The -σ²/2 term in the drift adjusts for the difference between arithmetic and geometric means. Without it, simulations would overestimate wealth, especially for volatile portfolios.
Formula
Drift = (μ - σ²/2) × dtMethodology
The Ito correction is a subtle but crucial detail in Monte Carlo simulation. It arises from Ito's Lemma, which describes how functions of stochastic processes evolve.
Under GBM, log returns follow: ln(S_t/S_0) ~ N((μ - σ²/2) × t, σ² × t)
This means: - The expected value of S_t is S_0 × exp(μ × t) - But the median of S_t is S_0 × exp((μ - σ²/2) × t)
The correction is especially important for: - High volatility portfolios (larger σ² adjustment) - Long time horizons (effect compounds over time)
For example, with μ = 10% and σ = 20%: - Without correction: median grows at 10%/year - With correction: median grows at 10% - 2% = 8%/year
This ensures the 50th percentile of our simulations matches the mathematical expectation.
Data Source
Derived from Ito's Lemma in stochastic calculus.
Reference
Itô, K. (1944). Stochastic Integral. Proceedings of the Imperial Academy, 20(8), 519-524
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For Educational Purposes Only
This analysis is not investment advice. Results are based on simplified models using historical data. Past performance does not guarantee future results. All investments carry risk of loss. Consult a qualified financial advisor before making investment decisions.