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Monte Carlo Option Pricing

Path Simulation · Longstaff-Schwartz · Glasserman (2004)
Family 02 Pricing Simulation $\mathbb{Q}$-measure

Boyle (1977) proposed simulating price paths under the risk-neutral measure $\mathbb{Q}$ and averaging the discounted payoff. For path-dependent or high-dimensional options where trees become unwieldy and PDEs hit the curse of dimensionality, Monte Carlo scales gracefully: error shrinks as $O(1/\sqrt{M})$ in the number of paths $M$, independent of the dimension. For American options, Longstaff & Schwartz (2001) used regression to approximate the continuation value, making Monte Carlo competitive with trees even for early-exercise payoffs.

Educational Purpose
This page is an educational demonstration of Monte Carlo option pricing. It is not investment or hedging advice. The $O(1/\sqrt{M})$ convergence rate is a central limit theorem statement; finite-sample variance and discretisation bias both matter in practice, and variance-reduction (antithetic, control variates, importance sampling) is essential in production pricers.

The method

Risk-neutral simulation, discounted payoff averaging, regression for early exercise

OR family: Derivative pricing Solver class: Monte Carlo simulation Measure: Risk-neutral $\mathbb{Q}$ Realism: ★★★ Unbiased (for European)

Notation

SymbolMeaning
$M$Number of simulated paths
$N$Time steps per path, $\Delta t = T/N$
$S^{(m)}_n$Underlying on path $m$ at step $n$
$Z^{(m)}_n$Standard-normal innovation on path $m$ at step $n$
$g(\cdot)$Option payoff at maturity (European) or intrinsic value (American)

Exact GBM step

Log-Euler scheme under $\mathbb{Q}$ (exact for GBM) $$ S^{(m)}_{n+1} = S^{(m)}_n \, \exp\!\Bigl[\bigl(r - \tfrac{1}{2}\sigma^2\bigr)\Delta t \;+\; \sigma \sqrt{\Delta t}\, Z^{(m)}_n\Bigr] $$
For constant $r, \sigma$ this step is exact (no discretisation bias). For general SDEs an Euler-Maruyama scheme has bias of order $\Delta t$.

European pricing

Monte Carlo estimator — European option $$ \hat{V}_0 \;=\; e^{-rT} \, \frac{1}{M} \sum_{m=1}^{M} g\!\left(S^{(m)}_N\right) $$
$\hat{V}_0$ is unbiased for $V_0$ by the risk-neutral pricing theorem. Standard error $\propto 1/\sqrt{M}$.

American pricing: Longstaff-Schwartz (LSM)

American options add an early-exercise decision at each time step. Longstaff & Schwartz (2001) approximate the continuation value by regressing discounted future payoffs on a small basis of functions of the current underlying value. Backward through time, at each step $n$:

LSM regression step $$ \text{Continuation}_n^{(m)} \;\approx\; \sum_k \hat{\beta}_k \, \phi_k\!\left(S^{(m)}_n\right) $$
$\phi_k$ are basis functions (polynomials, Laguerre, ...). Regress discounted next-step values onto $\{\phi_k\}$ to estimate $\hat\beta$. Exercise at $n$ if intrinsic value exceeds continuation value.

Convergence & variance reduction

Standard error of the MC estimator $$ \text{SE}(\hat V_0) \;=\; \frac{e^{-rT} \, \hat\sigma_g}{\sqrt{M}} $$
Halving the error requires quadrupling the paths. Variance-reduction (antithetic variates, control variates, stratified sampling, quasi-MC) can reduce constants by 10–100× on typical problems.

Interactive simulator

Sample paths and Monte Carlo price convergence

Path simulator

★★★ Exact GBM step, unbiased for European
100
100
1.00
3.00%
25.00%
2000
50
42
MC price
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Std error
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95% CI
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BSM (European)
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20 sample paths out of $M$. Horizontal dashed line = strike $K$. Colour is per-path.

Running estimate $\hat V_0$ as paths accumulate. Shaded band = $\pm 2\,\text{SE}$. BSM (dashed) for European reference.

Strengths and limits

When Monte Carlo wins, and when it loses

Strengths

  • Path-dependent payoffs. Asian, barrier, lookback, Parisian, cliquet — all natural in MC.
  • High-dimensional basket options. Error is dimension-independent (vs $O(h^d)$ for PDE).
  • Easy to incorporate rich dynamics. Jumps (Merton 1976), stochastic volatility (Heston), local vol — just simulate.
  • Embarrassingly parallel. Paths are i.i.d.; trivially distributable.

Limits

  • Early-exercise. Requires LSM regression; basis choice and bias analysis are non-trivial.
  • Slow convergence. Halving SE needs $4\times$ paths; variance reduction is essential for production.
  • Greeks are noisy. Naive bump-and-revalue has large variance; pathwise / likelihood-ratio methods are preferred (Broadie & Glasserman 1996).
  • Discretisation bias. Non-exact schemes (Euler-Maruyama on general SDEs) add $O(\Delta t)$ bias.

Key references

Seminal papers & textbooks

Boyle, P. P. (1977).
Options: A Monte Carlo Approach.
Journal of Financial Economics, 4(3), 323–338. doi:10.1016/0304-405X(77)90005-8
Longstaff, F. A. & Schwartz, E. S. (2001).
Valuing American Options by Simulation: A Simple Least-Squares Approach.
The Review of Financial Studies, 14(1), 113–147. doi:10.1093/rfs/14.1.113
Broadie, M. & Glasserman, P. (1996).
Estimating Security Price Derivatives Using Simulation.
Management Science, 42(2), 269–285. doi:10.1287/mnsc.42.2.269
Glasserman, P. (2004).
Monte Carlo Methods in Financial Engineering.
Springer. ISBN 978-0-387-00451-8.
Merton, R. C. (1976).
Option Pricing When Underlying Stock Returns Are Discontinuous.
Journal of Financial Economics, 3(1-2), 125–144. doi:10.1016/0304-405X(76)90022-2
Heston, S. L. (1993).
A Closed-Form Solution for Options with Stochastic Volatility.
Review of Financial Studies, 6(2), 327–343. doi:10.1093/rfs/6.2.327
Hull, J. C. (2021).
Options, Futures, and Other Derivatives, 11th ed., ch. 21.
Pearson. ISBN 978-0-13-693997-9.

Related sub-applications

See the two other canonical pricing engines for European & American options.

Finance landscape Black-Scholes Binomial tree MC-based VaR
Reminder — educational purpose only
Not investment or hedging advice. Finite-sample MC variance, discretisation bias, regression error in LSM, and model-misspecification error all matter in practice; historical model performance never implies future results.