The first chapter studies indifference pricing with Knightian uncertainty in an incomplete market.
A one period trinomial model is analysed assuming an exponential utility function.
Indifference pricing and hedging methods are extended to the case of Knightian uncertainty represented by the maxmin expected utility criterion by I. Gilboa & D. Schmeidler (1989, ).
Since the issuance of a contingent claim can change the reference belief, one must take into account how the worst case scenarios change with or without the contingent claim. In this paper, the optimal selection rule is characterized firstly, and some examples are presented
to show how to calculate indifference prices and optimal hedge ratios.
The second chapter investigates the problem of optimal hedging and pricing with Knightian uncertainty a la I. Gilboa & D. Schmeidler. We start with a simple one period model and extend it to multi-period models as Cox et al. did for the Black-Scholes model (1979, ).
We assume that the set of beliefs is convex and closed to utilize the min-max theorem. We also assume that for multi-period models, the set of beliefs satisfies the rectangularity defined as in Epstein & Schneider (2003, ) to ensure the time consistency of value functions.
In discrete time models, given the second moment of the risky asset the agent would choose his belief from risk neutral ones. So the hedging demand for the risky asset would be zero.
If the set of beliefs does not contain any risk neutral belief, he would choose the one which is closest to the set of risk neutral beliefs, which gives rise to hedging demand for the risky asset. If we admit the second moment of the risky asset vary, the problem would be more complex.
For the simplest case in which the set of beliefs consist of only risk neutral ones, the agent would choose one of the extreme beliefs according to comparision between the certainty equivalent values at future nodes. It turns out that the comparison is related to the second order derivative of the certainty equivalent value along spatial dimension. We could use the multi-period results and Taylor series expansion to derive the limitting equations, which are
formulated as a conjecture by the author. In doing this, we have to scale the set of beliefs according to the unit size of time steps. In the limiting case, it is demonstrated by an example that the process of indifference price satisfies the Black-Scholes-Barenblatt equation, which is a kind of G-equation introduced by Shige Peng (2006, ).
The third chapter is about the Pareto optimal allocation of Knightian uncertainty in a single period discrete model.
Maxmin expected utility by Gilboa and Schmeidler is also
adopted to deal with ambiguity of two agents. If two agents have the same belief on ambiguity in a complete market, ambiguity does not seem to be involved in their decisions. But if their beliefs are differerent, ambiguity might play some roles in their decision. If the market is not complete, the two agents could decide according to their beliefs even though they share the same belief on the common state.
All the proofs can be found in the appendix except the shorter ones located in the main text.