Monte Carlo Pricing (Basic)

Motivation: why this matters in quant finance

Risk-neutral valuation says the price of any European-style payoff $X$ is

V_0 = e^{-rT}\,\mathbb{E}^{\mathbb{Q}}[X(S_T)].

For Black-Scholes vanillas, this expectation has a closed form. For almost everything else — basket options, path-dependent payoffs (Asians, lookbacks, barriers), models with stochastic volatility or jumps, multi-asset products, models calibrated to vol surfaces with no analytic solution — there is no closed form, and the integral has to be computed numerically.

Monte Carlo is the workhorse: simulate

N

sample paths under

\mathbb{Q}

, compute the payoff on each, average and discount. The error converges as

O(N^{-1/2})

— slow, but independent of the dimension of the problem, which is why it dominates over deterministic quadrature in any setting with more than a few state variables.

This note develops the basic technique: how to discretise an SDE under $\mathbb{Q}$ , generate paths, estimate the price, and quantify the standard error. Variance-reduction techniques that improve on the basic method are covered in the antithetic, control variate, importance sampling, and quasi-MC lessons.

The informal idea

Three steps.

Simulate. Generate $N$ independent realisations $S^{(1)}_T, S^{(2)}_T, \dots, S^{(N)}_T$ of the underlying at maturity (or paths if path-dependent).
Evaluate. Compute payoffs $X_i = X(S^{(i)}_T)$ .
Average and discount.

\hat V_0 = e^{-rT} \cdot \frac{1}{N}\sum_{i=1}^N X_i.

By the law of large numbers, $\hat V_0 \to V_0$ as $N \to \infty$ . By the CLT, the standard error is

\text{SE}(\hat V_0) = e^{-rT} \cdot \frac{\sigma_X}{\sqrt{N}},

where $\sigma_X$ is the (estimated) standard deviation of the discounted payoff. A 95% confidence interval is $\hat V_0 \pm 1.96 \cdot \text{SE}$ .

Formal statement

Let $X$ be a payoff with finite second moment under $\mathbb{Q}$ , and let $X_1, \dots, X_N$ be i.i.d. $\mathbb{Q}$ -distributed copies. The Monte Carlo estimator is

\hat V_0 = e^{-rT} \bar X_N, \quad \bar X_N = \frac{1}{N}\sum_{i=1}^N X_i.

Properties.

Unbiased: $\mathbb{E}^{\mathbb{Q}}[\hat V_0] = V_0$ .
Consistent: $\hat V_0 \to V_0$ a.s. by the strong law.
Asymptotically normal: $\sqrt{N}(\bar X_N - \mathbb{E}[X])/\sigma_X \to N(0,1)$ in distribution.
MSE: $\mathbb{E}[(\hat V_0 - V_0)^2] = e^{-2rT}\sigma_X^2/N$ .

The standard deviation $\sigma_X$ is estimated from the sample as $\hat\sigma_X = \sqrt{\frac{1}{N-1}\sum (X_i - \bar X_N)^2}$ .

Convergence rate. The error decays as

O(N^{-1/2})

, which means: to halve the error, multiply

N

by 4. To gain one decimal digit (factor 10 reduction), multiply

N

by 100. This is slow compared to deterministic methods but does not deteriorate with dimension.

Algorithm: pricing a European call under Black-Scholes

Under $\mathbb{Q}$ , $S_T = S_0\exp((r - \sigma^2/2)T + \sigma\sqrt{T}Z)$ with $Z \sim N(0,1)$ .

For a European call with strike $K$ :

import numpy as np

def mc_european_call(S0, K, T, r, sigma, N=100_000, seed=42):
    rng = np.random.default_rng(seed)
    Z = rng.standard_normal(N)
    ST = S0 * np.exp((r - 0.5 * sigma**2) * T + sigma * np.sqrt(T) * Z)
    payoff = np.maximum(ST - K, 0.0)
    discounted = np.exp(-r * T) * payoff
    price = discounted.mean()
    se = discounted.std(ddof=1) / np.sqrt(N)
    return price, se

S0, K, T, r, sigma = 100, 100, 1.0, 0.05, 0.2
price, se = mc_european_call(S0, K, T, r, sigma, N=100_000)
print(f"MC price: {price:.4f} ± {1.96*se:.4f} (95% CI)")
# MC price: 10.4287 ± 0.0901 (95% CI)

The closed-form Black-Scholes price is $10.4506$ — well within the confidence interval.

Algorithm: path-dependent payoffs

For payoffs depending on the path

\{S_t\}_{t\in[0,T]}

— Asians (average), lookbacks (max/min), barriers (hitting times) — discretise the time axis and simulate the path.

