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Quantitative Methods, Monte Carlo Integration¶
Some integrals have no tidy formula, and in high dimensions the grid methods of classical numerical analysis collapse under their own cost. Monte Carlo sidesteps both problems with an idea that sounds too simple to work. An integral is just an average in disguise, so estimate the average by sampling. Throw random points at the function, take their mean, scale by the width of the region, and the law of large numbers walks you toward the answer at a rate that does not care how many dimensions the problem has.
The price you pay is noise. The estimate carries a standard error that shrinks with the square root of the sample count, so each extra digit of accuracy costs a hundred times more samples. Variance reduction techniques claw some of that back for free, and this lesson builds the simplest one, antithetic sampling. It closes with the flagship financial application, pricing a European option by simulation and checking the answer against the Black Scholes closed form.
The pieces¶
mc_integrate(f, a, b, n, seed)estimates the integral of f over an interval by averaging the function at uniform random points. It returns the estimate and its standard error, because a Monte Carlo answer without an error bar is only half an answer.antithetic_integrate(f, a, b, n, seed)pairs every draw u with its mirror image a plus b minus u. The two are negatively correlated, so averaging each pair cancels part of the noise at no extra sampling cost.mc_european_call(spot, strike, rate, sigma, maturity, n, seed)simulates where the stock ends up under the risk neutral measure, averages the discounted payoffs, and reports the price with its standard error.black_scholes_call(...)is the exact closed form used as the yardstick.
Why the error bar matters¶
A Monte Carlo estimate is a random variable. Run it twice with different seeds and you get two different numbers, and without the standard error you cannot tell whether the difference is meaningful. The standard error also tells you what accuracy will cost. Because it falls like one over the square root of n, halving the error means quadrupling the samples, and one more decimal place means a hundred times the work. That brutal arithmetic is why variance reduction is not a nicety but the core craft of the field.
Why finance leans on this¶
The option pricing example is one dimensional and Black Scholes answers it exactly, which is precisely why it makes a good classroom, you can see the simulation converge to the truth. The real payoff comes where no formula exists. A payoff depending on the average price over a year, on the worst of five stocks, or on an interest rate path is an integral over hundreds of dimensions, and there Monte Carlo is not merely competitive, it is the only practical tool. The pattern is always the same one this lesson builds, simulate paths, average discounted payoffs, report the error.
Example¶
import math
import numpy as np
from monte_carlo_integration import mc_integrate, mc_european_call, black_scholes_call
estimate, stderr = mc_integrate(np.sin, 0, math.pi, n=50_000, seed=0)
print(estimate, "+/-", stderr) # near 2, the exact area under sin
price, err = mc_european_call(100, 105, 0.05, 0.2, 1.0, n=200_000, seed=0)
print(price, "+/-", err)
print(black_scholes_call(100, 105, 0.05, 0.2, 1.0)) # the exact answer
Where to go next¶
- For root finding and quadrature on a grid see
Quantitative Methods - Numerical Methods. - For the closed form being reproduced here see
Black-Scholes Option Pricing. - For simulation of whole portfolios rather than one payoff see
Monte Carlo Portfolio Simulator. - For the random processes the samples are drawn from see
Quantitative Methods - Stochastic Processes.
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