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Contents of PMS, Vol. 30, Fasc. 2,
pages 247 - 258
 

QUANTILE HEDGING FOR AN INSIDER

PrzemysŁaw Klusik
Zbigniew Palmowski
Jakub Zwierz

Abstract: In this paper we consider the problem of the quantile hedging from the point of view of a better informed agent acting on the market. The additional knowledge of the agent is modelled by a filtration initially enlarged by some random variable. By using equivalent martingale measures introduced in [1] and [2] we solve the problem for the complete case, by extending the results obtained in [4] to the insider context. Finally, we consider the examples with the explicit calculations within the standard Black–Scholes model.

2000 AMS Mathematics Subject Classification: Primary: 60H30; Secondary: 60G44.

Keywords and phrases: Insider trading, quantile hedging, initial enlargement of filtrations, equivalent martingale measure.

References

[1]   J. Amendinger, Martingale representation theorems for initially enlarged filtrations, Stochastic Process. Appl. 89 (1) (2000), pp. 101–116.

[2]   J. Amendinger, P. Imkeller and M. Schweizer, Additional logarithmic utility of an insider, Stochastic Process. Appl. 75 (2) (1998), pp. 263–286.

[3]   D. Duffie and C. F. Huang, Multiperiod security markets with differential information, J. Math. Econom. 15 (1986), pp. 283–303.

[4]   H. Föllmer and P. Leukert, Quantile hedging, Finance Stoch. 3 (3) (1999), pp. 251–273.

[5]   H. Föllmer and P. Leukert, Efficient hedging: cost versus shortfall risk, Finance Stoch. 4 (2) (2000), pp. 117–146.

[6]   A. Grorud and M. Pontier, Probabilités neutres au risque et asymétrie d’information, C. R. Acad. Sci. Paris Sér. I Math. 329 (11) (1999), pp. 1009–1014.

[7]   T. Jeulin, Semi-martingales et grossissement d’une filtration, Lecture Notes in Math. No 833, Springer, Berlin 1980.

[8]   T. Jeulin and M. Yor (Eds.), Grossissements de filtrations: exemples et applications, Lecture Notes in Math. No 1118, Springer, Berlin 1985. ewblock Papers from the seminar on stochastic calculus held at the Université de Paris VI, Paris 1982/83.

[9]   I. Karatzas and I. Pikovsky, Anticipative portfolio optimization, Adv. in Appl. Probab. 28 (1996), pp. 1095–1122.

[10]   J. Zwierz, On existence of local martingale measures for insiders who can stop at honest times, Bull. Polish Acad. Sci. Math. 55 (2) (2007), pp. 183–192.

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