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Robust static super-replication of barrier options. (English) Zbl 1196.91007

Radon Series on Computational and Applied Mathematics 7. Berlin: de Gruyter (ISBN 978-3-11-020468-1/hbk). xii, 197 p. (2009).
The book deals with the hedging of barrier options with the help of a portfolio consisting of plane vanilla options with different strikes and maturities.
The author considers the simple market consisting of the bond and the base instrument – a stock. The dynamics of the bond prices is given by a constant risk-free rate, while under the equivalent martingale measure \(Q\) the dynamics of the stock is described by the stochastic volatility model based on two correlated Wiener processes. The broad class of such markets contains, as special cases, the Heston and Black-Scholes models.
The goal is to provide a static super-hedging strategy which will be only marginally more expensive than the barrier option itself and will be robust against model parameter uncertainty. The mathematical techniques used to prove appropriate existence, duality and convergence results range from financial mathematics, stochastic and semi-infinite optimization, convex analysis, stochastic and partial differential equations to semidefinite programming. Moreover, the author develops the whole “numerical toolbox” which enables to construct the strategy in the real time.

MSC:

91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
91G20 Derivative securities (option pricing, hedging, etc.)
91B24 Microeconomic theory (price theory and economic markets)
90C30 Nonlinear programming
90C34 Semi-infinite programming

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