P vs NP Problem in Portfolio Optimization: Integrating the Markowitz-CAPM Framework with Cardinality Constraints and Black-Scholes Derivative Pricing

Abstract

This paper makes the Millennium Prize problem P vs NP operational in quantitative finance by studying cardinality-constrained portfolio selection. Starting from the convex Markowitz mean-variance program with CAPM-based expected returns (Rf plus beta times ERP), we impose a hard sparsity rule that limits the portfolio to K assets out of approximately 94 industry portfolios (Damodaran). The constraint couples discrete subset selection with continuous weight optimization, yielding a mixed-integer quadratic program and an NP-hard search space that grows combinatorially with n and K. We therefore evaluate scalable approximation schemes (greedy screening, Monte Carlo sampling, and genetic algorithms) under a replication-oriented protocol with random-seed control, distributional performance summaries (median and quantiles), runtime profiling, and convergence diagnostics. Dependence structure is documented via correlation and covariance diagnostics and positive-semidefinite checks to link algorithm behavior to the geometry implied by the risk matrix. To support the title's derivatives component, we add a European call option priced by the Black-Scholes model and map it into CAPM-consistent moments using delta-based linearization, validated with a bump test and moneyness/maturity sensitivity. Results highlight how the cardinality constraint reshapes the attainable efficient frontier, why stability and computational-cost trade-offs matter more than single-best runs, and how common-factor dependence can limit diversification in K-sparse solutions. The study provides a reproducible template for NP-hard portfolio optimization with transparent inputs and extensible derivative overlays.