Option hedging for semimartingales. (English) Zbl 0735.90028

Summary: We consider a general stochastic model of frictionless continuous trading. The price process is a semimartingale and the model is incomplete. Our objective is to hedge contingent claims by using trading strategies with a small riskiness. To this end, we introduce a notion of local \(R\)-minimality and show its equivalence to a new kind of stochastic optimality equation. This equation is solved by a Girsanov transformation to a minimal equivalent martingale measure. We prove existence and uniqueness of the solution, and we provide several examples. Our approach contains previous treatments of option trading as special cases.


91B62 Economic growth models
91B26 Auctions, bargaining, bidding and selling, and other market models
93E03 Stochastic systems in control theory (general)
91B60 Trade models
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