euler step applied directly to the price process, producing a simple one-period return equation with drift and a volatility-scaled normal shock. This illustrates a discretization route, but it does not resolve the question’s log-return observation equation or establish equivalence to the stated discrete model. The treatment also sets the time step to one, so its parameter mapping is specific to that choice; readers should distinguish this approximation from an exact transition distribution.
Key ideas
- Itô’s lemma transforms the price and variance equations into dynamics for log price and log variance.
- Euler discretization of the log-variance process produces an autoregressive state equation.
- The log-return drift includes a term that depends on current variance.
- Discretizing price directly gives a simple return equation with a volatility-scaled shock.
- The provided answer does not fully derive the discrete log-return model or an exact parameter mapping.
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# Discretizing a Continuous Time Stochastic Volatility Model
# Discretizing a Continuous Time Stochastic Volatility Model
How does the discrete time stochastic volatility model arise from the continuous time one?
Also, forgive me for cross-posting.
I have the following continuous time SDE for a stochastic volatility model. $S_t$ is the price, and $v_t$ is a variance process. $$ dS_t = \mu S_tdt + \sqrt{v_t}S_t dB_{1t} \\ dv_t = (\theta - \alpha \log v_t)v_tdt + \sigma v_t dB_{2t} . $$ I'm more familiar with the discrete time version: $$ y_t = \exp(h_t/2)\epsilon_t \\ h_{t+1} = \mu + \phi(h_t - \mu) + \sigma_t \eta_t \\ h_1 \sim N\left(\mu, \frac{\sigma^2}{1-\phi^2}\right). $$ $\{y_t\}$ are the log returns, and $\{h_t\}$ are the "log-volatilites." Keep in mind there might be some confusion about parameters; for example the $\mu$s in each of these models are different.
How do I verify that the first discretizes into the second?
Here's my work so far. First I define $Y_t = \log S_t$ and $h_t = \log v_t$. Then I use Ito's lemma to get \begin{align*} dY_t &= \left(\mu - \frac{\exp h_t}{2}\right)dt + \exp[h_t/2] 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*}
I got the state/log-vol process piece. I use the Euler method to discretize, setting $\Delta t = 1$, to get \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*}
The observation equation is a little bit more difficult, however:
\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*}
Why is the mean return not $0$ or $\mu$? How should I have defined the transformations? I suspect it might have something to do with the meaning of parameters and random variables. In the discrete time model above, $y_t$ is the mean-adjusted log return. In the SDE above that, $\mu$ probably means the interest rate plus half the variance (Questions on continuously compounded return vs long term expected return).
## Answer by Taylor (score 0, accepted)
https://quant.stackexchange.com/a/43461
I guess you can discretize the raw price process too instead of the log price process. You get $$ S_{t+1} = S_t + \mu S_t + \sqrt{v_t} S_t Z_t $$ (where $Z_t$ is a standard normal variate), or $$ \frac{S_{t+1}}{S_t} - 1 = \mu + \sqrt{v_t} Z_t. $$ Got the idea from: https://arxiv.org/pdf/1707.00899.pdfShown 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.