Analytic Pricing of Employee Stock Options

We introduce a model that captures the main properties thatcharacterize employee stock options (ESO). We discuss the likelihoodof early voluntary ESO exercise, and the obligation to exerciseimmediately if the employee leaves the firm, except if thishappens before options are vested, in which case the optionsare forfeited. We derive an analytic formula for the price ofthe ESO and in a case study compare it to alternative methods.     

Dynamic Portfolio Choice with Parameter Uncertainty and the Economic Value of Analysts' Recommendations

We derive a closed-form solution for the optimal portfolio ofa nonmyopic utility maximizer who has incomplete informationabout the alphas or abnormal returns of risky securities. Weshow that the hedging component induced by learning about theexpected return can be a substantial part of the demand. Usingour methodology, we perform an "ex ante" empirical exercise,which shows that the utility gains resulting from optimal allocationare substantial in general, especially for long horizons, andan "ex post" empirical exercise, which shows that analysts’recommendations are not very useful. (JEL C61, G11, G24)     

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.     

Optimal replication of contingent claims under portfolio constraints

We determine the minimum cost of super-replicating a nonnegativecontingent claim when there are convex constraints on portfolioweights. We show that the optimal cost with constraints is equalto the price of a related claim without constraints. The relatedclaim is a dominating claim, that is, a claim whose payoffsare increased in an appropriate way relative to the originalclaim. The results hold for a variety of options, includingsome path-dependent options. Constraints on the gamma of thereplicating portfolio, constraints on the portfolio amounts,and constraints on the number of shares are also considered.     

Erratum to: Utility maximization in incomplete markets with random endowment

K. Larsen, M. Soner and G. ?itkovi? kindly pointed out to us an error in our paper (Cvitani? et al. in Finance Stoch. 5:259–272, 2001) which appeared in 2001 in this journal. They also provide an explicit counterexample in Larsen et al. (https://arxiv.org/abs/1702.02087, 2017).In Theorem 3.1 of Cvitani? et al. (Finance Stoch. 5:259–272, 2001), it was incorrectly claimed (among several other correct assertions) that the value function \(u(x)\) is continuously differentiable. The erroneous argument for this assertion is contained in Remark 4.2 of Cvitani? et al. (Finance Stoch. 5:259–272, 2001), where it was claimed that the dual value function \(v(y)\) is strictly concave. As the functions \(u\) and \(v\) are mutually conjugate, the continuous differentiability of \(u\) is equivalent to the strict convexity of \(v\). By the same token, in Remark 4.3 of Cvitani? et al. (Finance Stoch. 5:259–272, 2001), the assertion on the uniqueness of the element \(\hat{y}\) in the supergradient of \(u(x)\) is also incorrect.Similarly, the assertion in Theorem 3.1(ii) that \(\hat{y}\) and \(x\) are related via \(\hat{y}=u'(x)\) is incorrect. It should be replaced by the relation \(x=-v'(\hat{y})\) or, equivalently, by requiring that \(\hat{y}\) is in the supergradient of \(u(x)\).To the best of our knowledge, all the other statements in Cvitani? et al. (Finance Stoch. 5:259–272, 2001) are correct.As we believe that the counterexample in Larsen et al. (https://arxiv.org/abs/1702.02087, 2017) is beautiful and instructive in its own right, we take the opportunity to present it in some detail.