3.8.3 Monte Carlo Pricing

Basic Prerequisites: 01 · From Zero and BSM · The PDE & Feynman–Kac.

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1. Intuition & Practical Objective

Monte Carlo prices an option by manufacturing the risk-neutral expectation one path at a time. Draw a path of $S$ under $\mathbb{Q}$, evaluate the discounted payoff, repeat $n$ times, average. Its justification is the strong law; its error is the central limit theorem:

$${\hat{\alpha }}_n=\frac{1}{n}\sum _{i=1}^{n}f(U_i)\stackrel{\text{a.s.}}{\to }\alpha ,{\hat{\alpha }}_n-\alpha \approx \mathcal{N}(0,\frac{{\sigma }_f^2}{n}).$$

The practical objectives are three: (i) build the estimator and always attach a standard error; (ii) sample paths exactly when possible - for GBM the transition is lognormal so the simulation is exact, with zero discretisation bias; (iii) recognise the two structural costs - the $O(n^{-1/2})$ rate (four times the work per halving) and the fact that a payoff average is a path functional whose sampling requires care (Brownian bridge, monitoring dates).

Three “aha”s:

1. The error rate is dimension-free. $O(n^{-1/2})$ holds for an integral over $[0,1{]}^d$ for every $d$, whereas a product trapezoidal rule degrades as $O(n^{-2/d})$. At $d=10$ MC already wins outright - that is why basket and term-structure exotics are simulation problems.
2. For GBM there is no discretisation. $S(t_{i+1})=S(t_i)\exp [(r-\frac{1}{2}{\sigma }^2)\Delta t+\sigma \sqrt{\Delta t}Z]$ is the exact law, not an approximation (Glasserman eq. 3.20–3.22). Bias only enters through the payoff’s monitoring convention or through non-GBM dynamics.
3. The estimator’s error is a distribution, not a number. A 100,000-path run gave $10.4754$ where a 10,000-path run gave $10.4180$ - both consistent with the closed form $10.4506$ within their standard errors. Quoting an MC price without its $\pm$ is meaningless.

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2. Mathematical Ground Truth & Derivations

The pricing identity (Glasserman eq. 1.39, the same as Feynman–Kac’s integral form):

$$V(0)={\mathbb{E}}_{\beta }\left[\frac{V(T)}{\beta (T)}\right]=e^{-rT}{\mathbb{E}}^{\mathbb{Q}}[\text{payoff}(S_T)].$$

Simulate the risk-neutral dynamics - the drift is $r$, never $\mu$; the volatility is unchanged by the change of measure (Glasserman eq. 1.42: only the drift shifts under Girsanov).

The path sampler. For GBM the exact grid-point transition is

$$S(t_{i+1})=S(t_i)\exp \left[\right(r-\frac{1}{2}{\sigma }^2)(t_{i+1}-t_i)+\sigma \sqrt{t_{i+1}-t_i}{Z}_{i+1}],$$

and for a Gaussian short rate the exact transition is likewise available (Glasserman eq. 3.43–3.45). Nothing is discretised at the grid points for these models.

The Brownian bridge (Glasserman eq. 3.7–3.8) - the tool that makes a path’s intermediate values cheap and, later, makes barriers correct. Given $W(u)=x$ and $W(t)=y$ with $u<s<t$:

$$\mathbb{E}[W(s)\mid \cdot ]=\frac{(t-s)x+(s-u)y}{t-u},\mathrm{Var}[W(s)\mid \cdot ]=\frac{(s-u)(t-s)}{t-u}.$$

Two properties matter: (i) the conditional variance depends only on the interval lengths, not on the endpoint values - so bridge refinement is numerically stable; (ii) the first (coarsest) normal drives the largest share of path variance, which is why the bridge ordering is the standard dimension-reduction device for QMC (page 06).

Path-dependent payoffs (Glasserman §3.2): for discrete monitoring dates $t_1<\cdots <t_m$,

• arithmetic Asian: $\bar{S}=\frac{1}{m}{\sum }_{i}S(t_i)$, payoff $(\bar{S}-K{)}^{+}$ - no closed form;
• geometric Asian: $({\prod }_{i}S(t_i){)}^{1/m}$, lognormal, with a closed form - hence a perfect control variate (page 04);
• barrier (down-and-out): $1\{\tau (b)>T\}(S(T)-K{)}^{+}$ with $\tau (b)=\inf \{t_i:S(t_i)<b\}$ - discretely monitored barrier prices depend on the monitoring frequency, and a grid-based simulation misses crossings between dates (fixed on page 05/06 by Brownian interpolation).

