# How to calculate return rates with negative prices?

I'm dealing with electricity options and I'm considering the possibilty of negative prices. I want two estimate the historic volatility. However, an arithmetic mean doesn't feel appropriate and $\log(\frac{P_i}{P_{i-1}})$ doesn't work if $P_{i-1}$ is less or equal than 0.

For example:

20th July: $P_1$= 24 euros/MWh 21st July: $P_2$= -70 euros/MWh

what do you suggest to properly calculate the return rate? what is the correct interpretation?

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There is no negative prices even in electricity options. But aside that I second what Hebe suggested. Use simple returns rather than log returns. – Matt Wolf Jul 21 '13 at 11:23
Is P the price or the return? – Christian Fries Jul 21 '13 at 20:10
Yes, P is the price. Electricity, in some markets, can achieve negative prices. – Joao Serafim Jul 21 '13 at 21:13
You mean you are paid to consume electricity? Then we must be dealing with markets where the laws of supply and demand do not apply and hence conventional methods and models should not be applied either. But out of curiosity can you please share with us where that is the case and why? – Matt Wolf Jul 21 '13 at 22:50
here is an example at EEX platform: eex.com/en/Market%20Data/Trading%20Data/Power/… – Joao Serafim Jul 22 '13 at 2:58

In this case,you shouldn't use log return.You should calculate return as (P(i)/P(i-1))-1.

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Defining || as the absolute value, wouldn't it be more suitable to make an adaptation and use (Pi-Pi1)/|Pi-1|?? because if P1=-20 and P2=-10 you are not losing more money – Joao Serafim Jul 21 '13 at 14:48
@Hebe: Which does not work if P(i-1) is zero. Actually (P(i)/P(i-1))-1 is just a finite difference approximation of the log-return. – Christian Fries Jul 21 '13 at 17:29
@Joao Serafim: This does not solve the explosion for $P_{i-1}$ becoming close to zero. You won't consider wiggeling around zero as such a huge vol, or do you? – Christian Fries Jul 21 '13 at 17:30
@Christian Fries: Definitely not. But how can I solve that issue? – Joao Serafim Jul 21 '13 at 18:55

The answer depends on what model you assume for the underlying. The situation, that the underlying can become negative also occures for interest rate spreads and even for interest rates. Here some people use absolute changes, that is $X_{i} - X_{i-1}$ instead of relative changes $\frac{X_{i} - X_{i-1}}{X_{i-1}}$ or (which is almost the same as relative returns) log-returns $\log(\frac{X_{i}}{X_{i-1}})$

If you model the return as normal distributed, you are assuming that your underlying is log-normal. Hence you estimate vol from log-returns.

If you model the absolute changes as normal distributed, you would estimate the historic vol from absolute changes.

Since you are speaking of options and vol, you should know what your model assumption is.

PS: More details on this can also be found in http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2194917

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I'm assuming log normality of the underlying, and threfore, as you said, I should estimate from the log-returns but can´t used them because it may occur Pi/Pi-1<0. And I don't think the absolute changes are suitable for this problem. Therefore I may be out of alternatives – Joao Serafim Jul 21 '13 at 18:52
If $P_{i}/P_{i-1} < 0$, then the assumption of log-normality is clearly wrong. You should consider an alternative model. Why do you think that absolute changes are not suitable? Why do you believe in log-normality so strongly? You may also consider a displaced log-normal model, like in the cited paper. If data with $P_{i}/P_{i-1} < 0$ is just a rare sample error, you may consider leaving that data out. Put it appears as if this is not the right way here... – Christian Fries Jul 21 '13 at 20:09
Basically I'm estimating the volatility in order to determine the option value according to Black-Scholes and compare it to the empirical value and then analyze the differences, that's why I'm tried to use log normal returns – Joao Serafim Jul 21 '13 at 21:11
Black-Scholes seems to be not adequate. A displaced model may be more adequate. Of course, you may calculate an implied BS vol. What is your heeding strategy? What does your hedge do if prices are negative? The answers to these question should give the you the option prices. Not the blind believe in an inappropriate model... – Christian Fries Jul 21 '13 at 21:21
Exactly, I second what Christian said. It seems you try to force data into distributional nomenclature just to be able to apply an option pricing model of your own choosing rather than the other way around. But you still have not explained why a producer would pay money to deliver electricity rather than just "dump it" and why consumers would receive money for power consumption. I am afraid there is a misunderstanding on your end going on. – Matt Wolf Jul 21 '13 at 22:55

Abandon the idea to use lognormal (GBM) model for power spot market. Just don't do it. Spot power prices have totally different properties and Black-Scholes option prices will not make any sense. They won't be even remotely close to the correct option value!

Just a list of things that lognormal model does not capture and that are important for option pricing:

• Negative prices
• Price spikes (stochastic volatility)
• Mean reversion
• Strong periodicity (daily, weekly, yearly)