Rescaling Correlated Ornstein–Uhlenbeck Models for Hourly Exercise
Summary
This note considers an option on an electricity and gas spread that can be exercised hourly, even though liquid gas prices are observed only daily. The question proposes modeling seasonality-adjusted log prices as correlated mean-reverting Ornstein–Uhlenbeck processes, then asks whether daily estimated parameters can be converted to hourly drift, mean reversion, volatility, and correlation for a quadrinomial lattice.
The answer presents the continuous-time process equations and suggests Euler discretization: the mean-reversion adjustment scales with the time step, while the random shock scales with the square root of that step. It says annualized parameters need not be altered simply because the simulation time increment changes. This is a modeling explanation rather than a calibrated method for inferring hourly gas behavior from daily observations. It does not establish that daily data identify hourly dynamics, address estimation uncertainty or seasonal structure, or resolve the choice between the proposed quadrinomial and binomial lattices.
Key ideas
- The setup models electricity and gas prices as correlated mean-reverting processes.
- Euler discretization scales the drift adjustment with the time step and the noise with its square root.
- Annualized parameters can be used at different time increments through discretization.
- Daily observations alone do not establish that the model’s hourly gas dynamics are empirically identified.
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Full text
# How to use daily and hourly prices in same option model?
# How to use daily and hourly prices in same option model?
An option can be exercised hourly but depends on two prices - one is available daily and hourly, the other one only daily. How can I write an option model that uses a quadrinomial lattice with both these prices and can be exercised hourly?
Please take the following as given, as I already have a model that serves very well its purpose for daily granularity: I assumed that log-prices (after removing seasonalities) are mean reverting (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) and I estimated 2 correlated Ornstein-Uhlenbeck processes from historic spot prices - electricity and gas. Then I wrote an option model that calculates the value of the spread between the two (i.e. a gas power plant) with the possibility to exercise daily (binomial method is convenient, as there are many physical restrictions of the power plant). I want to do the same for HOURLY exercise, but for gas there are only DAILY liquid spot prices.
Is there a way that I can for example infer a hypothetical hourly drift, mean-reversion speed, volatility and correlation with electricity price for gas price from the daily parameters?
## Answer by will (score 1)
https://quant.stackexchange.com/a/34568
From the sounds of things, you have two processes, for electricity and gas prices, and have decided they are Ornstein-Uhlenbeck processes. Let's call then $E$ and $G$.
$$ \begin{eqnarray} \mathrm{d}E_t &= \theta_{{}_E} (\mu_{{}_E} -E_t)\mathrm{d}t + \sigma_{{}_E} \mathrm{d}W_t^{{}_E} \\ \mathrm{d}G_t &= \theta_{{}_G} (\mu_{{}_G} -G_t)\mathrm{d}t + \sigma_{{}_G} \mathrm{d}W_t^{{}_G} \\ \end{eqnarray} $$
and also we have $\langle\mathrm{d}W_t^{{}_E}\mathrm{d}W_t^{{}_G}\rangle = \rho\sqrt{\mathrm{d}t}$.
Where you've found all these parameters through some sort of optimization/regression (presumably).
In the above, $\mu_{{}_X}$ is the anualized drift for process $X$, and $\sigma_{{}_X}$ its anualized vol - you do not have term structures for them, so you don't need to worry about treating them differently when you change $\mathrm{d}t$.
Now, i don't know exactly how you're using these - you say you want a quadrinomial lattice in the question, and then mention that the binomial model is convenient due to physical restrictions.
If you want to fiddle with the numbers in excel or something, just use the Euler discretization:
$$ \begin{eqnarray} S_{t+\mathrm{d}t} &=& S_t + \int_t^{t+\mathrm{d}t} \theta (\mu - S_t) \mathrm{d}t + \int_t^{t+\mathrm{d}t} \sigma\mathrm{d}W_t \\ &=& S_t + \theta (\mu - S_t) \mathrm{d}t + \sigma \sqrt{\mathrm{d}t}\tilde{X} \end{eqnarray} $$
You should notice that you don't need to change the params to account for a different time discretization.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.