American Monte Carlo: Longstaff-Schwartz
Longstaff-Schwartz least-squares Monte Carlo (LSMC) prices Bermudan and American-style claims by learning an approximation to continuation value from simulated paths. It is a practical solution when a tree or PDE cannot accommodate the number of risk factors. The essential idea is simple: at each exercise date, compare immediate exercise to the expected value of continuing, estimated by regression.
The apparent simplicity hides a statistical problem. The regression is not a generic prediction task: it is used to make a discontinuous stopping decision, and the training targets depend on later decisions. Basis selection, state sufficiency, in-sample optimism, and exercise-date conventions all affect value and Greeks.
The backward optimal-stopping recursion
At exercise date tj, let hj(Xj) be intrinsic value and let Yj be the discounted cash flow obtained by following the current later-date policy. The continuation value is:
C_j(x) = E_Q[Y_j | X_j=x]
V_j(x) = max(h_j(x), C_j(x))
LSMC regresses Y_j on basis functions of the current state, usually using in-the-money paths. If intrinsic exceeds the fitted continuation, exercise; otherwise retain the future cash flow. Move backward from maturity to the first decision date.
| Step | Input | Output |
|---|---|---|
| Simulate | risk-neutral state paths | states at all dates |
| Initialize | terminal payoff | future cash flows |
| Regress backward | ITM states and discounted cash flows | continuation estimate |
| Exercise | intrinsic vs estimate | updated cash flows |
| Discount | time-zero path values | price estimate |
Minimal regression skeleton
import numpy as np
def basis(spot, strike):
x = spot / strike
return np.column_stack([np.ones_like(x), x, x*x])
# At a date j, cashflows are already valued at j+1.
itm = intrinsic[j] > 0
X = basis(spots[j, itm], strike)
y = discount[j] * cashflow[itm]
coef, *_ = np.linalg.lstsq(X, y, rcond=None)
continuation = X @ coef
exercise = intrinsic[j, itm] > continuation
This is a teaching fragment, not a full implementation. A production version tracks the exercise date per path, discounts a path’s cash flow from its actual exercise date, and handles no-ITM-path cases. It also makes the time grid, rates, dividends, and simulation scheme explicit.
Choose a sufficient state
For a Black-Scholes put, spot and time are sufficient. For a stochastic-volatility option, continuation depends on spot and variance; for a callable rate product it can depend on multiple curve factors; for a path-dependent claim it may require running averages or extrema. Regressing on spot alone because it is convenient creates a systematically wrong exercise policy.
| Product/model | Candidate regression state |
|---|---|
| Equity put, GBM | spot |
| Stochastic-volatility put | spot, variance |
| Bermudan swaption | rates/factors, annuity, swap rate |
| Asian Bermudan | spot, running average |
| Callable note | all contractual path variables |
Normalize inputs and use economically scaled bases. Orthogonal polynomials, splines, or carefully regularized machine learning can help, but flexibility increases overfit risk near the boundary. A low regression error averaged over all paths is not the relevant metric; accuracy around h=C is.
In-sample optimism and dual validation
The same simulated paths are often used to fit the exercise policy and value it. This can produce optimistic exercise choices and a biased high estimate, especially with rich bases. Cross-fitting divides paths into folds: fit on other folds and exercise on the held-out fold. An even clearer approach trains the policy on one path set and evaluates its realized discounted payoff on an independent set.
The resulting lower-bound estimator is the value of a feasible stopping rule. Dual methods can construct upper bounds using martingales, giving an interval around the true price. If the gap remains wide, more simulation or a better policy is needed.
| Validation tool | What it establishes |
|---|---|
| Binomial/PDE benchmark | low-dimensional price accuracy |
| Independent policy evaluation | overfit control |
| Basis sensitivity | approximation stability |
| More paths | sampling convergence |
| Dual upper bound | residual stopping-rule gap |
Exercise dates and market conventions
American exercise means any time in theory, but LSMC works on a discrete Bermudan grid. Increase decision frequency and show convergence; do not label a monthly Bermudan approximation an American price without qualification. Cash dividends require dates placed exactly on the grid or handled with explicit jumps. Settlement lags, call notices, and business-day calendars may matter more than an extra polynomial term.
The economic decision also includes financing, transaction charges, and operational cutoffs. The theoretical boundary is a valuation input, not necessarily an instruction to submit exercise. The general exercise logic is covered in American options and early exercise.
Greeks and risk
LSMC Greeks are difficult because small bumps can change the estimated exercise boundary. A frozen-policy Greek revalues paths using the original policy; it is stable and often useful for hedging, but omits policy response. Refit-policy bumps include more effects but add regression noise. Use common random numbers, multiple bump sizes, and confidence intervals. Pathwise or adjoint techniques require differentiating the regression and stopping decision carefully; see pathwise Greeks and adjoint differentiation.
Key takeaways
- LSMC estimates continuation values by backward regression and applies a feasible stopping rule.
- The regression state must contain the variables that drive continuation, not merely spot.
- Independent policy evaluation or cross-fitting is vital for controlling in-sample optimism.
- Basis, path count, exercise grid, and dividend conventions need sensitivity testing.
- Report price bounds and policy-aware Greek conventions, not a single unexplained estimate.