The MSE framework (Glasserman §1.1.3) - the honest budget statement. With bias $b{\delta }^{\beta }$, per-path cost $c{\delta }^{-\eta }$ and $n$ paths,

$$\mathrm{MSE}=\underset{O({\delta }^{2\beta })}{\underbrace{{\text{bias}}^2}}+\underset{O(1/n)}{\underbrace{\text{variance}}},\mathrm{RMSE}=O\left(s^{-\beta /(2\beta +\eta )}\right),$$

where $s$ is the work budget; unbiased simulation ($\beta \to \infty$) recovers $s^{-1/2}$, and the discretisation-aware case is page 04’s (§2 efficiency rule) $s^{-\beta /(2\beta +1)}$.

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3. Computational Implementation - estimator, error rate, bridge, Asian

Four checks in one script: the $O(n^{-1/2})$ RMSE law, a standard error from one batch, the bridge variance identity, and a path-dependent price against its exact geometric counterpart.

Three verifications: the RMSE ratio between $n$ and $10n$ is $3.54$ and between $10^4$ and $10^5$ is $2.93$ - both $\approx \sqrt{10}=3.16$, and RMSE*sqrt(n) stays flat at $14$–$15.6$ (the theoretical ${\sigma }_f$); the bridge variance matches $s(t-s)/t=0.25$ to three decimals; and the MC geometric-Asian price $5.6471$ lands on its closed form $5.6374$ (within the $\approx 0.05$ standard error of $2\times 10^5$ paths). The arithmetic Asian - no closed form - prints $5.8640$, correctly above the geometric value (AM ≥ GM under $\mathbb{Q}$).

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4. Failure Modes & First-Principles Breakdowns

1. Discounting under the wrong measure. ${\mathbb{E}}^{\mathbb{P}}[e^{-rT}(S_T-K{)}^{+}]$ depends on $\mu$ and is not the price; the drift must be $r$ (Glasserman eq. 1.41). The change of measure moves the drift only - volatility is invariant.
2. Silent discretisation bias on a path functional. A crude Euler path for GBM has weak order 1, and for a running maximum only $O(h^{1/2})$ - even for Brownian motion (Asmussen–Glynn–Pitman, cited in Glasserman §6.4). Averaging more paths does not remove bias; only a better scheme or Brownian interpolation does.
3. Missing the barrier between monitoring dates. A discrete grid under-detects crossings, so a discretely-sampled knock-out is over-valued and the continuous-barrier limit is approached only as $h\to 0$. Fix: Brownian-interpolation survival probabilities (page 06).
4. Reading the sample mean without the sample error. ${\sigma }_f/\sqrt{n}$ is itself estimated; the reported standard error uses $s_f=\sqrt{\frac{1}{n-1}\sum (Y_i-\bar{Y}{)}^2}$ (Glasserman §1.1) and the interval is only asymptotically valid - heavy tails and importance sampling (page 04) break it badly.
5. Payoff discontinuities destroy the naive error estimate. Indicator payoffs (digital, barrier) have ${\sigma }_f^2$ driven by a thin region; the CLT still holds but convergence is governed by rare events, and plain MC needs enormous $n$ for deep-OTM contracts - the entry point for importance sampling.
6. The long-horizon likelihood-ratio pathology. For path-functionals the Radon–Nikodym derivative of a long path degenerates: under the twisted measure the average log-ratio converges to a strictly negative constant, so the weight falls to zero a.s. while its mean stays $1$ (Glasserman §4.6). The estimator remains unbiased but has an unusable variance - this is the structural failure of naive importance sampling on long horizons.

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Hull, Options, Futures, and Other Derivatives
• Duffy, Finite Difference Methods in Financial Engineering
• Haug, Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Back: 02 · Finite Differences · Index Hub
• Forward: 04 · Variance Reduction & Efficiency → 05 · Failure Modes → 06 · Advanced Extensions (American MC, QMC)
• Base: BSM · Pricing Formulas · Stochastic Calculus & Itô