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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.

MSC:
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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