Estimating Volatility in a Mean-Reverting SDE: Discretization Choices
Summary
The document compares two discrete approximations for estimating the volatility parameter in a mean-reverting stochastic differential equation with a deterministic time-varying mean. One approach begins from the continuous-time solution and approximates the stochastic integral; the other discretizes the SDE directly. Both yield regressions of deviations from the mean, but their residual standard deviations are scaled differently, leading to inconsistent volatility estimates in the questioner’s calculations.
The text raises whether the discrepancy comes from the choice of stochastic-integral approximation, including left- versus right-endpoint rules. It presents no answer or empirical validation, so it does not establish which estimator is preferred or whether the proposed endpoint change is valid. The core lesson is that discretization choices affect the noise term’s scale, and that estimating volatility requires matching the regression residual variance to the specific discrete model.
Key ideas
- The SDE’s continuous-time solution and direct discretization produce different discrete regression forms.
- Both approaches regress deviations from the deterministic mean on their lagged values.
- The stochastic term’s scaling depends on how the integral is approximated.
- Residual standard deviation must be adjusted for the coefficient multiplying the noise term.
- The document leaves the preferred approximation unresolved.
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# Approximating an SDE for Volatility Estimation
# Approximating an SDE for Volatility Estimation
Consider the SDE $$ dT(t) = ds(t) + a(s(t) - T(t))dt + \sigma dW(t) $$ where $s(t)$ is a deterministic function that turns out to be the long-term mean (this SDE is used to model daily temperature, so $s$ consists of sines and cosines). The solution is $$ T(t) = s(t) + (T(0) - s(0))e^{-at} + \int_0^t \sigma e^{-a(t-u)} \, dW(u) $$ I'm trying to estimate the volatility parameter $\sigma$, and I've tried two methods, but they give quite inconsistent results and I'd like to understand the differences. The first method is to note we can use the solution to write the difference $\Delta T(t) := T(t+1) - T(t)$ as $$ \Delta T(t) = \Delta s(t) - (1 - e^{-a})(T(t) - s(t)) + \sigma e^{-a}\int_t^{t+1}e^{-a(t-u)}\, dW(u) $$ If we then approximate the integral using the left-hand rule, we get $$ \Delta T(t) \approx \Delta s(t) - (1 - e^{-a})(T(t) - s(t)) + \sigma e^{-a}\Delta W(t) $$ and rearranging a bit (and assuming equality) gives
> $$ T(t+1) - s(t+1) = e^{-a}(T(t) - s(t)) + e^{-a}\sigma \epsilon(t) $$
where $\epsilon(t) \sim \mathcal{N}(0,1)$, i.i.d. So we have a regression on $T - s$ with regression coefficient $e^{-a}$, and can estimate $\sigma$ by computing the sample standard deviation of the residuals and dividing by $e^{-a}$, where the residuals are $[T(t+1) - s(t+1)] - [e^{-a}(T(t) - s(t))]$.
The second method is to discretize the SDE from the start: $$ T(t+1) - T(t) = s(t+1) - s(t) + a(s(t) - T(t)) + \sigma \epsilon(t) $$ Rearranging also gives a regression equation:
> $$ T(t+1) - s(t+1) = (1-a)(T(t) - s(t)) + \sigma \epsilon(t) $$
So we again have a regression on $T - s$ with regression coefficient $1-a$, and can again estimate $\sigma$ by computing the sample standard deviation of the residuals $[T(t+1) - s(t+1)] - [(1-a)(T(t) - s(t))]$, with no need to divide by anything at the end.
The problem is that we get the same sample standard deviation estimate for both methods, and thus the two estimates of $\sigma$ will be off by a factor of $e^{-a}$. I recognize that, had I used the right-hand rule to approximate the stochastic integral, the factor of $e^{-a}$ multiplying $\sigma$ in the first method would disappear, and we'd get the same estimates for $\sigma$. So, I'd like to know which method of the two is preferred, and whether switching to a right-hand rule is kosher.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.