Question 1

How does Monte Carlo estimate p?

Accepted Answer

Random points (x,y) are sampled uniformly in the unit square [0,1]. A point is 'inside' the quarter-circle if x+y=1. Since the quarter-circle has area p/4 and the square has area 1, the fraction of inside points converges to p/4, so p4(inside/total). With N=10,000 samples the typical error is 4/(2vN)0.02, consistent with the 1/vN convergence law.

Question 2

What does stratified sampling do?

Accepted Answer

Stratified sampling divides the domain into equal sub-cells and places exactly one sample per cell. This eliminates the 'clumping and gaps' that occur in pure random sampling, reducing variance by roughly a factor equal to the number of strata. For a 1D integral with N strata, the variance decreases as N? instead of N?, making convergence dramatically faster for smooth integrands.

Question 3

What are quasi-random (low-discrepancy) sequences?

Accepted Answer

Quasi-random sequences like Halton, Sobol, and Fibonacci lattices fill space much more uniformly than pseudo-random generators. The Halton sequence in base b generates points by reversing the base-b digit expansion of integers: 1/2, 1/4, 3/4, 1/8, 5/8,  for base 2. The discrepancy measure (maximum deviation from uniform) drops as O(log(N)^d/N) instead of O(N^{-1/2}), giving faster convergence for smooth integrands in low dimensions.

Question 4

How does the option pricing preset work?

Accepted Answer

Each sample simulates a geometric Brownian motion path for the stock price: S(T)=S0exp((r-s/2)T + svTZ) where Z is a standard normal random variable. The call option payoff is max(S(T)-K,0) discounted at e^{-rT}. The Monte Carlo estimate is the average payoff over all samples. With S0=100, K=105, T=1yr, r=5%, s=20%, the Black-Scholes closed-form price is approximately $8.02  the simulation converges to this as sample count increases.

🧪 Monte Carlo Simulator Pi, Integration & Randomness

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