DFG-Forschungszentrum "Mathematik für Schlüsseltechnologien - MATHEON": Securitization: assessment of external risk factors (Teilprojekt E 2)

The financial and insurance risk in commodities and energy products generated by natural phenomena such as weather and climate is exogenous from the perspective of financial markets. Contracts such as derivatives written on non-tradable risk of this type therefore can be seen as instruments of securitization or insurance, serving the task of transfering exogenous risk to capital markets. Illiquidity is a major issue in these markets that calls for an investigation of the micro-economic structure, from which the usual meso-scopic models arise in the context of market equilibria. Dealing with pricing and hedging of these products, which constitutes the primary mathematical focus of this project, leads to archetypical models of incomplete financial markets. The project developed a pivotal stochastic approach of risk indifference based cross hedging in incomplete finance and insurance markets by means of quadratic growth Backward Stochastic Differential Equations (BSDE), founded on the martingale optimality concept and stochastic calculus of variations. Generalizing the Black-Scholes delta hedge formula to the incomplete setting, it allows an explicit description of investment strategies exhibiting the correlation of market and risk in the setting of exponential utility. A major issue for practical applications is a numerical analysis of these quadratic growth BSDEs. Our project presents first results in this direction. We were able to extend the stochastic calculus of variations to obtain explicit delta hedge formulas on stochastic bases on which the Brownian motion is replaced by a continuous Markovian martingale. To develop further the numerical approach for quadratic growth BSDE (e.g. enhancement of the convergence speed), and to extend the cross hedging approach to general utility functions constitute focal points of future research of the project. For publications, talks and further information please see the homepages of the project members and E2.