zbMATH — the first resource for mathematics

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.