A Regression Beats a Tree: Longstaff and Schwartz on American Options

Some papers win by being profound. This one won by being simple, and it is now, by some distance, the most widely implemented numerical method in derivatives pricing.

The problem it solves had been open for twenty years. Monte Carlo simulation can price anything except options with early exercise, because simulation runs forwards in time and the early-exercise decision has to be reasoned backwards. Broadie and Glasserman had produced a rigorous but computationally heavy answer in 1997. Longstaff and Schwartz produced, in 2001, an answer that a competent programmer can implement in an afternoon and which runs fast enough to use on a trading desk. It is called Least Squares Monte Carlo, and it is everywhere.

The problem: the holder needs to know what the future is worth

An American option can be exercised at any time. So at every moment, the holder faces a decision: take the money now, or hold on?

Exercising now gives you a payoff you can see: the stock is at 80, the put is struck at 100, exercising gives you 20. Simple.

Holding on gives you the continuation value: the expected value, given everything you know right now, of keeping the option alive. And that is the hard part. It is an expectation about the future, conditional on the present.

A binomial tree computes this naturally, because a tree knows all the future nodes reachable from where you are. A Monte Carlo simulation does not. It knows one path, and using what happened later on that path to decide whether to exercise now is cheating: the real holder cannot see the future, and an option priced with foresight is worth far more than any option really is.

So the entire problem reduces to one question: standing at this point on this path, without peeking, what is the option worth if I hold it?

The key idea via analogy: ask what usually happens to people like you

Longstaff and Schwartz's answer is almost embarrassingly practical.

You cannot see your future. But you have ten thousand other simulated paths sitting in memory, and you have already computed what the option ended up being worth on each of them. So look across all the paths that, at this moment in time, were in roughly the same situation as you are, meaning the stock was at a similar price. Look at what the option actually ended up delivering on those paths. That is your estimate of the continuation value.

You are not predicting your own future. You are asking: for people in my position, what typically happens next?

And the way you extract that answer is with the humblest tool in statistics: a regression.

At each exercise date, working backwards from expiry:

1. Take all the simulated paths where the option is currently in the money. (Out-of-the-money paths are irrelevant, since you would never exercise, and excluding them sharpens the fit. This is a small detail that matters a lot in practice.)
2. Regress the discounted future payoff actually received on those paths against a few simple functions of the current stock price. Longstaff and Schwartz used polynomials: the price, the price squared, and so on. A handful of basis functions is usually enough.
3. The fitted regression is now your estimate of the continuation value as a function of the current stock price. It is a rule: "if the stock is here, holding on is worth about this much."
4. Compare, path by path: is the immediate exercise payoff bigger than the fitted continuation value? If yes, exercise. If no, hold.
5. Step back to the previous exercise date and repeat.

When you reach today, average across paths. That is the price.

The regression is doing the job that the tree's backward induction used to do, but statistically, from the simulated data itself. That is the entire idea, and its elegance is that it uses only information available at the decision point, so it does not cheat.

Why it took over

• It scales to high dimensions, and that is the whole point. A tree on twenty underlying factors is impossible. A Monte Carlo on twenty factors is routine, and Longstaff-Schwartz inherits that. The original paper demonstrates the method on an American swaption in a twenty-factor model of the yield curve, a problem that no lattice method could touch. This is exactly the world of Bermudan swaptions and callable structured notes, which is a very large market.
• It handles path dependence naturally. The regression's inputs do not have to be just the current stock price. You can include the running average, the running maximum, the level of volatility, or anything else that describes your current state. So an American option on a path-dependent underlying, which is otherwise a nightmare, is barely harder.
• It is simple enough that people actually implement it. This is underrated. A method that is elegant but hard to code does not get used. Longstaff-Schwartz is roughly thirty lines of code, and it is in every derivatives library on earth.
• It handles jumps, stochastic volatility, whatever. Since the underlying is just simulated, the model can be as complicated as you like. The paper explicitly demonstrates this with a jump-diffusion example.

The honest limitations

• The price is biased low, and there is no error bound. This is the fundamental honest caveat. The regression gives you an approximate exercise rule. An approximate rule is a suboptimal rule. A holder following a suboptimal rule captures less value than an optimal one. So the resulting price is systematically an underestimate, and the method offers no way to tell you by how much. This is precisely the gap that Broadie-Glasserman's upper bound, and its later duality-based refinements, exist to close. Serious practitioners run Longstaff-Schwartz for the price and a dual method for the bound.
• The answer depends on your choice of basis functions, and there is no theory to guide it. How many polynomials? Which ones? Chebyshev, Laguerre, plain powers? Add too few and your exercise rule is crude. Add too many and you overfit the simulation noise, which quietly reintroduces look-ahead bias through the back door: the regression starts fitting the specific random draws rather than the underlying relationship. There is no principled rule for choosing, and practitioners rely on convention and testing. The method's greatest weakness is that its most important design choice is left to taste.
• Convergence is proved only under conditions that are hard to check. The theoretical guarantees require the number of basis functions and the number of paths to grow together in a particular way. In practice you pick both by hand and hope.
• Greeks are still noisy. As with all Monte Carlo, the sensitivities are harder to get than the price, and the presence of an exercise boundary makes gamma particularly ill-behaved.
• It is slow compared with a tree, when a tree will do. For a plain American put on one stock, a binomial tree is faster, more accurate and simpler. Longstaff-Schwartz earns its keep only in dimensions where trees fail.
• Its results can be quietly wrong in a way that looks fine. A poorly specified regression gives you a plausible-looking price with no warning sign. That is a dangerous property for a method this widely deployed.

The one-line takeaway

Longstaff and Schwartz cracked American options by simulation with a trick of disarming simplicity: estimate the value of holding on by regressing what actually happened on other simulated paths that were in a similar position, a method that scales to problems no tree can handle, runs in thirty lines of code, and is now the industry default, at the cost of a price that is always slightly too low by an amount it cannot tell you.