Abstract
This paper proposes a consistent approach to the pricing of weather derivatives. Since weather derivatives are traded in an incomplete market setting, standard hedging based pricing methods cannot be applied. The growth optimal portfolio, which is interpreted as a world stock index, is used as a benchmark or numeraire such that all benchmarked derivative price processes are martingales. No measure transformation is needed for the proposed fair pricing. For weather derivative payoffs that are independent of the value of the growth optimal portfolio, it is shown that the classical actuarial pricing methodology is a particular case of the fair pricing concept. A discrete time model is constructed to approximate historical weather characteristics. The fair prices of some particular weather derivatives are derived using historical and Gaussian residuals. The question of weather risk as diversifiable risk is also discussed.
Similar content being viewed by others
References
Bajeux-Besnainou, I. and Portait, R. (1997) The numeraire portfolio: A new perspective on financial theory, Eur. J. Finance 3, 291–309.
Becherer, D. (2001) The numeraire portfolio for unbounded semimartingales, Finance Stoch. 5, 327–341.
Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities, J. Polit. Econ. 81, 637–659.
Borch, K. H. (1968) The Economics of Uncertainty, Princeton University Press, Princeton.
Bühlmann, H. and Platen, E. (2003) A discrete time benchmark approach for insurance and finance, ASTIN Bull. 33(2), 153–172.
Cao, M. and Wei, J. (2001) Equilibrium Valuation of Weather Derivatives, Working Paper, York University and University of Toronto.
Chau, T., Makita, S., Riopel, G., and Wang, T. (2000) Modelling and hedging electricity generation profits. Financial Engineering Proseminar, MIT Sloan School of Management.
Cochrane, J. H. (2001) Asset Pricing, Princeton University Press, Princeton.
Delbaen, F. and Schachermayer, W. (1994) A general version of the fundamental theorem of asset pricing, Math. Ann. 300, 463–520.
Fama, E. F. and MacBeth, J. D. (1974) Long-term growth in a short-term market, J. Finance 29, 857–885.
Goll, T. and Kallsen, J. (2003) A complete explicit solution to the log-optimal portfolio problem, Adv. Appl. Probab. 13(2), 774–799.
Happ, E. (1999) Indices for exploring the relationship between temperature and grape and wine flavour, Wine Industry J. 14(4), 1–5.
Heath, D., Platen, E., and Schweizer, M. (2001) A comparison of two quadratic approaches to hedging in incomplete markets, Math. Finance 11(4), 385–413.
Hofmann, N., Platen, E., and Schweizer, M. (1992) Option pricing under incompleteness and stochastic volatility, Math. Finance 2(3), 153–187.
Kariya, T. (2003) Weather risk swap valuation, Proceedings JAFFEE International Conference, March 15–16, 2003, Tokyo, Hitotsubashi University, pp. 466–481.
Kelly, J. R. (1956) A new interpretation of information rate, Bell Syst. Techn. J. 35, 917–926.
Korn, R. and Schäl, M. (1999) On value preserving and growth-optimal portfolios, Math. Methods Operations Res. 50(2), 189–218.
Leggio, K. B. and Lien, D. (2002) Hedging gas bills with weather derivatives, J. Econ. Finance 26(1), 88–100.
Long, J. B. (1990) The numeraire portfolio. J. Financ. Econ. 26, 29–69.
McIntyre, R. and Doherty, S. (1999) Weather risk –an example from the UK, Energy and Power Risk Management June.
Merton, R. C. (1973) Theory of rational option pricing, Bell J. Econ. Manage. Sci. 4, 141–183.
Platen, E. (2002) Arbitrage in continuous complete markets, Adv. Appl. Probab. 34(3), 540–558.
Platen, E. (2004a) A benchmark framework for risk management. In Stochastic Processes and Applications to Mathematical Finance, Proceedings of the Ritsumeikan International Symposium, World Scientific, Singapore, pp. 305–335.
Platen, E. (2004b) Modeling the volatility and expected value of a diversified world index, Int. J. Theor. Appl. Finance 7(4), 511–529.
Platen, E. (2004c) Pricing and hedging for incomplete jump diffusion benchmark models, In Mathematics of Finance, Vol. 351 of Contemporary Mathematics, American Mathematical Society, pp. 287–301.
Protter, P. (1990) Stochastic Integration and Differential Equations, Springer, New York.
Ralston, A. and Rabinowitz, P. (1978) A First Course in Numerical Analysis (2nd ed.), Dover Publications, New York.
Samuelson, P. A. (1971) The fallacy of maximising the geometric mean in long sequences of investing or gambling, Proceedings of the National Academy of Science, pp. 2493–2496.
Author information
Authors and Affiliations
Corresponding author
Additional information
1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20
JEL Classification: C16, G10, G13
Rights and permissions
About this article
Cite this article
Platen, E., West, J. A Fair Pricing Approach to Weather Derivatives. Asia-Pacific Finan Markets 11, 23–53 (2004). https://doi.org/10.1007/s10690-005-4252-9
Issue Date:
DOI: https://doi.org/10.1007/s10690-005-4252-9