Monte Carlo pricing and sampling error

Definition Fix a model with a pricing measure $\mathbb{Q}$ and a constant continuously compounded rate $r$, and consider a contract that pays $g(S_T)$ currency units at $T$. Its discounted payoff is $Y = e^{-rT} g(S_T)$ and its model price is $V = \mathbb{E}^{\mathbb{Q}}[Y]$. More generally $Y$ is the payoff divided by the money-market account $B_T$. Let $Y_1, \dots, Y_N$ be independent random variables, each with the law of $Y$ under $\mathbb{Q}$, and compute all expectations and variances of the draws under that law. The Monte Carlo estimator is

If $\mathbb{E}^{\mathbb{Q}}[\lvert Y \rvert] < \infty$, linearity of expectation gives $\mathbb{E}[\widehat V_N] = V$, so the estimator is unbiased, and the strong law of large numbers gives $\widehat V_N \to V$ almost surely as $N \to \infty$. If also $\mathbb{E}^{\mathbb{Q}}[Y^2] < \infty$, then

because independence makes every covariance with $j \ne k$ vanish and leaves $N$ copies of $\operatorname{Var}(Y)$.

Definition Risk-neutral sampling draws the inputs of the payoff from their law under the pricing measure $\mathbb{Q}$, with the same contract terms and discounting as the price. In the Black-Scholes model with constant $r$, $q$ and $\sigma$, the ex-dividend price satisfies $dS_t = (r - q) S_t \, dt + \sigma S_t \, dW^{\mathbb{Q}}_t$ under $\mathbb{Q}$, and the exact transition is

with $Z$ standard normal. Sampling $S_T$ this way from independent standard normal draws gives draws of $Y$ with exactly the law needed for a payoff that depends only on $S_T$, with no time-discretization error. A path-dependent payoff needs the path at the dates, or the crossings, that the contract specifies, and a stochastic discount factor must be generated jointly with the payoff on the same path. Draws made under the physical measure $\mathbb{P}$, where $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$, estimate $\mathbb{E}^{\mathbb{P}}[Y]$ instead. That is not the price unless each draw is multiplied by the density

This density defines $\mathbb{Q}$ on $\mathcal{F}_T$, and by Girsanov’s theorem $W^{\mathbb{Q}}_t = W_t + \lambda t$ is a Brownian motion under $\mathbb{Q}$. Here $\lambda$ is constant, so Novikov’s condition holds.

Definition The sampling error of a run is $\widehat V_N - \mathbb{E}[\widehat V_N]$, the deviation of the estimate from the expectation that the sampler actually targets. For independent draws with $\mathbb{E}[Y^2] < \infty$ it has mean zero and standard deviation $\sqrt{\operatorname{Var}(Y)/N}$. The payoff variance is estimated without bias by the sample variance

and the Monte Carlo standard error is

It estimates the standard deviation of the estimator across repeated runs, not the spread of individual payoffs. It is consistent, and slightly biased low, since $\mathbb{E}[s_N] \le \sqrt{\operatorname{Var}(Y)}$ by Jensen’s inequality. Quadrupling $N$ halves the standard deviation of the estimator. The formula needs independent draws. With dependent samples, such as antithetic pairs, Markov chain output or the points of one randomized quasi-Monte Carlo set, the covariances in the variance no longer vanish, and the standard error has to be computed from units that are independent, for example the pair averages of antithetic draws.

Definition Suppose the draws are independent and $0 < \operatorname{Var}(Y) < \infty$. The central limit theorem and the consistency of $s_N$ give

in distribution as $N \to \infty$, where $\mathcal{N}(0, 1)$ is the standard normal law. Hence the probability that the interval

contains $V$ tends to $2\Phi(z) - 1$, and $z = 1.96$ gives the approximate 95 percent Monte Carlo confidence interval. The exact quantile is $\Phi^{-1}(0.975) \approx 1.959964$. The statement is asymptotic, and for a finite $N$ the probability can differ from $0.95$, especially with heavy tails, rare events or small samples. A Student $t$ interval is exact only for normally distributed draws, and a call payoff with an atom at zero and a skewed tail is not normal, so it carries no exact small-sample guarantee here either. If $\operatorname{Var}(Y) = 0$, the payoff is constant, the estimate is exact and the interval is a single point, a case outside the theorem.

Definition The coverage of an interval procedure is the probability, over fresh independent runs, that the interval it produces contains the target $V$. A 95 percent Monte Carlo interval has nominal coverage $0.95$, and under the hypotheses of the previous card its actual coverage tends to $0.95$ as $N \to \infty$. For a fixed $N$ the actual coverage can be lower or higher. Coverage describes the procedure. Once a run is done, its interval either contains $V$ or does not, and saying that $V$ lies in that one interval with probability $0.95$ confuses the two. Over independent runs the number of misses is binomial with the per-run miss probability, which allows a simulator to be tested against a known price. Coverage also presumes that the sampler targets $V$. If the sampler has a fixed bias, so that $\mathbb{E}[\widehat V_N] - V$ is the same nonzero number for every $N$, the coverage tends to $0$ as $N$ grows, because the interval shrinks around the wrong value.

Definition Suppose the code averages independent draws of a discounted payoff $\tilde Y$ whose law may differ from that of $Y$, and $V$ is the price in the stated model. The error of the estimate splits as

The first term is the sampling error, which the standard error measures and which shrinks like $1/\sqrt{N}$. The second is a bias that does not depend on $N$. It contains discretization error, from time stepping, discrete monitoring of a continuous condition or any other approximation of the model’s law, and specification error, from a wrong measure, payoff or discount factor. Model error is a further gap, between $V$ and the quantity the model price is meant to represent, such as the price in a richer model or a quoted price. Neither bias nor model error appears in the standard error, and a narrow interval can concentrate around a wrong value. Refining the time grid or using an exact transition addresses discretization, checking the contract and the measure addresses specification, and comparison with prices or hedging behavior addresses the model.