e review several points of view on pricing, hedging and making decisions in
incomplete markets. We start from brief summary of what we want from a good
model of incomplete market.
By the assumption of incompleteness we cannot hedge and/or replicate
perfectly. Hence, we would like to develop a recipe for identification of good
investment opportunities. Hopefully, such a recipe should be stated in terms
of a "price" of personal nature, depending on portfolio, risk preference,
credit rating and so fourth. The following are some of the desirable features.
1. The model should include all observable quantitative data. In particular,
we do not want to discard historical information.
2. The model should be dynamic. We want to hedge and may want to exit the
position early. Consequently, the liquidity information should be included.
3. The model should deliver inventory control. It should not recommend taking
unlimited exposure when a good opportunity is discovered. Hence, the
attractiveness of an investment decision should vanish as we accumulate
exposure in line with such investment decision. Consequently, current position
should be a part of the model.
4. The set of "good" opportunities should include those leading to possible
losses. We may take risk if the somehow evaluated potential gain makes the
5. Absence of "good" opportunities for all market participants should cause
existence of generally accepted pricing.
6. All sources of incompleteness (lack of liquidity, transaction costs,
trading restrictions, unheadgeable risks...) should increase bid/ask spread.
Willingness to take risk should decrease bid/ask spread.