Monte Carlo Simulation for Derivative Pricing › Monte Carlo Error and Convergence

Understanding the error in Monte Carlo estimates is essential for practical implementation. Unlike deterministic methods that give the same answer every time, Monte Carlo produces a random estimate that varies from run to run. If we repeat our simulation many times, each estimate $\hat{V}$ will differ slightly due to random sampling. This variability is not a bug but an inherent feature of the method, and it must be quantified so that you understand the precision of your estimates.

Let $X_1, X_2, \ldots, X_N$ be independent samples of the discounted payoff. Each $X_i$ represents the discounted payoff from one simulated path, and these samples are independent because we draw fresh random numbers for each path. The Monte Carlo estimator is simply the sample mean:

$$\hat{V} = \frac{1}{N} \sum_{i=1}^{N} X_i$$

where:

• $\hat{V}$: Monte Carlo estimate of the option value
• $N$: number of simulations
• $X_i$: discounted payoff from the $i$-th simulation

This estimator is unbiased, meaning that its expected value equals the true option price. However, any single realization of $\hat{V}$ will deviate from the true value due to sampling variation. The central limit theorem tells us how this variation behaves: for large $N$, the sampling distribution of $\hat{V}$ is approximately normal with mean equal to the true value and variance inversely proportional to $N$:

$$\hat{V} \sim \mathcal{N}\left(V, \frac{\sigma_X^2}{N}\right)$$

where:

• $\hat{V}$: Monte Carlo estimate of the option value
• $V$: true option value
• $\sigma_X^2$: variance of the individual discounted payoffs
• $N$: number of simulations
• $\mathcal{N}$: normal distribution notation

Regardless of the distribution of individual payoffs (which may be highly skewed or otherwise non-normal), the average of many payoffs follows an approximately normal distribution. The approximation improves as $N$ increases, typically becoming excellent by $N = 1000$ or so.

The standard error of our estimate quantifies the typical magnitude of sampling error:

$$\text{SE}(\hat{V}) = \frac{\sigma_X}{\sqrt{N}}$$

where:

• $\text{SE}(\hat{V})$: standard error of the estimator
• $\sigma_X$: standard deviation of individual discounted payoffs
• $N$: number of simulations

The standard error measures the standard deviation of the sampling distribution, telling us how much our estimate typically varies from run to run. Since we don't know the true $\sigma_X$, we estimate it from our sample using the sample standard deviation formula:

$$\hat{\sigma}_X = \sqrt{\frac{1}{N-1} \sum_{i=1}^{N} (X_i - \hat{V})^2}$$

where:

• $\hat{\sigma}_X$: sample standard deviation estimator
• $N$: number of simulations
• $X_i$: discounted payoff of the $i$-th path
• $\hat{V}$: sample mean (Monte Carlo estimate)

The division by $N-1$ rather than $N$ in this formula is Bessel's correction, which makes the estimator unbiased for the true variance.

With the standard error in hand, we can construct confidence intervals that express uncertainty about the true price. A 95% confidence interval for the true price is:

$$\hat{V} \pm 1.96 \cdot \text{SE}(\hat{V})$$

where:

• $\hat{V}$: Monte Carlo estimate
• $\text{SE}(\hat{V})$: standard error of the estimator
• $1.96$: approximate $z$-score for a 95% two-sided confidence interval

This formula does not mean there is a 95% probability that the true price lies in this interval. Rather, it means that if we repeated the entire simulation many times, constructing a new confidence interval each time, 95% of those intervals would contain the true price. The distinction is subtle but important for statistical rigor.

The critical insight for practical applications is that $\text{SE} \propto 1/\sqrt{N}$, meaning the standard error is inversely proportional to the square root of the number of simulations. This square-root relationship determines the computational cost of greater accuracy. To reduce the error by half, we must quadruple the number of simulations. To achieve one additional decimal place of accuracy (reducing error by a factor of 10), we need 100 times more simulations. To gain two decimal places, we need 10,000 times more paths.

This slow convergence rate motivates the search for variance reduction techniques, which we explore in the next chapter. Methods like antithetic variates, control variates, and importance sampling can dramatically reduce variance without increasing the number of paths, effectively making each simulation more informative and accelerating convergence.