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Variance-optimal hedging for processes with stationary independent increments. (English) Zbl 1189.91206

Summary: We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace- or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H05 Stochastic integrals
44A10 Laplace transform
60G51 Processes with independent increments; Lévy processes
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