## Crash Hedging Strategies and Optimal Portfolios

• In traditional portfolio optimization under the threat of a crash the investment horizon or time to maturity is neglected. Developing the so-called crash hedging strategies (which are portfolio strategies which make an investor indifferent to the occurrence of an uncertain (down) jumps of the price of the risky asset) the time to maturity turns out to be essential. The crash hedging strategies are derived as solutions of non-linear differential equations which itself are consequences of an equilibrium strategy. Hereby the situation of changing market coefficients after a possible crash is considered for the case of logarithmic utility as well as for the case of general utility functions. A benefit-cost analysis of the crash hedging strategy is done as well as a comparison of the crash hedging strategy with the optimal portfolio strategies given in traditional crash models. Moreover, it will be shown that the crash hedging strategies optimize the worst-case bound for the expected utility from final wealth subject to some restrictions. Another application is to model crash hedging strategies in situations where both the number and the height of the crash are uncertain but bounded. Taking the additional information of the probability of a possible crash happening into account leads to the development of the q-quantile crash hedging strategy.
• Crash Hedging Strategien und Optimale Portfolios

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Verfasserangaben: Olaf Arnd Menkens urn:nbn:de:hbz:386-kluedo-18010 Ralf Korn Dissertation Englisch 2004 2004 Technische Universität Kaiserslautern Technische Universität Kaiserslautern 30.11.2004 21.01.2005 Betrachtung des Schlimmstmöglichen Falles; Crash Hedging; Gleichgewichtsstrategien; Portfolio Optimierungchanging market coefficients; crash hedging; equilibrium strategies; portfolio optimization; worst-case scenario Hamilton-Jacobi-Differentialgleichung ; Portfolio Selection; Stochastische dynamische Optimierung Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J15 Optimal control problems involving ordinary differential equations 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX] / 49Jxx Existence theories / 49J20 Optimal control problems involving partial differential equations 60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Hxx Stochastic analysis [See also 58J65] / 60H15 Stochastic partial differential equations [See also 35R60] 91-XX GAME THEORY, ECONOMICS, SOCIAL AND BEHAVIORAL SCIENCES / 91Bxx Mathematical economics (For econometrics, see 62P20) / 91B70 Stochastic models Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011

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