A Closed-Form Approximation for Pricing Temperature-Based Weather Derivatives

Affiliation(s)

School of Economics and Finance, Queensland University of Technology, Brisbane, Australia.

School of Economics and Finance, Queensland University of Technology, Brisbane, Australia.

ABSTRACT

This paper develops analytical distributions of temperature indices on which temperature derivatives are written. If the deviations of daily temperatures from their expected values are modelled as an Ornstein-Uhlenbeck process with timevarying variance, then the distributions of the temperature index on which the derivative is written is the sum of truncated, correlated Gaussian deviates. The key result of this paper is to provide an analytical approximation to the distribution of this sum, thus allowing the accurate computation of payoffs without the need for any simulation. A data set comprising average daily temperature spanning over a hundred years for four Australian cities is used to demonstrate the efficacy of this approach for estimating the payoffs to temperature derivatives. It is demonstrated that expected payoffs computed directly from historical records are a particularly poor approach to the problem when there are trends in underlying average daily temperature. It is shown that the proposed analytical approach is superior to historical pricing.

KEYWORDS

Weather Derivatives; Temperature Models; Cooling-Degree Days; Distributions for Correlated Variables

Weather Derivatives; Temperature Models; Cooling-Degree Days; Distributions for Correlated Variables

Cite this paper

A. Clements, A. Hurn and K. Lindsay, "A Closed-Form Approximation for Pricing Temperature-Based Weather Derivatives,"*Applied Mathematics*, Vol. 4 No. 9, 2013, pp. 1347-1360. doi: 10.4236/am.2013.49182.

A. Clements, A. Hurn and K. Lindsay, "A Closed-Form Approximation for Pricing Temperature-Based Weather Derivatives,"

References

[1] J. Tindall, “Weather Derivatives: Pricing and Risk Management Applications,” Institute of Actuaries of Australia, Unpublished Manuscript, 2006.

[2] M. Garman, C. Blanco and R. Erickson, “Weather Derivatives: Instruments and Pricing Issues,” Environmental Finance, 2000.

[3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, 1973, pp. 637-659. doi:10.1086/260062

[4] L. Zeng, “Pricing Weather Derivatives,” Journal of Risk Finance, Vol. 81, No. 3, 2000, pp. 72-78. doi:10.1108/eb043449

[5] E. Platen and J. West, “Fair Pricing of Weather Derivatives,” Quantitative Finance Research Centre, University of Technology Sydney, Research Paper Series, 106, 2003.

[6] S. D. Campbell and F. X. Diebold, “Weather Forecasting for Weather Derivatives,” Journal of the American Statistical Society, Vol. 100, 2005, pp. 6-16.

[7] D. S. Wilks, “Statistical Methods in the Atmospheric Sciences,” Academic Press, New York, 1995.

[8] S. Jewson and R. Caballero, “The Use of Weather Forecasts in the Pricing of Weather Derivatives,” Meterological Applications, Vol. 10, No. 4, 2003, pp. 377-389. doi:10.1017/S1350482703001099

[9] F. E. Benth and J. Saltyne-Benth, “Stochastic Modelling of Temperature Variations with a View Toward Weather Derivatives,” Applied Mathematical Finance, Vol. 12, No. 1, 2005, pp. 53-85.

doi:10.1080/1350486042000271638

[10] M. H. A. Davis, “Pricing Weather Derivatives by Marginal Value,” Quantitative Finance, Vol. 1, 2001, pp. 1-4. doi:10.1080/713665730

[11] P. Alaton, B. Djehiche and D. Stillberger, “On Modelling and Pricing Weather Derivatives,” Applied Mathematical Finance, Vol. 9, No. 1, 2002, pp. 1-20. doi:10.1080/13504860210132897

[12] B. M. Bibby and M. Sorensen, “Martingale Estimation Functions for Discretely Observed Diffusion Processes,” Bernoulli, Vol. 1, No. 1/2, 1995, pp. 17-39. doi:10.2307/3318679

[1] J. Tindall, “Weather Derivatives: Pricing and Risk Management Applications,” Institute of Actuaries of Australia, Unpublished Manuscript, 2006.

[2] M. Garman, C. Blanco and R. Erickson, “Weather Derivatives: Instruments and Pricing Issues,” Environmental Finance, 2000.

[3] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, 1973, pp. 637-659. doi:10.1086/260062

[4] L. Zeng, “Pricing Weather Derivatives,” Journal of Risk Finance, Vol. 81, No. 3, 2000, pp. 72-78. doi:10.1108/eb043449

[5] E. Platen and J. West, “Fair Pricing of Weather Derivatives,” Quantitative Finance Research Centre, University of Technology Sydney, Research Paper Series, 106, 2003.

[6] S. D. Campbell and F. X. Diebold, “Weather Forecasting for Weather Derivatives,” Journal of the American Statistical Society, Vol. 100, 2005, pp. 6-16.

[7] D. S. Wilks, “Statistical Methods in the Atmospheric Sciences,” Academic Press, New York, 1995.

[8] S. Jewson and R. Caballero, “The Use of Weather Forecasts in the Pricing of Weather Derivatives,” Meterological Applications, Vol. 10, No. 4, 2003, pp. 377-389. doi:10.1017/S1350482703001099

[9] F. E. Benth and J. Saltyne-Benth, “Stochastic Modelling of Temperature Variations with a View Toward Weather Derivatives,” Applied Mathematical Finance, Vol. 12, No. 1, 2005, pp. 53-85.

doi:10.1080/1350486042000271638

[10] M. H. A. Davis, “Pricing Weather Derivatives by Marginal Value,” Quantitative Finance, Vol. 1, 2001, pp. 1-4. doi:10.1080/713665730

[11] P. Alaton, B. Djehiche and D. Stillberger, “On Modelling and Pricing Weather Derivatives,” Applied Mathematical Finance, Vol. 9, No. 1, 2002, pp. 1-20. doi:10.1080/13504860210132897

[12] B. M. Bibby and M. Sorensen, “Martingale Estimation Functions for Discretely Observed Diffusion Processes,” Bernoulli, Vol. 1, No. 1/2, 1995, pp. 17-39. doi:10.2307/3318679