Discretizing a Mean-Reverting Commodity Price Model
Summary
The document examines how to convert a continuous-time mean-reverting model for a commodity price into a discrete-time equation for log price. It applies Itô's lemma to the log price and an integrating factor to solve the resulting linear stochastic differential equation. The paper's discrete form includes a Gaussian innovation whose variance is given, along with a correction term involving half that variance.
The questioner calculates the innovation variance but obtains a different expression for the drift correction, and asks how to reconcile the discrepancy. This highlights that the correction arises from the log transformation and must be integrated with the same time weighting as the stochastic term. The document does not provide a resolution, so it serves as a problem statement rather than a complete derivation. Its equations assume the model's parameters and other inputs can be treated as constant, or approximated as constant, over each time step; results may differ when they vary through time.
Key ideas
- Applying Itô's lemma to a price process introduces a correction when expressing the model in log prices.
- An integrating factor can solve the resulting linear mean-reverting stochastic differential equation.
- The discrete-time innovation is Gaussian, with variance determined by integrating volatility over the step.
- The question concerns consistency between the variance integral and the log-price correction term.
- The derivation approximates some time-varying inputs as constant within each time step.
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# Confused about discretization
# Confused about discretization
I am reading a paper here: https://pdfs.semanticscholar.org/5f91/2d46b02b03230a4ffaaa42d655b2b6147d56.pdf The following is my confusion. The paper has the following continuous time model for the price of a commodity: $$\frac{dS(t)}{S(t)}=a(lnL(t)+\mu(t)-lnS(t))dt+\sigma(t)dW(t)$$ Then the paper converts this continuous time model into a discrete time model: $$lnS(t+\Delta t)=e^{-a\Delta t}lnS(t)+(1-a^{-a\Delta t})(lnL(t)+\mu(t))-\frac{\eta_t^2}{2}+\tilde{W}_{\Delta t}$$ where $$\eta_t^2=\sigma^2(t)(\frac{1-e^{-2a(\Delta t)}}{2a})$$ $$\tilde{W}_{\Delta t}\sim N(0, \eta_t^2)$$ My confusion is how to get the second to the last term $-\frac{\eta_t^2}{2}$. Here is what I have tried to discretize the continuous model, and I did not get the 2 in front of $a$: I let $Y(t)=lnS(t)$. Then $$dY(t)=(a(lnL(t)+\mu(t)-Y(t))-\frac{1}{2}\sigma^2(t))dt+\sigma(t)dW(t)$$ And $$d(Y(t)e^{at})=e^{at}((a(lnL(t)+\mu(t))-\frac{1}{2}\sigma^2(t))dt+\sigma(t)dW(t))$$ So $$Y(t+\Delta t)e^{a(t+\Delta t)}=Y(t)e^{at}+\int_{t}^{t+\Delta t}e^{as}(a(lnL(s)+\mu(s))-\frac{1}{2}\sigma^2(s))ds+\int_t^{t+\Delta t}e^{as}\sigma(s)dW(s)$$ I approximated $lnL(s)+\mu(s)\approx lnL(t)+\mu(t)$ on $[t,t+\Delta t]$, and obtained $$lnS(t+\Delta t)=Y(t+\Delta t)\approx e^{-a\Delta t}Y(t)+(1-e^{-a\Delta t})(lnL(t)+\mu(t))$$ $$-e^{-a(t+\Delta t)}\frac{1}{2}\int_t^{t+\Delta t}e^{as}\sigma^2(s)ds+e^{-a(t+\Delta t)}\int_t^{t+\Delta t}e^{as}\sigma(s)dW(s)$$ The last term has variance $$e^{-2a(t+\Delta t)}\int_t^{t+\Delta t}e^{2as}\sigma^2(s)ds\approx \frac{\sigma^2(t)}{2a}(1-e^{-2a(\Delta t)})=\eta_t^2$$ This agrees with the result - the variance of the last term is indeed $\eta_t^2$. However, for the second to the last term, it seems the approximate for that is instead $$-\frac{\sigma^2(t)}{2a}(1-e^{a(\Delta t)})$$ The result in the paper says this should be $$-\frac{1}{2}\eta_t^2=-\frac{\sigma^2(t)}{4a}(1-e^{-2a(\Delta t)})$$ Can someone help me reconcile the discrepency?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.