Methods of Partial Hedging

In this article we survey methods of dealing with the following problem: A financial agent is trying to hedge a claim C, without having enough initial capital to perform a perfect (super) replication. In particular, we describe results for minimizing the expected loss of hedging the claim C both in complete and incomplete continuous-time financial market models, and for maximizing the probability of perfect hedge in complete markets and markets with partial information. In these cases, the optimal strategy is in the form of a binary option on C, depending on the Radon-Nikodym derivative of the equivalent martingale measure which is optimal for a corresponding dual problem. We also present results on dynamic measures for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C. This revised version was published online in August 2006 with corrections to the Cover Date.     

Beliefs regarding fundamental value and optimal investing

Standard optimal portfolio selection models take no account of the special information that active investors believe they possess. For example, active investors who believe they can place bounds on the price of a security will want to use that information when assessing risk and expected return in order to construct an optimal portfolio. In this paper, we use two continuous-time models to analyze how placing boundaries on the price of a stock affects assessed risk, expected returns, and the optimal holdings of an active investor, and how those vary as a function of the relation between the stock price and the boundaries. In particular, the optimal strategy takes significant long/short positions as the price nears its lower/upper boundary.     

Price impact and portfolio impact

We study survival, price impact, and portfolio impact in heterogeneous economies. We show that, under the equilibrium risk-neutral measure, long-run price impact is in fact equivalent to survival, whereas long-run portfolio impact is equivalent to survival under an agent-specific, wealth-forward measure. These results allow us to show that price impact and portfolio impact are two independent concepts: a nonsurviving agent with no long-run price impact can have a significant long-run impact on other agents' optimal portfolios.     

Dynamic programming approach to principal–agent problems

We consider a general formulation of the principal–agent problem with a lump-sum payment on a finite horizon, providing a systematic method for solving such problems. Our approach is the following. We first find the contract that is optimal among those for which the agent’s value process allows a dynamic programming representation, in which case the agent’s optimal effort is straightforward to find. We then show that the optimization over this restricted family of contracts represents no loss of generality. As a consequence, we have reduced a non-zero-sum stochastic differential game to a stochastic control problem which may be addressed by standard tools of control theory. Our proofs rely on the backward stochastic differential equations approach to non-Markovian stochastic control, and more specifically on the recent extensions to the second order case.     

Co-development ventures: Optimal time of entry and profit-sharing

We find the optimal time for entering a joint venture by two firms, and the optimal linear contract for sharing the profits. We consider risk-sharing, timing-incentive and asymmetric decisions contract designs. If the firms are risk-neutral and the cash payments are allowed, all three designs are equivalent. With risk aversion, the optimal contract parameters may vary significantly across the three designs and across varying levels of risk aversion. We also analyze a dataset of joint biomedical ventures, in which, in agreement with our theoretical predictions, both royalty and cash payments are mostly increasing in the smaller firm's experience, and the time of entry happens sooner for more experienced small firms.