Oxford

We study an equilibrium model for a defaultable bond in the asymmetric dynamic information setting. The market consists of noise traders, an insider and a risk neutral market maker. Under the assumption that the insider observes the firm value continuously in time we study the optimal strategies for the insider and the optimal pricing rules for the market maker. We show that there exists an equilibrium where the insider’s trades are inconspicuous. In this equilibrium the insider drives the total demand to a certain level at the default time. The solution follows from answering the following purely mathematical question which is of interest in its own: Suppose Z and B are two independent Brownian motions with B(0)=0 and Z(0) is a positive random variable. Let T be the first time that Z hits 0. Does there exists a semimartingale X such that

1) it is a solution to the SDE

dX(t) = dB(t) + g(t,X(t),Z(t))dt

with X(0) = 1, for some appropriate function g,

2) T is the first hitting time of 0 for X, and

3) X is a Brownian motion in its own filtration?

We consider the hedging of a claim on a non-traded asset using a correlated traded asset, when the agent does not know the true values of the asset drifts, a partial information scenario. The drifts are taken to be random variables with a Gaussian prior distribution. This is updated via a linear filter. The result is a full information model with random drifts. The utility infdifference price and hedge is characterised via the dual problem, for an exponential utility function. An approximation for the price and hedge is derived, valid for small positions in the claim. The effectiveness of this hedging strategy is examined via simulation experiments, and is shown to yield improved results over the Black-Scholes strategy which assumes perfect correlation.

In a financial market, the appreciate rates are very difficult to estimate precisely, and in general only some confidence interval will be estimated. This paper is devoted to the portfolio selection with the appreciation rates being in a certain closed convex set rather than some precise point. We study the problem in both expected utility framework and mean-variance framework, and robust solutions are given explicitly in both frameworks.