Non-concave Corporate Management with Option Incentives under Value-at-Risk Constraint

Battle Count: 78

The paper is primarily about corporate risk management and managerial decision-making rather than trading strategies. However, it is relevant to quantitative finance in several ways: (1) VaR constraints are central to risk management in trading desks and financial institutions; (2) the martingale and quantile formulation techniques are widely used in portfolio optimization; (3) the non-concave optimization with concavification is applicable to derivative pricing and hedging; (4) understanding how risk constraints affect behavior is relevant for designing trading risk limits. The paper's insights on how VaR constraints reshape payoff distributions and induce gambling-for-recovery behavior have parallels in trading contexts.

The paper involves advanced mathematical finance techniques: stochastic differential equations, martingale representation, concavification of non-concave functions, quantile formulation, and Lagrange duality. The nine-case classification of optimal terminal firm value requires careful handling of parameter regimes. Computing the tangency points (xi1, xi2), slopes (beta1, beta2), and Lagrange multipliers requires numerical root-finding. The explicit formulas for time-t firm value and project choice (Proposition 4.1) involve normal distribution functions and their derivatives. While the formulas are explicit, correctly identifying which of the nine cases applies and computing all threshold values requires substantial mathematical sophistication.

The paper provides fully explicit analytical solutions (Propositions 3.2, 3.3, 4.1) with all formulas derived. Baseline parameters for numerical illustrations are given (T=1, r=0.05, theta=0.4, V(0)=1, delta=0.1, K=1, n=0.05, w=0.02, gamma=1.5). However, no code or computational scripts are provided. The mathematical derivations are complete with proofs in the appendix, making theoretical reproduction feasible but requiring significant expertise in stochastic calculus and optimization.