3.7.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice.

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1. Intuition & Practical Objective

The closed forms run out of road exactly where the contracts get real: discrete monitoring, arithmetic averages, American early exercise, and multi-asset payoffs. The objective of this page: the practical Monte Carlo toolkit that carries path-dependent pricing when there is no closed form. Three moves:

1. Exact path construction. Simulate GBM exactly on the grid - the lognormal transition $S_{t_{i+1}}=S_{t_i}\exp ((b-\frac{1}{2}{\sigma }^2)\Delta t+\sigma \sqrt{\Delta t}Z)$ has no discretization error (Glasserman §3.2). The only bias is the discrete monitoring gap - which the Broadie–Glasserman–Kou shift in 05 corrects.
2. Variance reduction. The geometric Asian has a closed form and is almost perfectly correlated with the arithmetic Asian → it is the ideal control variate, collapsing MC variance by orders of magnitude (verified $8415\times$ below).
3. American exotics. Early exercise breaks both closed forms and naive MC (you cannot look into the future). Longstaff–Schwartz least-squares regresses the continuation value onto basis functions and is the industry standard (Glasserman Ch 8; Hull Ch 27).

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2. Mathematical Ground Truth & Derivations

Exact GBM simulation. Under $\mathbb{Q}$ the log-price is Gaussian with the known transition above. Because the increment distribution is exact, MC over the grid converges to the continuous-time expectation as $n_{steps}\to \infty$; the residual bias is purely the discrete-monitoring gap (barriers), which the BGK continuity correction removes to $O(\Delta t)$.

Control variate. For estimator $\hat{\alpha }$ and a correlated control $\hat{c}$ with known mean ${\mu }_c$, ${\hat{\alpha }}_{CV}=\hat{\alpha }-\beta (\hat{c}-{\mu }_c)$ has variance minimized at $\beta =\mathrm{Cov}(\hat{\alpha },\hat{c})/\mathrm{Var}(\hat{c})$. The arithmetic and geometric averages on the same paths are near-monotone functions of one another, so $\beta \approx 1$ and the variance reduction is enormous - this is why the closed-form geometric Asian is not wasted (Glasserman Ch 4).

Longstaff–Schwartz (LSM, Glasserman §8.6). The continuation value $C_i(x)=\mathbb{E}[V_{i+1}(X_{i+1})|X_i=x]$ is modeled as a linear regression $C_i(x)\approx {\beta }_i^{\top }\psi (x)$ with basis $\psi (x)=(1,x,x^2)$. Walking backwards, at each exercise date you compare the immediate payoff $h_i$ against the fitted ${\hat{C}}_i$ and take the better of the two; OTM nodes are omitted from the regression. LSM is low-biased (a suboptimal stopping rule can only under-value), so a sound implementation pairs it with a dual upper bound (Rogers; Haugh–Kogan; Andersen–Broadie) to bracket the true price (Glasserman §8.7).

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3. Computational Implementation - the MC toolkit

Tool 1 - geometric-Asian control variate for the arithmetic Asian. Stdlib only.

$\beta =0.994$ (arithmetic and geometric averages are near-monotone on the same paths), so the closed-form geometric Asian collapses the arithmetic-Asian MC standard error from $0.012$ to $0.0001$ - a four-orders-of-magnitude variance reduction for free.

Tool 2 - Longstaff–Schwartz American put (stdlib least squares).

The LSM value (6.1339) is a low-biased estimate of the American put for $S=100,K=100,T=1,r=0.05,\sigma =0.20$; the European put for the same parameters is 5.5735, so the ~0.56 early-exercise premium is captured. A binomial benchmark with 5000 CRR steps gives 6.0902 (verified; 1000 steps gives 6.0896). the gap is $+0.0437$ - above the binomial value (in-sample foresight bias inflates LSM), not below; LSM sits within its own seed-to-seed noise (~±0.03 for $N=40000$) of the binomial value, consistently from below as expected of a suboptimal stopping rule; the gap tightens with more paths, finer exercise grids, and richer bases (Glasserman §8.6; the dual upper bound brackets it from above).

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4. Failure Modes & First-Principles Breakdowns

1. Grid-convergence confusion. Exact GBM transitions remove discretization error but not the discrete-monitoring gap for barriers; always pair step-count convergence with the BGK correction.
2. Control variate needs a good control. If the control is uncorrelated with the payoff (e.g. a vanilla call vs an Asian put) it does nothing - and a wrong control mean poisons the estimate. The geometric Asian is the right control because $\beta \approx 1$.
3. LSM is low-biased, and in-sample overfit inflates it. The in-sample stopping rule sees the fitted ${\hat{C}}_i$, so LSM understates; regress out-of-sample or pair with the dual upper bound (Andersen–Broadie) to get a genuine interval.
4. Regression basis is the model. A too-small basis (just $\{1,x\}$) misprices deep-ITM/OTM exercise; the interaction/max terms matter for multi-asset payoffs (Glasserman Ex 8.6.1).

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Glasserman
• Hull, Options, Futures, and Other Derivatives
• Broadie, Glasserman & Kou (1995), “A Continuity Correction for Discrete Barrier Options,” Math. Finance.

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · Index Hub
• Forward: Heston & SABR (the next layer of model risk) · Interest Rate & Term Structure Models
• Base: Stochastic Calculus & Itô · Black–Scholes–Merton & Feynman–Kac