Utility Maximisation with Model-Independent Constraints

Battle Count: 60

Highly relevant to quantitative trading, particularly for: (1) portfolio margin management where regulators use conservative model-independent valuations; (2) optimal position sizing in options trading under worst-case constraints; (3) semi-static hedging strategies for exotic derivatives; (4) understanding how intrinsic value constraints affect the economics of trading mispriced options. The paper directly addresses the tension between maximizing expected returns and maintaining solvency under pessimistic valuation, which is central to real-world trading desk management. The numerical results on optimal lambda (position size) and the trade-off between mean and variance of terminal wealth are directly actionable for quantitative portfolio managers.

Implementation requires: (1) Computing the greatest convex minorant of payoff functions; (2) Deriving the max-plus decomposition of the intrinsic wealth constraint process; (3) Computing hitting time densities for Brownian motion with drift via Laplace transform inversion; (4) Solving the integral equation (4.18) for the optimal threshold M numerically; (5) Evaluating the optimal utility via numerical integration. The theoretical framework is sophisticated, involving supermartingale theory, change of measure techniques, and stochastic calculus. The numerical implementation for the BSM case is tractable but requires careful handling of the hitting time distributions and the decreasing property of z(u;lambda).

The paper provides complete mathematical derivations, explicit formulas for the Black-Scholes-Merton case, and numerical parameters for all figures (S0=1.2, K=0.85, T=2, sigma=0.5, r=0.01, alpha=0.4, p=0.75, theta=0.05). However, no code repository is provided. Numerical results require computing hitting time densities for Brownian motion with drift via Laplace transform inversion, which is non-trivial but well-documented in the literature. The theoretical framework is fully self-contained with proofs.