Quantitative Finance
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Foundations
What a Simulation Actually Does
Instead of solving an equation for one exact answer, a Monte Carlo simulation draws thousands of random, equally-plausible future paths for a variable (a stock price, a portfolio's return), then looks at the resulting distribution of outcomes. The average of those outcomes approximates the answer; the spread tells you how uncertain that answer is.
Why Finance Needs It
Closed-form formulas like Black-Scholes only work for simple, well-behaved payoffs. The moment a problem involves path-dependency (did the price ever cross a barrier?), multiple correlated assets, or irregular cash flows, there's usually no clean formula — simulation handles arbitrary complexity in exchange for computation time instead of algebra.
Core Applications
Prices exotic, path-dependent derivatives (Asian, barrier, lookback options) by simulating the underlying's price path thousands of times and averaging the discounted payoff — the workhorse method whenever there's no closed-form alternative.
Simulates portfolio value across thousands of scenarios to estimate Value-at-Risk (VaR) and Conditional VaR — how bad the tail of the loss distribution actually is, not just the expected case.
Simulates thousands of possible market paths to show the full range of retirement/portfolio outcomes (e.g. "85% probability the portfolio lasts 30 years") instead of a single deterministic projection.
Simulates counterparty default scenarios and exposure paths over the life of a trade to price the Credit Valuation Adjustment — the cost of the counterparty potentially failing to pay.
How a Simulation Is Built
Random Sampling & Stochastic Processes
A process (commonly Geometric Brownian Motion for stock prices) specifies how the variable evolves; at each time step a random shock — usually drawn from a normal distribution — is added, and the whole path is repeated thousands of times to build a distribution of outcomes.
Convergence & Number of Paths
Simulation accuracy improves with the square root of the number of paths — doubling precision requires roughly 4x the simulations, not 2x. This is why naive Monte Carlo can be computationally expensive for high-precision pricing.
Variance Reduction Techniques
Methods like antithetic variates (pairing each random path with its mirror image) and control variates (using a related, known quantity to cancel out noise) cut the number of paths needed for a target accuracy, since plain random sampling converges slowly on its own.
Common Pitfalls
Related & Advanced Topics
Derivatives Pricing
Monte Carlo is one of several pricing methods alongside closed-form models and binomial trees — see Derivatives Pricing for when each approach fits best.
Statistical Analysis
The distributional assumptions that drive a simulation — mean, volatility, correlation, fat tails — come from statistical estimation; see Statistical Analysis.
Risk Management
Simulation output (VaR, CVaR, drawdown distributions) directly feeds portfolio risk limits and stress testing — see Risk Management.
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