Value of optionality in ‘Direkt vermarktung’

Alternative valuation method in the sustainable energy industry

Value of optionality in ‘Direkt vermarktung’

Zanders has been involved in financing issues and transactions related to sustainable energy for several years now, supporting developers, power companies, and start-ups in a variety of countries.

As part of one of its consultancy projects, Zanders advised a Dutch developer on the valuation of its German wind farms. For this particular client, Zanders calculated the value of the implicit option embedded in the complex financing of wind energy projects.

Under the Erneuerbaren Energien Gesetz (EEG), a German act to promote sustainable energy, the owners of German wind parks are entitled to a minimum selling price per megawatt hour produced (MWh), which is significantly higher than the price for traditional electricity. The right to this price is valid for an average of 15 years.

At the beginning ofeach month, the owner is free to decide whether he will sell his energy during that month at the price per MWh established when the wind farm became operational, or at the current price on the power exchange. Since this pre-established contract price is currently significantly above the market price, investors generally include only the fixed price in their cash-flow projections.

However, the choice to be made each month to sell at the fixed or the variable price represents a value with the characteristics of a financial option, and can be valued as such.

“By including these additional revenues in the cash-flow projections, a higher valuation and, by extension, a higher transaction value in the event of the acquisition or sale of the wind farm is justified.”

Taking into account the volatility observed in the prices of fossil fuels and energy in recent years, Zanders decided to examine the value of this implicit optionality. In the absence of trade in forward contracts with similar maturities, the company set out to
find an alternative valuation method. This article describes the analytical process on which Zanders based its valuation.

Energy as a commodity

As European energy markets were deregulated only a few years ago, the trade in energy derivatives, valuation methods, and the debate related to these issues have all yet to reach maturity. Although commodity futures in products such as grain have been traded for decades, the similarities with power are limited. For one, electric power differs from commodities in that it cannot be stored.

This means that:

  • the secondary market for energy is extremely volatile because a surplus must be sold elsewhere, and
  • the basic need fulfilled by energy creates a strong political and geographical bias in local energy markets.

In terms of valuation, electricity prices have a number of distinct characteristics. Figure 1 shows the price of a single MWh ofenergy in the German spot market (the EEX Intraday Spot for major companies) since the market was deregulated in January 2004.

Figure 1: Closing price per MWh and volatility since the deregulation of the German energy market

The graph clearly shows the irregularity of the price movement. On several occasions, the price per MWh doubled or halved in a single day. The consistent return to average price levels (i.e. mean reversion) and the higher prices during the winter (i.e. seasonality) are typical. The graph also includes an estimate of volatility (shown in light blue on the right axis), which reveals a high degree of volatility clustering: the level of price fluctuations, both high and low, appears to remain stable for an extended
period of time.

Modeling techniques

Since energy is partly generated by fossil fuels, its market price is strongly correlated to the price of gas, and since there is a thriving forward market for gas with maturities ofup to three years, a valuation model for energy prices might be based on the correlation between both products, while the model could be calibrated on the forward curve for gas.

“In the absence of forward contract trading in correlated derivatives with comparable maturities, we decided to model the energy price as an independent stochastic process.”

However, unfortunately this method is inadequate. Aside from the imperfect correlation between energy and gas (for example, because the latter can be stored), for this particular problem revenues were considered over a 20-year horizon. A model based on the price movements of gas derivatives would be seriously inadequate for the past 17 years of our forecast, which happens to be the period during which the optionality in the supply agreement could become interesting.

In the absence of forward contract trading in correlated derivatives with comparable maturities, we decided to model the energy price as an independent stochastic process. Our model is based on the Mean- Reverting Jump Diffusion Model developed by Cartea and Figueroa*.

Jump diffusion style modeling

The Jump Diffusion Model divides the price level St into a determinist function Gt, which models the seasonal mean, and a variable stochastic process Yt, which triggers the uncertainty and volatility in the price level. In our model, Yt is described by means of the following differential equation.


In this equation, α determines the mean reversion rate; the Poisson process q, along with the log-normal stochastic process J, triggers the irregular shocks; and Z represents a Brownian motion driven by volatility σt. The equation shows that Yt has a long-term average level of 0. By ensuring that this correlates exponentially with seasonal level Gt, a log-normal process is created with averages at the seasonal level.

Implementing the model

For Gt, we are looking for a continuous function that provides a seasonal price for any random moment. An obvious candidate is a Fourier method (in our case, of the fifth order), which we will estimate based on the average price per month for the past five years. The result is shown in Figure 2.

Figure 2: Fourier method for seasonal price level ofenergy

In order to estimate the parameters for irregular shocks, we will iteratively explain the largest relative price movements in price history until this no longer contains any days with a positive or negative return of more than 3 standard deviations of 0. The number of excesses in relation to the number of measurements represents the frequency parameter for the Poisson process q, and we will use the mean and the standard deviation of the excesses as division parameters for the log-normal stochastic process J.

Since the stochastic process is modeled in relation to the average price level, it is easy to embed an inflation effect into the model – all we need to do is gradually increase the seasonal average Gt.


The option value in our problem is created when the operator of the wind farm has the opportunity to sell the energy at the market price if this yields more than the contract price. The option value can be calculated by means of a simulation with hundreds of option value price paths. Since each path in the simulation has the same probability, the average of the additional yield per path represents the value of the optionality in the problem as a whole.

Figure 3 shows an example of such a simulated path.

Figure 3: Example of a simulated path over a 20-year horizon

It is important to understand that simulating a process such as the energy price is not an exact science. Based on the assumptions of the model, the structure of the price movements over the next 20 years is presumed to remain equal to that of the past five years. This is reflected in factors such as the frequency and intensity of the shocks and the level of volatility clustering.

However, in the absence of forward trading with long maturities in correlated products, the chosen approach is currently the best alternative, as it at least enables us to include the value of the long-term optionality in the valuation of the contract price alone. Since the contract price is generally significantly above the current market price – for the wind farm in question, it was approximately EUR 90 per MWh, versus a market price ofeUR 40 per MWh currently – the option value is extremely limited during the first several years. However, over time and when inflation becomes a factor, the value will increase.

By including these additional revenues in the cash-flow projections, a higher valuation and, by extension, a higher transaction value in the event of the acquisition or sale of the wind farm is justified. In addition, the option value can be separated from the operation based on the fixed prices, and additional funding can be secured through the sale of the option value. This could be extremely interesting to developers and utility companies if energy prices continue to rise.

* ‘Pricing in electricity markets: A mean reverting jump diffusion model with seasonality’, A. Cartea & M.G. Figueroa, Birkbeck College, University of London. Applied Mathematical Finance, Vol. 12, No. 4, December 2005