ملٹی اثاثہ ایکویٹی سیمولیشنز میں مارکیٹ ویرینس کی اصلاح

# Variance-Corrected Multi-Asset Equity Simulation with Hybrid Hidden Markov Marginals


# Variance-Corrected Multi-Asset Equity Simulation with Hybrid Hidden Markov Marginals









Synthetic multi-asset equity data must reproduce each asset's return distribution and its relationship with the market. Reusing a generator fitted to full asset returns creates a problem: adding its draws to a market factor counts market variance twice. We derived a correction that centers and rescales each draw over a fixed horizon before adding the market factor, allowing reuse without fitting a second generator to regression residuals. We tested the correction on 423 non-market assets in a 424-asset United States equity and exchange-traded-fund universe, using hidden Markov generators with heavy-tailed emissions. The corrected paths retained heavy tails and recovered the calibrated market loadings with low error. On 416 complete asset histories held out from 2025, the correction improved the mean Kolmogorov-Smirnov pass rate over naive composition and brought the median ratio of synthetic to observed variance close to one. Its one-day left-tail 99% Value-at-Risk exceedance rate, using thresholds pooled across separate simulated paths, was 1.12%, compared with 1.16% for the residual-fit hidden Markov comparator and a nominal rate of 1%. We also tested a jump-duration mechanism that extended visits to extreme-return states and improved volatility clustering for the broad-market exchange-traded fund with ticker SPY. The multi-asset correction remained effective with jumps enabled, but transferring the SPY jump settings improved temporal fit in training and worsened it in the 2025 holdout. The method provides a way to reuse fitted asset generators for daily-fit comparisons. Its zero-sum residual constraint removes terminal residual uncertainty, and unmodeled dependence among asset-specific residuals can lead to overstated rebalancing returns. Code, cached inputs, result summaries, and instructions for fitting the per-asset models accompany the paper.