Risk management methods based on Value-at-Risk estimate the lowest quantile of possible profits and losses over a fixed time horizon. To calculate this value there is a need to construct an approximation of the probabilistic distribution of P&L. One of the most popular techniques is based on an assumption that the portfolio value can be expressed as a deterministic function of some basic market parameters. Having a distribution of these parameters one can construct the distribution of the value function. The most popular method is delta approach. Here a first order expansion of the value function is used in order to approximate the distribution at the end of the period. Typically the time period is assumed short, in which case the changes in market parameters are distributed almost normally and under this linear approximation the value of the approximated portfolio is also normally distributed. Value-at-Risk methods based on a delta approximation can not take into account different forms of convexity. An appropriate solution to this problem is to consider a longer series expansion, for example the so-called delta-gamma approximation. However the delta-gamma approximation loses a very useful property of “delta only” approach ‐ linearity. This linearity property is very convenient computationally, since it guarantees that as soon as the market factors are distributed normally, the resulting changes in the portfolio value are also normally distributed. Denote by x a vector of n market parameters that can be easily measured and their historical distributions are known. For example stock prices, interest rates, exchange rates. Denoting the calendar time by t we can price a portfolio of assets V.t, x/. The Value-at-Risk measures the lowest 1% (sometimes 5%) quantile of the distribution of profits and losses of the fixed portfolio over a fixed time horizon (in banking for example 10 business days). The standard assumption of this measurement is that over a short time horizon the changes in the market factors 1x are normally distributed. If the value of the portfolio is linear in the market factors then the P&L distribution is normal as well and any quantile can be expressed analytically through its mean and standard deviation. However the assumption of linear dependence is often very restrictive, a higher order approximation is required to reflect convexity. Consider the value functionV.x,t/around the current market valuesx. As soon as the market changes are small and the function V smooth, we can use the Taylor expansion. However the variablex is stochastic. Thus instead of the standard series

[We introduce a model that captures the main properties that characterize employee stock options (ESO). We discuss the likelihood of early voluntary ESO exercise, and the obligation to exercise immediately if the employee leaves the firm, except if this happens before options are vested, in which case the options are forfeited. We derive an analytic formula for the price of the ESO and in a case study compare it to alternative methods.]

[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.]

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.

We propose a new approach to dynamic representation of different groups of stakeholders on the board of directors. This approach is based on a simple economic model of the firm, with an objective function to maximize its market value. We look at the marginal claim of each stakeholder on the assets of the firm. It divides the voting rights based on the change in value of each stakeholder with a one dollar change in the value of the firm as a whole. We translate these conditions to relative voting powers on the board. While there are many claims in the academic and popular literature on sharing voting rights on the board, our paper is the first to propose a quantitative dynamic model of the power sharing in the corporation.

We introduce a model that captures the main properties that characterize employee stock options (ESO). We discuss the likelihood of early voluntary ESO exercise, and the obligation to exercise immediately if the employee leaves the firm, except if this happens before options are vested, in which case the options are forfeited. We derive an analytic formula for the price of the ESO and in a case study compare it to alternative methods. Since the mid-1980s, stock options have been a substantial component of com-pensation packages for employees. For example, in 1999, 94 % of companies in the S&P 500 offered stock options to their top employees (see Murphy, 1999; Hall and Murphy, 2002). In 1995, the Financial Accounting Standards Board (FASB) (with FAS 123) set a standard that required firms to expend stock-based compensation at the moment the compensation was granted (see FASB, 1995). Firms were encour-aged to use the “fair value ” of the stock option to compute the value of the compensation, but were allowed to use the “intrinsic value”—market price of the stock minus strike price. Since employee stock options (ESOs) are typically granted at the money, the intrinsic value is zero, which results in no expense recorded at the time of the grant, and this is probably one of the reasons that helped their popularity.

We study a regulation that increased mutual funds’ risk salience through name change. Using daily fund flow data and several identification strategies, we find that requiring certain fixed income mutual funds to affix an exclamation mark ("!") to their names caused a statistically and economically significant decline in their net flows, with a larger effect on fund inflows than outflows. The exclamation mark's impact stems from retail investors, both those that seek financial advice and those that invest independently. Mutual funds “defamed” by the exclamation mark designation actually increased their exposure to the particular risk highlighted by the regulator.

Many bankruptcy codes implicitly or explicitly contain net-worth covenants, which provide the firm’s bondholders with the right to force reorganization or liquidation if the value of the firm falls below a certain threshold. In practice, however, default does not necessarily lead to immediate change of control or to liquidation of the firm’s assets by its debtholders. To consider the impact of this on the valuation of corporate securities, we develop a model in which liquidation is driven by a state variable that accumulates with time and severity of distress. We model a dynamic grace period for the liquidation event. Recent or severe distress events may have greater impact on the liquidation trigger. Our model can be applied to a wide array of bankruptcy codes and jurisdictions.

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.