For geometric Brownian motion, the exact discretisation is

S_{t_{k+1}} = S_{t_k}\exp((r - \sigma^2/2)\Delta t + \sigma\sqrt{\Delta t}Z_k),

with independent $Z_k \sim N(0,1)$ . No discretisation error — the multiplicative log-normal structure means each step is exactly the GBM increment.

def mc_asian_call(S0, K, T, r, sigma, M=50, N=100_000, seed=42):
    rng = np.random.default_rng(seed)
    dt = T / M
    Z = rng.standard_normal((N, M))
    increments = (r - 0.5 * sigma**2) * dt + sigma * np.sqrt(dt) * Z
    log_paths = np.log(S0) + np.cumsum(increments, axis=1)
    paths = np.exp(log_paths)
    avg = paths.mean(axis=1)
    payoff = np.maximum(avg - K, 0.0)
    discounted = np.exp(-r * T) * payoff
    return discounted.mean(), discounted.std(ddof=1) / np.sqrt(N)

price, se = mc_asian_call(100, 100, 1.0, 0.05, 0.2, M=50, N=100_000)
print(f"Asian call: {price:.4f} ± {1.96*se:.4f}")
# Asian call: 5.7720 ± 0.0467

Algorithm: SDEs without closed-form increments

For more general SDEs

dS_t = \mu(S_t,t)dt + \sigma(S_t,t)dW_t

, Euler-Maruyama gives

S_{t_{k+1}} = S_{t_k} + \mu(S_{t_k},t_k)\Delta t + \sigma(S_{t_k},t_k)\sqrt{\Delta t}Z_k.

Strong convergence rate $O(\sqrt{\Delta t})$ , weak convergence rate $O(\Delta t)$ . For options pricing, weak convergence is what matters.

For Heston (stochastic vol), Euler is unstable for the variance process; Andersen's QE scheme or Lord-Koekkoek-Dijk's truncation is preferred. For jump-diffusions, simulate Poisson arrivals and inject jumps. These are model-specific corrections to the basic recipe.

Key properties

Dimensional independence. Convergence rate $O(N^{-1/2})$ does not depend on the dimension. A 100-asset basket option converges at the same rate as a 1-asset option (per sample), though each sample is more expensive.
Bias from discretisation. For non-GBM models discretised by Euler, MC includes both Monte Carlo error ( $O(N^{-1/2})$ ) and discretisation bias ( $O(\Delta t)$ ). The true error decomposes; total compute scales as $N \cdot M$ .
Variance reduction. Antithetic variates, control variates, importance sampling, stratified sampling, and quasi-MC all aim to reduce $\sigma_X$ without increasing $N$ .
Greeks via MC. Greeks (delta, vega, etc.) can be computed by bumping inputs, by pathwise derivatives (when payoff is differentiable), or by likelihood-ratio estimators (when not). All have trade-offs in bias/variance.
Confidence interval $\ne$ tolerance. A 95% CI half-width of $\$ 0.10 $does not mean the price is correct to$ $0.10 $— it's a statistical bound. For a deterministic guarantee you'd need$ N$ much larger or to use QMC.

Worked example: 95% CI sample size

Target: estimate a Black-Scholes ATM call price of magnitude $\sim\$ 10 $to within$ \pm $0.05$ at 95% confidence.

Empirical $\sigma_X \approx \$ 15$ for ATM call (sample std of discounted payoff).

Need $1.96 \cdot \sigma_X/\sqrt{N} \le 0.05 \Rightarrow N \ge (1.96 \cdot 15 / 0.05)^2 = 588^2/0.0025 = ~346,000$ .

So roughly $N = 350{,}000$ paths give a $\pm\$ 0.05 $95% CI. To halve the CI to$ \pm$0.025 $, need$ N = 1.4M$. Slow.

This is exactly why variance reduction is mandatory for production pricing — naive MC at $N = 10^7$ for a complex multi-asset model takes minutes per pricing, which kills any iterative use (calibration, real-time risk).

Common confusions and pitfalls

Using $\mathbb{P}$ instead of $\mathbb{Q}$ . Pricing must be under the risk-neutral measure. Real-world drift gives wrong prices.
Forgetting the discount factor. $V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[X]$ , not $\mathbb{E}^{\mathbb{Q}}[X]$ .
Random seed. Reproducibility matters — set seeds explicitly. Don't use np.random.seed and the default global RNG; use a Generator.
Discretisation bias. For path-dependent options on Black-Scholes, exact GBM discretisation has zero bias. For Heston, Euler can produce negative variance — kills the simulation. Use a proper scheme.
Antithetic doesn't always help. If the payoff is monotone in $Z$ , antithetic gives big variance reduction. For symmetric payoffs (straddles, butterflies), antithetic provides little benefit.
Reporting "exact" prices. Always report the SE alongside the point estimate. A point estimate without uncertainty is misleading.
In-the-money paths only. For deeply OTM options, most paths contribute zero — variance per sample is dominated by rare ITM paths. Importance sampling fixes this.

Where this goes next

Antithetic variates — pairing $Z$ with $-Z$ .
Control variates — subtract a correlated known-mean.
Importance sampling — re-weight paths toward the payoff region.
Quasi-Monte Carlo — replace pseudo-random with low-discrepancy sequences.