Extracting pricing densities for weather derivatives using the maximum entropy method

Abstract

In this paper we propose the use of the maximum entropy method to extract pricing densities directly from the weather market prices. The proposed methodology can overcome the data sparsity problem that governs the weather derivatives market and it is model free, non-parametric, robust and computationally fast. We propose a novel method to infer consistent pricing probabilities, and illustrate the method with a motivating example involving market prices of temperature options. The probabilities inferred from a smaller subset of the data are found to consistently reproduce out-of-sample prices, and can be used to value all other possible derivatives in the market sharing the same underlying asset. We examine two sources of the out-of-sample valuation error. First, we use different sets of possible physical state probabilities that correspond to different temperature models. Then, we apply our methodology under three scenarios where the available information in the market is based on historical data, meteorological forecasts or both. Our results indicate that different levels of expertise can affect the accuracy of the valuation. When there is a mix of information available, non-coherent sets of prices are observed in the market.

Keywords:

• Extracting probability densities
• weather derivatives pricing
• temperature options
• maximum entropy
• out-of-sample valuation

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their constructive comments that helped to substantially improve the final version of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Computations can be performed in around 0.15 s with a 2.8Ghz dual core PC.

2 The time τ is the exercise time of the option. At that time, the holder has the option to buy a futures contract with a measurement period $[{\tau }_1,{\tau }_2].$ In reality $\tau \le {\tau }_2,$ however, the most interesting case is when $\tau <{\tau }_1.$ Otherwise, the transaction takes place after the start of the futures contract and the measurement period. Please see Chapter 6 of Alexandridis and Zapranis (2013) for a rigorous treatment of these cases.

3 For example one might use any of the temperature models for weather derivatives pricing proposed in Alexandridis and Zapranis (2013), Benth and Saltyte-Benth (2013) or Alaton et al. (2002).

4 Note, that we assume a constant interest rate for simplicity however this is not restrictive for the proposed method.

5 The high cost involved in obtaining historical weather derivatives prices prohibit us from using more recent and diverse data.

6 For an analytical description of the HBA we refer to Alexandridis and Zapranis (2013).

7 Full set of results available online as supplementary material.