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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.

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  • $\begingroup$ If you are using daily data, then $t+1$ may should be replaced by $t+1/365$. $\endgroup$ – Gordon Nov 2 '16 at 20:38
  • $\begingroup$ You could use that the Ito integral of a deterministic integrand is normally distributed, i.e. \begin{equation} \int_t^{t + 1} \sigma e^{-a (t - u)} \mathrm{d}W_u \sim \mathcal{N} \left( 0, \int_t^{t + 1} \sigma^2 e^{-2 a (t - u)} \mathrm{d}u \right). \end{equation} $\endgroup$ – LocalVolatility Nov 2 '16 at 20:55

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