Least-Squares Monte Carlo

Plain Monte Carlo prices European options easily: simulate the underlying to expiry, average the payoff, discount. American options are harder because at every point before expiry the holder can exercise early, and the optimal choice depends on comparing an immediate, known payoff against the expected value of waiting — a quantity that depends on the unknown future. Simulated paths naturally tell you what happened on each path, but not what a rational holder, standing at an earlier date without seeing the future, would have expected the continuation to be worth. Least-squares Monte Carlo (the Longstaff-Schwartz method) fills that gap with a regression.

An analogy: deciding whether to sell a house now or wait

Imagine many parallel-universe copies of a homeowner deciding, at some point in time, whether to sell now at today's offer or hold out for a possibly better one later. You can't ask any single homeowner what "the future" holds, but if you have thousands of simulated universes, you can look at all the homeowners currently in a similar situation (similar house value, similar market conditions) and see what selling later actually turned out to be worth on average for the ones who held. That average, estimated by fitting a simple relationship between "current situation" and "eventual realized value from waiting," is a stand-in for the true expected continuation value.

The method, one symbol at a time

Simulate $M$ paths of the underlying asset up to expiry. Work backward from the last exercise date. At each earlier exercise date $t$, for every path where the option is currently in the money, the continuation value $C_t$ — the expected discounted value of not exercising now — is estimated by regressing the (already known, from later in the backward recursion) realized discounted cash flow each path eventually received against a small set of basis functions of the current state, typically $1, S_t, S_t^2$ (a polynomial in the current asset price $S_t$). The fitted regression, evaluated at each path's own $S_t$, gives that path's estimated continuation value. Compare it to the immediate exercise value $I_t$: if $I_t > \hat{C}_t$, the path exercises now and that becomes its cash flow at $t$; otherwise it continues, carrying forward whatever cash flow it already had. Stepping backward through all exercise dates this way builds up, path by path, the value of following the estimated-optimal exercise strategy.

Worked example 1: the regression step by hand

At one exercise date, suppose three in-the-money paths have current prices $S = (10, 12, 14)$ and known discounted continuation cash flows (from later steps) of $Y = (1.0, 1.5, 1.0)$. Fit a simple linear regression $Y = a + bS$. With these three points, the least-squares line has slope $b = \frac{\sum (S_i - \bar S)(Y_i-\bar Y)}{\sum (S_i-\bar S)^2}$; here $\bar S = 12, \bar Y \approx 1.167$, giving numerator $(-2)(-0.167)+(0)(0.333)+(2)(-0.167) = 0.334 - 0.334 = 0$, so $b \approx 0$ and $a \approx 1.167$ — the fitted continuation value is roughly flat at 1.167 across these three prices, because the realized outcomes didn't actually trend with price in this small sample.

Worked example 2: the exercise decision

Suppose the immediate exercise payoff (a put with strike 15) at these same three prices is $I = (5, 3, 1)$. Comparing to the fitted continuation value $\hat{C} \approx 1.167$ at each: at $S=10$, $I=5 > 1.167$, so exercise now, cash flow $=5$. At $S=12$, $I=3 > 1.167$, exercise now, cash flow $=3$. At $S=14$, $I=1 < 1.167$, so continue — the path keeps its previously known future cash flow of $1.0$ instead. Averaging the resulting cash flows across all simulated paths (discounted back to today) and discounting once more to time 0 gives the option's estimated price.

The line above is exactly the kind of fit Longstaff-Schwartz runs at every exercise date — here the "x" is the underlying price and the "y" is each path's known future payoff, and the fitted line stands in for the unknown true continuation value.

What this means in practice

Least-squares Monte Carlo is the standard technique for pricing American and Bermudan-style derivatives, mortgage prepayment options, and other path-dependent instruments with early-exercise features where simulation is otherwise the only tractable pricing method (multiple correlated underlyings, complex payoff structure). Its accuracy depends on the choice of basis functions for the regression — too few terms underfit the continuation value and mis-time exercise; too many overfit to simulation noise and produce an upward-biased price, since a regression fit to a finite sample can spuriously curve-fit unrealized value.