General financial market model defined by a liquidation value process

Abstract

Financial market models defined by a liquidation value process generalize the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. The solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex, e.g. in presence of proportional transaction costs and fixed costs. Therefore, the classical duality principle based on the Hahn–Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. Using an alternative method based on the concepts of essential supremum and maximum, we provide a characterization of European and American contingent claim prices under the absence of arbitrage opportunity of the second kind.

Keywords:

• Financial markets
• liquidation value
• transaction costs
• European and American options
• hedging
• partial order

AMS Subject Classifications:

• 60G44
• G11-G13

Notes

No potential conflict of interest was reported by the authors.

1 We may see the bid/ask prices as , where is the mid-price and is the proportional transaction costs coefficient.

2 Observe that we may replace by defined by

which satisfies .

3 is the set of all non negative integers.

4 A set is -decomposable if and implies that .

5 The condition is for all .

6 Or equivalently with respect to since if and are -measurable, then means that , .

8 Note that the NA condition as defined for the Kabanov model and the one we introduce for more general models coincide.