Black-Equivalent Volatility for Mean-Reverting Energy Prices

Summary

The document addresses how to choose a volatility input for real-option valuation when historical observations are unavailable and annual present values are described as Gaussian. Its answer points to a mean-reverting model for the logarithm of spot prices, a framework used in energy markets such as oil and electricity. In the model, the log price moves toward a long-run equilibrium at a specified reversion rate while also receiving random shocks scaled by volatility.

It then gives a conversion from the model’s volatility to a Black-equivalent volatility for a horizon T. The adjustment multiplies the model volatility by the square root of a factor involving the mean-reversion rate and horizon, reflecting that mean reversion changes uncertainty over time. The response refers readers to a book for solving or estimating the model parameters. It does not explain how to infer those parameters without data or establish that Gaussian annual present values alone determine the appropriate volatility, so the conversion’s relevance depends on the model assumptions and inputs.

Key ideas

• The proposed energy-price model makes the logarithm of spot price mean-reverting toward a long-term level.\nThe model includes a reversion rate and a volatility parameter governing random shocks.\nBlack-equivalent volatility adjusts model volatility using both the reversion rate and valuation horizon.\nThe answer does not provide a parameter-estimation method for settings without historical data.

Tags

# Appropriate measure of Volatility for economic returns from an asset?


# Appropriate measure of Volatility for economic returns from an asset?












In order to use Real Option Valuation (ROV), using Black-Scholes equation, I must know the volatility of the economic returns for T years. Knowing this information what could be the appropriate measure of computing volatility of the economic returns from my reservoir?

The distribution of PV for any particular year is coming out to be gaussian. There is no historical data known.

## Answer by dragunov (score 7, accepted)

https://quant.stackexchange.com/a/255

I think this has something to do with my question ("Black Equivalent Volatility"). I just realized that the answer might be your question:

> Knowing this information what could be the appropriate measure of computing volatility of the economic returns?

In Energy Markets, like oil and electricity, one model we use is the mean reversion in the natural log of the spot prices.

$$d(ln(S)) = a(b-ln(S))de + vdz$$ where:

$S$ = spot price

$t$ = time of observation

$a$ = rate of mean reversion

$v$ = volatility

$b$ = long-term equilibrium

$dz$ = random stochastic variable

Now there are books that would show you how to solve for the volatility in that equation but i think the best one for me is Dragana Pilipovic's book entitled "Energy Risk 2nd Ed" (chapter 5, page 108)

And I think the black-equivalent volatility is a short-form equation that you can use off the bat. So here is my answer to your question:

black-equivalent volatility = volatility x $\sqrt{(1-e^{-2aT})/2aT }$

where:

$T$ = period of time (20 years)

$a$ = rate of mean reversion

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.