Comonotonic and moment matching approximations for sums of lognormal random variables

Battle Count: 62

The paper is primarily relevant to actuarial science and risk management rather than direct quantitative trading. However, the approximation of sums of lognormal random variables is directly applicable to: (1) pricing and risk management of Asian options, (2) portfolio risk aggregation when asset returns are lognormal, (3) computing VaR and CVaR for portfolios of dependent lognormal assets, and (4) pension fund liability valuation. The comonotonic structure enables fast computation of distortion risk measures, which is valuable for real-time risk monitoring. The right-tail improvement is particularly relevant for tail risk management in trading portfolios.

The theoretical framework is mathematically sophisticated (weighted distributions, convex cones, distortion functions), but the actual implementation is relatively straightforward once the starting comonotonic approximation is chosen. Key steps: (1) compute moments of S and T, (2) compute conditional moments b_ij, (3) solve the convex cone feasibility problem (linear programming), (4) construct the step-weight function, (5) compute quantiles/risk measures via the distortion function G_{A,Q}. The main computational challenge is the convex cone check and solving for lambda weights, which is a small linear system for m=3.

The paper provides detailed mathematical formulations, explicit parameter choices (n=20, mu=0.075, sigma=0.15, alpha_i=1), specific Q vectors, and computed lambda and A values for all three starting approximations. Monte Carlo benchmark uses 500,000 paths with antithetic variables. However, no code or supplementary data files are mentioned. The numerical tables (Tables 1-6) are referenced but individual values are not fully reproduced in the text extract.