Squeezing the Answer From Both Sides: Broadie and Glasserman on American Options

For twenty years after Boyle brought Monte Carlo to finance, there was one thing it flatly could not do, and it happened to be something the market needed constantly: price an American option.

The obstruction is not a technical detail. It is a genuine, structural conflict between how simulation works and how early exercise works, and it took two decades and some real ingenuity to get around.

The problem: simulation runs the wrong way

Monte Carlo works forwards. You start at today, draw random shocks, and walk the price path out to expiry. Then you look at the payoff. Simple, general, wonderful.

Early exercise works backwards. At any moment, the holder of an American option must decide: exercise now and take the immediate payoff, or hold on? To make that decision rationally you need to know the value of continuing, which depends on what you would do at all future moments, which depends on their continuation values, and so on back from expiry. It is a backward recursion. Trees do it naturally, precisely because a tree is a backward recursion device.

Now try to combine them. You are standing at some point on a simulated path, halfway to expiry, and you must decide whether to exercise. To decide, you need to know the option's continuation value from here. But your simulation only tells you what happened on this one path. You cannot use that, because that would mean deciding whether to exercise using knowledge of the future, and that is cheating: it is called look-ahead bias or perfect foresight, and it gives you an option that is worth far more than any real option can be, because no real holder knows what the stock will do tomorrow.

So the naive fix, run one simulation, look at each path, exercise at whichever moment happened to be best, produces a badly inflated price. And the obvious remedy, run a sub-simulation from every point on every path to estimate the continuation value, is a nested Monte Carlo whose cost explodes exponentially.

The key idea via analogy: two flawed referees who disagree in known directions

Broadie and Glasserman's solution is one of the most elegant pieces of thinking in computational finance. Instead of trying to build one unbiased estimator, which is extremely hard, they deliberately built two biased ones, and made sure the biases point in opposite directions.

The high estimator. Give the simulated option holder a small amount of foresight. Let them peek, a little, at the outcomes of the sub-simulations when deciding whether to exercise. A holder with any foresight makes better decisions than a real one, and therefore extracts more value from the option. So this estimator is biased high, and, crucially, we can prove it is biased high. It is an upper bound.

The low estimator. Now do the opposite. Use one set of simulated sub-paths to decide whether to exercise, but a completely separate, independent set to value the consequence of that decision. Because the exercise decision is now made using information that is unrelated to the outcome being valued, the holder is making decisions using a somewhat noisy, imperfect rule. An imperfect exercise rule is worse than the optimal one, and a suboptimal holder captures less value. So this estimator is biased low, provably. It is a lower bound.

Now you have the true price trapped between two computable numbers. And here is the payoff: as you increase the number of simulations, both bounds converge toward the true value, squeezing from above and below. You do not merely get an estimate. You get an estimate with a guaranteed interval around it, and you can watch the interval tighten as you spend more computation.

That is a rare and valuable thing. Most numerical methods hand you a number and a vague hope. This one hands you a number and a proof that the answer lies between here and there.

Why the bounding idea matters more than the algorithm

The specific simulated-tree algorithm Broadie and Glasserman proposed is expensive. It is a nested simulation, and its cost grows unpleasantly with the number of exercise opportunities. In raw production terms, it was largely superseded four years later by Longstaff and Schwartz's least-squares method, which is vastly cheaper and is what most desks actually run.

But Longstaff-Schwartz has no error bound. It gives you a price, and it is a lower bound in expectation, because it uses a fitted, imperfect exercise rule. It does not tell you how far below the truth you are.

The Broadie-Glasserman upper bound is what closes that gap, and the "generate a candidate exercise policy cheaply, then bound its error rigorously" pattern, refined later into the duality-based upper bound methods, is now the standard way serious practitioners validate American Monte Carlo prices. The algorithm aged. The idea did not.

Why it mattered

• It cracked a genuinely hard problem. American-style optionality is everywhere: listed equity options, Bermudan swaptions, callable bonds, mortgage prepayment, and every real-options problem in corporate finance. Before this, if your product had both early exercise and high dimensionality, you were stuck: trees could not handle the dimensions, and Monte Carlo could not handle the exercise.
• It named and killed the foresight trap. The observation that a naive path-by-path exercise rule secretly uses information the holder cannot have is the single most important thing to understand about American Monte Carlo. It is the reason a beginner's implementation always produces a suspiciously high price.
• It made confidence intervals part of the answer. Giving a valid bound rather than a point estimate raised the standard for what counts as an acceptable numerical price on a derivatives desk.
• It seeded the dual approach. The idea that you can characterise the American option's value from above, by considering hedging strategies rather than exercise strategies, grew directly out of this line of work and is one of the more beautiful pieces of the modern theory.

The honest limitations

• The original algorithm is computationally brutal. Nested simulation means simulating from every node of a simulated tree. The cost grows fast with the number of exercise dates, which makes the method impractical for a Bermudan with monthly exercise over ten years, exactly the kind of thing rates desks trade.
• The bounds can be wide. Trapping the answer is only useful if the trap is tight. For difficult problems the gap between the high and low estimators can be uncomfortably large, and closing it costs a lot of computation.
• It was overtaken in practice. Longstaff-Schwartz is faster, simpler and good enough for most purposes, which is why most production systems use it. Broadie-Glasserman's legacy is conceptual, and the bounding techniques used today are refinements rather than direct implementations.
• Both estimators are still biased. They converge to the truth, but at any finite sample size neither is correct. You are relying on the bound, not the estimate.
• It inherits Monte Carlo's usual weaknesses. Slow convergence, noisy Greeks, and complete dependence on the model being right.

The one-line takeaway

Broadie and Glasserman confronted the fact that simulation runs forwards while early exercise reasons backwards, and rather than forcing them together, they built two deliberately biased estimators, one that gives the option holder illicit foresight (too high) and one that decides using noisy information (too low), trapping the true price between provable bounds, an idea that outlived their algorithm and still underpins how American Monte Carlo prices are validated today.