intermediate
8 min read
Finance & economics · Topic
Monte Carlo Methods in Finance
Monte Carlo simulation is indispensable in quantitative finance for pricing path-dependent derivatives, computing risk measures, and stress-testing portfolios. For exotic options with no closed form, it is often the only practical pricing method.
Pricing by simulation
To price a derivative with payoff $H = g(S_{t_1},\ldots,S_{t_m})$:
- Simulate $N$ paths of the asset under $\mathbb{Q}$
- Compute $H^{(k)}$ for each path $k$
- Estimate: $\hat{V}0 = e^{-rT}\frac{1}{N}\sum{k=1}^N H^{(k)}$
Standard error $\sim \sigma_H / \sqrt{N}$ — $10\times$ accuracy improvement requires $100\times$ more paths.
Discretising GBM
The Euler-Maruyama scheme for $dS = \mu S\,dt + \sigma S\,dW$:
\[S_{t+\Delta t} = S_t \exp\!\left[\left(\mu - \tfrac{\sigma^2}{2}\right)\Delta t + \sigma\sqrt{\Delta t}\,Z\right], \quad Z\sim\mathcal{N}(0,1)\]The log-Euler scheme is exact for GBM (no discretisation error).
Variance reduction techniques
| Technique | Idea | Typical speed-up |
|---|---|---|
| Antithetic variates | Use $(-Z_i)$ paired with $Z_i$ | $2\times$ |
| Control variates | Subtract correlated known-price instrument | $5$–$50\times$ |
| Importance sampling | Sample more tail scenarios | High for rare events |
| Quasi-Monte Carlo | Replace random with low-discrepancy sequences (Sobol) | $10\times$+ |
American options: Longstaff-Schwartz
American options allow early exercise — solved by least-squares Monte Carlo: regress continuation value on basis functions of the state at each time step, working backwards.