# 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?

• Why do you want to use a "quadrinomial" model? L why do you think it will be better than a tribunal model? Do you have market option prices? There are so m a NY questions that need answering before your question can be answered... – will Jun 4 '17 at 20:49
• Good point! It would be great if you could take the following as given (as I already have a model that serves very well its purpose for daily granularity): I estimated 2 correlated mean reverting (O-U) 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. – LenaH Jun 5 '17 at 9:53
• I'll try to put something down today sure - one question I have though, for the price that is only available daily - at what value is it exercised? Or is it that there is a price for it at all times, but your data only contains daily prices and you want to do the best given what you have? – will Jun 6 '17 at 7:49
• Hm, I hope I understand your question: So, for gas I use the daily EEX Gas Spotmarkt NCG Settlement Price, because within day trading is not very liquid. The price is determined as the average of the trades closed from 17:15 to 17:30 on the trading day preceding the delivery day. There might be hourly prices available, but I think it's not a good idea to use them. For electricity I use the hourly Phelix Day-ahead Auction Price of the 12pm auction on the trading day preceding the delivery day, and for the daily model, I use an average over the day. – LenaH Jun 6 '17 at 8:42
• The daily model is already OK – as both prices can be used daily – but not the hourly one, as I can’t simply assume 24 times the same hourly price each day for gas… Ah and of course LOG-prices are modelled Ornstein-Uhlenbeck (en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) (after removing seasonalities). Let me know if anything is unclear! – LenaH Jun 6 '17 at 8:42

## 1 Answer

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.

• That makes sense, thank you! That is also why dt is expressed in terms of years!... There is one open question for me: for the parameters of electricity (i.e. the price that is available both daily and hourly), would you use the ones that I estimated from the hourly prices, or the one that I estimated - together with gas and their correlation - from the daily prices? For the hourly ones I used maximum likelihood and for the daily ones multivariate maximum likelihood to also get the correlation. – LenaH Jun 7 '17 at 6:39
• For the correlation they need to be in synch, so you must use returns that line up in time. Since you're estimating a constant correlation though, you can still use it for your hourly simulated processes. – will Jun 7 '17 at 6:42
• OK, so I would do rather this: A) DAILY MODEL: do multivariate ML with daily prices to get parameters of electricity and gas, incl. correlation. Use parameters to create lattice with daily time steps. B) HOURLY MODEL: Take all parameters from A (for electricity, gas and correlation). Use these parameters to create lattice with hourly time steps. I.e. I am not using hourly electricity prices at all... – LenaH Jun 7 '17 at 7:35
• You can still use your hourly prices to obtain the parameters for your model of course. You just have one model that's calibrated to more data than the other... – will Jun 7 '17 at 7:38
• Sorry will, do I indeed have to express dt in terms of years, both in the daily and hourly model in order to use the same parameters? – LenaH Jul 5 '17 at 10:21