Stochastic Model Genetic Programming: Deriving Pricing Equations for Rainfall Weather Derivatives

Rainfall derivatives are in their infancy since starting trading on the Chicago Mercantile Exchange (CME) in 2011. Being a relatively new class of financial instruments there is no generally recognised pricing framework used within the literature. In this paper, we propose a novel Genetic Programming (GP) algorithm for pricing contracts.

Our novel algorithm, which is called Stochastic Model GP (SMGP), is able to generate and evolve stochastic equations of rainfall, which allows us to probabilistically transform rainfall predictions from the risky world to the risk-neutral world.

In order to achieve this, SMGP’s representation allows its individuals to comprise of two weighted parts, namely a seasonal component and an autoregressive component. To create the stochastic nature of an equation for each SMGP individual, we estimate the weights by using a probabilistic approach. We evaluate the models produced by SMGP in terms of rainfall predictive accuracy and in terms of pricing performance on 42 cities from Europe and the USA.We compare SMGP to 8 methods: its predecessor DGP, 5 well-known machine learning methods (M5 Rules, M5 Model trees, k-Nearest Neighbors, Support Vector Regression, Radial Basis Function), and two statistical methods, namely AutoRegressive Integrated Moving Average (ARIMA) and Monte Carlo Rainfall Prediction (MCRP). Results show that the proposed algorithm is able to statistically outperform all other algorithms.

The daily rainfall data used includes a total of 20 cities from around Europe and 22 from around the United States of America (USA). The data was retrieved from NOAA NCDC. The 20 European cities are: Amsterdam (Netherlands), Arkona (Germany), Basel (Switzerland), Bilbao (Spain), Bourges (Germany), Caceres (Spain), Delft (Netherlands), Gorlitz (Germany), Hamburg (Germany), Ljubljana (Slovenia), Luxembourg (Luxembourg), Marseille (France), Oberstdorf (Germany), Paris (France), Perpignan (France), Potsdam (Germany), Regensburg (Germany), Santiago (Portugal), Strijen (Netherlands), and Texel (Netherlands). The 22 USA cities are: Akron, Atlanta, Boston, Cape Hatteras, Cheyenne, Chicago, Cleveland, Dallas, Des Moines, Detroit, Jacksonville, Kansas City, Las Vegas, Los Angeles, Lousville, Nashville, New York City, Phoenix, Portland, Raleigh, St Louis, and Tampa.

The rainfall prediction results showed that the SMGP was the most suitable algorithm, which significantly outper formed all other machine learning algorithms on all data sets. It achieved the lowest predictive error and is favourable for rainfall derivatives, based on the correlation between predictive error and the pricing accuracy. Whilst we observed evidence that this statement is true, we were unable to fully test this, because of the unavailability of daily prices. However, we noticed that the SMGP predicted the actual rainfall for each contract more accurately than all other algorithms. The results achieved contribute significantly both to the literature and to the practice of rainfall derivatives. The methodology is able to provide more certainty for future events by a more accurate predictive model.

This study has been published in Swarm and Evolutionary Computation and it is a collaborative work of Dr. Antonis K. Alexandridis, Dr. Sam Cramer, Dr. Michael Kampouridis, and Dr. Alex Freitas.

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