When the underlying price process is a one‐dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non‐Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.

The institutional brokerage industry faces an ever-increasing pressure to lower trading costs, which has already driven down average commissions and shifted volume toward low-cost execution venues. However, traditional full-service brokers that bundle execution with services remain a force and their commissions are still considerably higher than the marginal cost of trade execution. We hypothesize that commissions constitute a convenient way of charging a prearranged fixed fee for long-term access to a broker’s premium services. We derive testable predictions based on this hypothesis and test them on a large sample of institutional trades from 1999 to 2003. We find that institutions negotiate commissions infrequently, and thus commissions vary little with trade characteristics. Institutions also concentrate their order flow with a relatively small set of brokers, with smaller institutions concentrating their trading more than large institutions and paying higher per-share commissions. These results are stable over time, are consistent with our predictions, and cannot be explained by cost-minimization alone. Finally, we discuss the evolution of the institutional brokerage market within the proposed framework and make informal predictions about future developments in the industry.