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.
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Notes
already presented as an example in Barles and Souganidis (1991)
References
Barcilon A, Fang Z, Wang B (1999) Stochastic dynamics of El Nino-Southern Oscillation. J Atmos Sci IPRC-31 56:5–23
Barles G (1994) Solutions de Viscosité des Equations de Hamilton-Jacobi. Mathématiques et Applications 17. Springer, Berlin Heidelberg New York
Barles G, Souganidis PE (1991) Convergence of Approximation schemes for fully nonlinear second order equations. asymptotic analysis, vol. 4. Elsevier North-Holland, New York, pp 273–283
Battisti DS (1989) On the role of off-equatorial oceanic Rossby waves during ENSO. J Phys Oceanogr 19:551–559
Borkar VS (1989) Optimal control of diffusion processes. Research notes in mathematics, vol 203. Pitman, London
Bouchard B, Touzi N (2004) Discrete time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stoch Proc Appl 111:175–206
Chaumont S (2002) Gestion optimale de bilan de compagnie d’assurance. PhD Thesis, Université Nancy I
Davies EB (1989) Heat kernels and spectral theory. Cambridge University Press, Cambridge
Crandall MG, Ishii H, Lions P-L (1992) User’s guide to Viscosity Solutions of 2nd Order PDE. Bull Am Math Soc 27(1):1–67
Fleming WH, Soner HM (1993) Controlled Markov processes and viscosity solutions. Applications of mathematics, vol 25. Springer, Berlin Heidelberg New York
Friedman A (1964) Partial differential equation of parabolic type. Prentice-Hall, Englewood Cliffs
Gaol JL, Manurung D (2000) El Nino Southern Oscillation impact on sea surface temperature derived from satellite imagery and its relationships on tuna fishing ground in the South Java seawaters. AARS
Gilbarg D, Trudinger NS (1977) Elliptic partial differential equations of second order. Springer, Berlin Heidelberg New York
Hu Y, Imkeller P, Müller M (2003) Partial equilibrium and market completion. IJTAF (to appear)
Herrmann S, Imkeller P (2005) The exit problem for diffusions with time periodic drift and stochastic resonance. Ann Appl Probab 15:39–68
Herrmann S, Imkeller P, Pavlyukevich I (2003) Stochastic resonance: non-robust and robust tuning notions. In: Haman K, Jakubiak B, Zabczyk J (eds) Probabilistic problems in atmospheric and water sciences. Opublikowano przy udziale Centrum Dosconalosci IMPAN-BC, Warsaw 2003
Imkeller P, Pavlyukevich I (2002) Stochastic resonance: a comparative study of two-state models. In: Dalang R, Russo E (eds) Seminar on stochastic analysis, random fields and applications IV. Progr Prob 58:141–154, Birkhäuser 2004
Karatzas I, Lehoczky JP, Shreve SE (1987) Optimal portfolio and consumption decision for a small investor on a finite horizon. SIAM J Control Optim 25:1157–1586
Karatzas I, Shreve SE (1988) Brownian motion and stochastic calculus. Springer, Berlin Heidelberg New York
Kobylanski M (2000) Backward stochastic differential equations and partial differential equations with quadratic growth. Ann Probab 28(2):558–602
Kramkov D, Schachermayer W (1999) The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann Appl Probab 9:904–950
Krylov NV (1980) Controlled diffusion processes. Springer, Berlin Heidelberg New York
Kushner HJ, Dupuis PG (2001) Numerical methods for stochastic control problems in continuous time, 2nd edn. Application of mathematics, vol 24. Springer, Berlin Heidelberg New York
Mizuno K (1995) Variabilities of thermal and velocity field of north of Australia Basin with regard to the Indonesia trough flow. In: Proceedings of the international workshop on trough flow studies in around Indonesian waters. BPPT, Indonesia
Penland C (1996) A stochastic model of Indo Pacific sea surface temperature anomalies. Physica D 98:534–558
Taylor ME (1997) Partial differential equations III—nonlinear equations. Applied mathematical sciences, vol 117. Springer, Berlin Heidelberg New York
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This work was partially supported by the DFG research center ‘Mathematics for key technologies’ (FZT 86) in Berlin.
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Chaumont, S., Imkeller, P. & Müller, M. Equilibrium trading of climate and weather risk and numerical simulation in a Markovian framework. Stoch Environ Res Ris Assess 20, 184–205 (2006). https://doi.org/10.1007/s00477-005-0001-x
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DOI: https://doi.org/10.1007/s00477-005-0001-x