Quadratic hedging in an incomplete market derived by an influential informed investor

Abstract

In this paper a model with an influent and informed investor is presented. The studied problem is the point of view of a non informed agent hedging an option in this influenced and informed market. Her lack of information makes the market incomplete to the non informed agent. The obtained results, by means of Malliavin calculus and Clark-Ocone Formula, as well as Filtering Theory are the expressions and a comparison between the strategy of the non informed trader, and the strategy of the informed agent. An expression of the residual risk a non informed trader keeps by detaining an option in this influenced and informed market is derived using a quadratic approach of hedging in incomplete market. Finally, the analysis leads to a measure of the lack of information that makes the incompleteness of the market. The financial interpretation is explained throughout the theoretical analysis, together with an example of such influenced informed model.Keywords Enlargement of filtration ⋅ FBSDE ⋅ quadratic hedging ⋅ risk minimization ⋅ insider trading ⋅ influent investor ⋅ asymmetric information ⋅ martingale representation ⋅ Clark-Ocone formula. AMS Classification (2000): 60H10, 60G44, 60H07, 60J75, 91G20, 91B70, 93E11. JEL Classification: C60,G11,G14.

How Does Asymmetric Information Create Market Incompleteness?

“…Moreover, it is possible to complete the market by simply adding the knowledge of the private information that is missing to the average agent (small investor investing on the market, but who has only access to the information generated by prices, and not to the private information). The study of this incomplete market from the mathematical point of view was the aim of [6], which was devoted to the study of the non-informed agent's strategy in this inuenced market.…”

“…Dierent methods may be used in order to solve the hedging problem for any contingent claim under such a ltration: we choose the Kunita-Watanabe (K-W) Decomposition in order to replace the lack of exact martingale representation theorem, as well as methods of quadratic hedging in incomplete markets as in Follmer-Schweizer [7]. We present here the main results of [6].…”

“…How does investors' information inuence their strategies when investing on markets? This question has already been studied in the litterature: from an optimization point of view at rst (see Grorud [9], Grorud and Pontier [10,11,12]), and from a hedging point of view more recently (see Eyraud-Loisel [4,5,6]).…”

“…Last, in [6], the model is studied from the point of view of a non informed investor trying to invest on such an inuenced market model. For a deep mathematical presentation of this modeling, the reader can refer to the papers cited before.…”

Option hedging by an influential informed investor

“…Our model is an example of complete market that becomes incomplete from a small non‐informed investor's point of view. The deeper study of investment in such a market uses tools of quadratic hedging in incomplete market, or under incomplete information, and will be developed separately in a further work, see Eyraud‐Loisel 20.…”