Principle:Avhz RustQuant Monte Carlo Option Pricing
| Knowledge Sources | |
|---|---|
| Domains | Monte_Carlo, Option_Pricing, Numerical_Methods |
| Last Updated | 2026-02-07 20:00 GMT |
Overview
A numerical method for pricing derivatives by simulating many random paths of the underlying asset, computing the payoff along each path, and averaging the discounted payoffs.
Description
Monte Carlo option pricing estimates the expected discounted payoff of a derivative under the risk-neutral measure by sampling from the distribution of the underlying asset's future paths. This method is especially powerful for:
- Path-dependent options (Asian, barrier, lookback) where the payoff depends on the entire price history
- High-dimensional problems where the curse of dimensionality makes lattice methods infeasible
- Complex payoff structures that lack closed-form solutions
The basic algorithm:
- Simulate N paths of the underlying using a stochastic process
- Compute the payoff for each path
- Average the payoffs
- Discount to present value
The convergence rate is O(1/sqrt(N)) regardless of dimension.
Usage
Use Monte Carlo pricing when analytic solutions are unavailable, when dealing with path-dependent or exotic options, or when you need a flexible pricing framework that can accommodate any payoff structure.
Theoretical Basis
The fundamental pricing equation under the risk-neutral measure Q:
The Monte Carlo estimator:
where S_T^{(i)} is the terminal (or path) value of the i-th simulated path.
Standard error: sigma / sqrt(N), where sigma is the standard deviation of the payoffs.
Pseudo-code:
// Abstract algorithm (NOT real implementation)
paths = process.generate(config)
discount_factor = exp(-rate * T)
average_payoff = sum(payoff(path) for path in paths) / N
price = discount_factor * average_payoff