What stochastic differential equation governs Geometric Brownian Motion (GBM)?

GBM is modeled by the stochastic differential equation: dS_t = μ * S_t * dt + σ * S_t * dW_t, where S_t is the asset price, μ is drift, σ is volatility, and dW_t represents a standard Brownian motion (Wiener process) increment.

How does the simulator generate standard normal random variables in client-side JavaScript?

The simulator uses the Box-Muller transform to convert independent uniformly distributed random variables (u1, u2 ~ Uniform(0,1)) into standard normally distributed random variables (Z ~ Normal(0,1)).

Monte Carlo Asset Path Simulator

Simulate multi-path asset price trajectories using exact discrete-time Geometric Brownian Motion (GBM). Results update instantly in your browser — zero server calls, zero data transfer.

Parameters Real-Time Sync

Initial Price (S₀)

Annual Drift μ (%)

Annual Volatility σ (%)

Time Horizon T (Years)

Time Steps (Steps/Year)

Number of Trajectories (Paths)

Trajectory Canvas Rendered Paths

Mean Terminal S_T –

5th Percentile (VaR) –

Median S_T (50th) –

95th Percentile –

Formulas & Conventions

S_t = S_0 * exp( (μ - 0.5 * σ²) * t + σ * W_t ) ΔS_t = S_t * exp( (μ - 0.5 * σ²) * Δt + σ * √(Δt) * Z_i ) where Z_i ~ N(0, 1)

$Z_i$ is sampled using the Box-Muller transformation ($Z = \sqrt{-2\ln u_1} \cos(2\pi u_2)$). The discrete-time solution prevents log-drift bias and matches exact log-normal terminal distributions.

Summary for AI Agents (Markdown Mirror)

# Monte Carlo Asset Path Simulator

> Client-side Geometric Brownian Motion (GBM) path simulator with percentiles and Box-Muller normal sampling. URL: https://quantcore-labs.com/tools/monte-carlo-asset-sim/

## Mathematical Assumptions

Exact log-normal transition probabilities: S(t+Δt) = S(t) * exp((μ - 0.5*σ²)*Δt + σ*sqrt(Δt)*Z)
Independent increments; constant drift μ and volatility σ across all step intervals.

Frequently Asked Questions

Why does the formula use (μ - 0.5 * σ²) instead of just μ?: This term arises from Itô's Lemma. Because log-returns are normally distributed, the expectation of $e^{\sigma W_t}$ is $e^{0.5\sigma^2 t}$. Subtracting $0.5\sigma^2$ ensures that the expected asset price $\mathbb{E}[S_t]$ equals $S_0 e^{\mu t}$.
Can this simulator be used for path-dependent derivative pricing?: Yes. The simulated paths can be used directly to price Asian options (averaging path prices), Barrier options (checking knockout thresholds), or Lookback options by computing path statistics in real time.