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).

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    $\begingroup$ Hi Taylor. First notice that the limiting behaviour of discrete time processes and their continuous time counterpart is quite tricky (see the famous 1990 paper by Neslon). Second, the first SDE prevails under a certain probability measure which you have yet to define. This measure will give you the meaning of the parmeter $\mu$. Under $\Bbb{P}$ (real world measure) it's the expected return of the stock, under $\Bbb{Q}$ it's its funding rate (e.g. risk-free rate minus dividends minus repo rate). $\endgroup$
    – Quantuple
    Commented Jul 11, 2018 at 7:45
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    $\begingroup$ Hi @Quantuple. Thanks for the note. Care to elaborate further in an answer? Also, which Nelson paper? There were two in 1990. $\endgroup$
    – Taylor
    Commented Jul 11, 2018 at 16:55
  • $\begingroup$ could you possibly tell me if the the SDE you wrote is representing some known continuous SV model (like Heston)? I am writing Masters thesis on discrete SV models and i want to include a reference to continuous model which leads to discrete version. Thanks in advance! $\endgroup$ Commented Feb 19, 2023 at 22:31
  • $\begingroup$ @AlexSlavik I don't believe it is one. I am more familiar with discrete-time, so what I was doing was starting with discrete-time model in terms of log-variance, then writing that in a continuous-time form, then using Ito's to come up with a traditional continuous-time model in terms of $v_t$. The problem, though, is as you observed: the cts-time model in terms of (not log) variance that I wrote in the second equation doesn't appear to be anything traditional, and I have not investigated its properties. In short, I would leave it out of your thesis :) $\endgroup$
    – Taylor
    Commented Feb 23, 2023 at 20:05

1 Answer 1


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.pdf

  • $\begingroup$ However, your approach the probability of the stock price being negative is non-zero (though very small). One does not have such a problem when discretising the solution of the SDE. $\endgroup$
    – FunnyBuzer
    Commented Jan 13, 2019 at 12:11

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