Simulating Geometric Brownian Motion in Python | Quantitative Finance Tutorial

To estimate model parameters from real financial data, one first calculates log returns from historical stock prices using the formula ln(S_t/S_{t-1}). The mathematical expectation of these log returns estimates the drift parameter mu. Volatility (sigma) is estimated as the standard deviation of log returns, but must be annualized by multiplying by the square root of trading days per year (typically 242). Since there are three unknowns (q, u, d) but only two equations, assumptions like symmetric probabilities (q = 1/2) or symmetric limits (d = 1/u) are needed to solve the system.

To estimate parameters μ and σ from financial data, take the logarithm of geometric Brownian motion: ln(X(t)) = ln(X₀) + μt + σB(t). The difference ln(X(t)) - ln(X(t-1)) = μ + σ(B(t)-B(t-1)). Since B(t)-B(t-1) has mean 0 and variance 1, the expected value of log differences equals μ, and their variance equals σ². Therefore, the sample mean of log returns estimates μ, and the sample variance estimates σ². This provides a practical method for calibrating the model to real financial time series data.

Accurate Monte Carlo simulations require proper estimation of drift and volatility parameters from historical data. Drift represents expected return and can be calculated as the average of historical returns over a chosen period (one month, one year, five years). Volatility represents price fluctuation magnitude and equals the standard deviation of historical returns. Multiple approaches exist: uniform weighting gives equal importance to all periods, while exponential weighting emphasizes recent data. Historical volatility uses past return data, while implied volatility derived from options markets is generally more accurate for pricing applications.

Historical volatility is estimated from price data by calculating the standard deviation of log returns: σ = √(∑(r_i - μ)² / (n-1)). This historical volatility serves as an input for option pricing models. The Black-Scholes model requires calibration of parameters including risk-free rate, volatility, and time to maturity. These parameters can be estimated from market data or historical price series.

To calculate the drift (expected daily rate of change), use the historical mean of periodic daily returns minus half the variance over time. This is done by taking historical closing prices, calculating daily returns for each period, finding their average, and then eroding that average by volatility at the rate of 1/2 variance per time period.

The Heston Model, developed by finance professor Steven Heston in 1993, is a stochastic volatility model used to price European options that addresses a key limitation of the Black-Scholes model by assuming volatility is arbitrary (randomly varying) rather than constant; unlike the Black-Scholes model which holds volatility constant, the Heston Model incorporates stochastic variables to provide more realistic option pricing estimates, and it also functions as a volatility smile model that graphically represents how volatility increases for options that are increasingly in-the-money or out-of-the-money relative to at-the-money options.

The R-measure applies to stochastic volatility models like Heston, SVJ, and CGMY (Levy models). For Heston, call prices under R move up, while under SVJ and SVJ they go down, which is interesting because negative expected call prices would violate risk aversion. The R-transform is a generalization of Q-transforms, where the only change is that small t becomes h and q becomes r. This framework can solve expected prices under SVJ, CGMY, and other complex models that would otherwise require complicated calculations.

The Heston model extends GBM by allowing volatility to vary over time rather than remaining constant. Key parameters include: κ (mean reversion speed - how quickly volatility returns to long-term average), θ (long-term average variance), ρ (correlation between stock price and volatility), and σ_v (volatility of volatility). This model captures mean-reverting volatility behavior observed in real markets, where high volatility tends to decrease toward a long-term average. It also incorporates risk-free interest rates, making it more realistic than GBM for option pricing and scenario generation.

Stochastic volatility models treat the volatility of an underlying security as a random process governed by state variables such as price level, mean reversion, and variance, addressing the Black-Scholes model's limitation of assuming constant volatility; these models include the Heston model (with square root variance dynamics), CEV model (relating volatility to price), SABR model (for reproducing volatility smiles), GARCH models (autoregressive conditional heteroskedasticity), and the Chen model (first stochastic mean and volatility model), with calibration typically performed using maximum likelihood estimation or Bayesian methods.

Stochastic volatility models extend the Black-Scholes framework by assuming that both asset prices and their variances follow stochastic processes, addressing the limitations of constant volatility and normal return distribution assumptions; these models introduce parameters such as volatility of volatility (which controls tail risk and creates volatility smiles), correlation between asset returns and variance changes (which generates skewness in return distributions), and mean-reversion parameters (as in the Heston model) to better capture real-world market phenomena like fat tails and varying implied volatilities across strike prices and maturities.

Basic historical simulation generates risk estimates by creating multiple simulated portfolio outcomes using a mean-variance framework. The process involves specifying an expected return (drift, μ) and volatility (σ), then randomizing volatility by multiplying it with a standard normal random variable to create profit and loss (P/L) values. Losses are converted to positive values (LP format) for easier interpretation and sorted from worst to best. This framework enables calculation of risk metrics like VaR and ES by analyzing the distribution of simulated outcomes.

The historical simulation approach is a nonparametric method for estimating Value-at-Risk (VaR) and Expected Shortfall (ES) that uses actual historical market data to generate scenarios without assuming any particular distribution for risk factors; the process involves identifying portfolio risk factors, converting historical market data into percentage or absolute changes, applying these historical scenarios to current portfolio values to simulate future portfolio values, calculating losses as the difference between initial and simulated values, sorting losses in ascending order, and then determining VaR at a given confidence level by finding the loss corresponding to the accumulated probability mass of alpha (where alpha = 1 - confidence level), with Expected Shortfall being the average of all losses exceeding the VaR threshold; for multi-day horizons, daily VaR and ES can be scaled by multiplying with the square root of the number of days under the independent and identically distributed (iid) assumption, and stressed risk measures can be computed by using historical data from particularly stressful periods instead of recent data.

This section covers fundamental risk measures in quantitative finance. Value at Risk (VaR) at confidence level q is the threshold where the probability of losses exceeding this value is (1-q). For 95% confidence, VaR is the value below which 5% of outcomes fall. For normal distributions, VaR = μ - z_q × σ. Expected Shortfall (ES), or Conditional VaR, is the average loss given that losses exceed the VaR threshold, measuring the severity of losses in the worst (1-q)% of cases. Unlike VaR, which only gives a threshold, ES provides information about how bad the worst cases actually are. The implementation requires filtering data below the VaR threshold and computing the average.

Portfolio risk is measured using Value at Risk (VaR) and Expected Shortfall (ES). VaR is the minimum loss value such that the probability of exceeding this loss is no more than (1-alpha), representing the maximum expected loss at a given confidence level. ES is the expected loss given that losses exceed the VaR threshold, providing a more comprehensive measure of tail risk. For stock data, returns are used instead of losses, with VaR and ES calculated from the distribution of portfolio returns.

Value-at-Risk (VaR) is a widely-used market risk measure that estimates the maximum potential loss of a portfolio over a specific time horizon with a given confidence level, calculated using historical data, Monte Carlo simulations, or stress scenarios; however, VaR is not a coherent risk measure because it fails the sub-additivity property, meaning it may not properly account for diversification benefits, which is why Expected Shortfall (ES) is preferred as a coherent alternative that calculates the expected loss given that losses exceed the VaR threshold, providing a more conservative assessment of tail risk.