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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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##### References:
 [1] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. political econom., 81, 637-659, (1973) · Zbl 1092.91524 [2] Davis, M.H.A., Functionals of diffusion processes as stochastic integrals, Math. proc. Cambridge philos. soc., 87, 157-166, (1980) · Zbl 0424.60063 [3] Elliott, R.J., Stochastic calculus and its applications, (1982), Springer New York [4] Föllmer, H.; Schweizer, M., Hedging by sequential regression: an introduction to the mathematics of option trading, The ASTIN bulletin, 18, 2, 147-160, (1989) [5] Föllmer, H.; Schweizer, M., Hedging of contingent claims under incomplete information, () · Zbl 0738.90007 [6] Föllmer, H.; Sondermann, D.; Hildenbrand, W.; Mas-Colell, A., Hedging of non-redundant contingent claims, Contributions to mathematical economics, 205-223, (1986) [7] Hakansson, N., The fantastic world of finance: progress and the free lunch, J. financial quant. anal., 14, 717-734, (1979) [8] Harrison, J.M.; Pliska, S.R., Martingales and stochastic integrals in the theory of continuous trading, Stochastic process. appl., 11, 215-260, (1981) · Zbl 0482.60097 [9] Harrison, J.M.; Pliska, S.R., A stochastic calculus model of continuous trading: complete markets, Stochastic process. appl., 15, 313-316, (1983) · Zbl 0511.60094 [10] Jacod, J., Calcul stochastique et problèmes de martingales, () · Zbl 0414.60053 [11] Novikov, A.A., On conditions for uniform integrability for continuous exponential martingales, (), 304-310 [12] Schweizer, M., Hedging of options in a general semimartingale model, () [13] Schweizer, M., Risk-minimality and orthogonality of martingales, Stochastics and stochastic reports, 30, 123-131, (1990) · Zbl 0702.60049 [14] Sondermann, D., Reinsurance in arbitrage-free markets, () · Zbl 0739.62078
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