Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework

  • Sébastien Chaumont
  • Peter Imkeller
  • Matthias Müller
Original paper

Abstract

We consider financial markets with agents exposed to external sources of risk caused, for example, by short-term climate events such as the South Pacific sea surface temperature anomalies widely known by the name El Nino. Since such risks cannot be hedged through investments on the capital market alone, we face a typical example of an incomplete financial market. In order to make this risk tradable, we use a financial market model in which an additional insurance asset provides another possibility of investment besides the usual capital market. Given one of the many possible market prices of risk, each agent can maximize his individual exponential utility from his income obtained from trading in the capital market, the additional security, and his risk-exposure function. Under the equilibrium market-clearing condition for the insurance security the market price of risk is uniquely determined by a backward stochastic differential equation. We translate these stochastic equations via the Feynman–Kac formalism into semi-linear parabolic partial differential equations. Numerical schemes are available by which these semilinear pde can be simulated. We choose two simple qualitatively interesting models to describe sea surface temperature, and with an ENSO risk exposed fisher and farmer and a climate risk neutral bank three model agents with simple risk exposure functions. By simulating the expected appreciation price of risk trading, the optimal utility of the agents as a function of temperature, and their optimal investment into the risk trading security we obtain first insight into the dynamics of such a market in simple situations.

2000 AMS subject classifications:

primary 60 H 30, 91 B 70 secondary 60 H 20, 91 B 28, 91 B 76, 91 B 30, 93 E 20, 35 K 55 

Keywords

External risk Climate risk ENSO Incomplete financial market Utility maximization Exponential utility Market equilibrium Market clearing Backward stochastic differential equation Semi-linear parabolic pde Non-linear Feynman–Kac formula HJB equation Numerical schemes for pde Numerical simulation 

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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Sébastien Chaumont
    • 1
  • Peter Imkeller
    • 1
  • Matthias Müller
    • 1
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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