Fitting the Smile Without Breaking the Tree: Rubinstein's Implied Binomial Trees

1994 was the year the volatility smile got solved three separate times. Dupire did it with calculus. Derman and Kani did it with a tree grown forwards. And Mark Rubinstein, delivering his presidential address to the American Finance Association, did it with a tree grown backwards, and in doing so avoided the problem that plagued the other tree.

It is worth noting who was speaking. Rubinstein was one of the three authors of the original binomial model in 1979. He was returning, fifteen years later, to fix the assumption his own model had rested on.

The problem: the tree assumes a bell curve, and the market says otherwise

The standard binomial tree, Rubinstein's own creation, has a hidden consequence. Because every step is the same size and every up-move has the same probability, the distribution of the stock price at expiry is a binomial distribution, which converges to a lognormal. A neat, symmetric, thin-tailed bell curve.

The market's actual implied distribution, as Breeden and Litzenberger had shown you could measure, is nothing like that. It is skewed to the left and it has a fat left tail. Investors price a real chance of a serious crash and a smaller chance of a huge rally. The smile is the visible fingerprint of that asymmetry.

Rubinstein's question: can we keep the tree, which is wonderfully practical, but let its ending distribution be whatever the market says it is?

The key idea via analogy: start with the destination, not the journey

Derman and Kani grew their tree forwards, layer by layer, solving for each node's position as they went. It works, but the equations at each node can go bad, sometimes producing negative probabilities.

Rubinstein reversed the whole construction.

Step one: get the ending distribution. Look at the market prices of options expiring at some maturity. Extract from them the risk-neutral probabilities of the stock ending at each possible final level. This is the Breeden-Litzenberger idea, applied to a discrete set of tree nodes: you are asking, what probability does the market assign to each terminal node?

Step two: smooth it. Real option prices are noisy, sparse and sometimes mutually inconsistent. If you force an exact fit, you will get a jagged, implausible, possibly negative distribution. So Rubinstein does not force an exact fit. He solves an optimisation: find the probability distribution that is as close as possible to a smooth, sensible prior (a lognormal, say) while still being consistent with the observed option prices, in the sense of reproducing them within the bid-ask spread.

That is the pivotal design decision. Derman-Kani says "fit the data exactly and hope it is clean." Rubinstein says "the data is not clean, so fit it as well as it deserves to be fitted, and prefer smoothness where the data is silent." In practice that is an enormously more robust engineering choice, and it is why Rubinstein's tree does not produce negative probabilities: non-negativity is imposed as a constraint in the optimisation.

Step three: grow the tree backwards. Once you have the terminal probabilities, you can work backwards through the lattice. The probability of being at any interior node is just the sum of the probabilities of the terminal nodes reachable from it, and from those node probabilities you can recover the transition probability at every branch. The whole tree, including its local volatilities everywhere, is determined by the ending distribution plus the requirement that it recombine.

The result is a tree whose branches have different volatilities in different regions, high where the stock has fallen, low where it has risen, exactly reproducing the market's skew, but built by a procedure that cannot blow up.

Why it mattered

• It made implied trees usable in production. The negative-probability problem was not theoretical. It genuinely broke Derman-Kani trees on ordinary market data. Rubinstein's formulation, by making non-negativity a constraint rather than an outcome, is well-posed by construction.
• It introduced the "fit smoothly, not exactly" philosophy. This has become the standard approach to volatility surface construction across the industry. Nobody fits raw quotes exactly any more. Everybody fits a smooth, arbitrage-free surface that respects the data without being enslaved to it. Rubinstein is where the discipline learned that lesson.
• It made the implied distribution the central object. Rubinstein's framing puts the terminal risk-neutral distribution front and centre, rather than treating it as a by-product. That is now how practitioners think: the smile is the distribution, and the model is just a way of getting there.
• It priced American and exotic options consistently with the smile. Like Derman-Kani, and unlike Dupire's PDE, it retains the tree's natural handling of early exercise.
• It documented the crash-shaped distribution. Applying the method to post-1987 S&P index options, Rubinstein showed a distinctly skewed, fat-left-tailed implied distribution that looks nothing like the pre-crash one. The market's fear of crashes is measurable, and it changed on a specific date.

The honest limitations

• The smoothness prior is a choice, and it shapes the answer. How much smoothness? Relative to what prior? Different choices give different distributions, especially in the tails where there is little data. The method is robust, but it is not objective: the answer depends on how much you trust the market versus how much you trust your smoothing.
• The single-maturity construction is a real constraint. The basic method is built around the options expiring at one date. Getting a consistent picture across the whole surface, many maturities at once, is harder, and the multi-maturity extensions are not as clean.
• It has the same dynamics problem as all local-volatility approaches. The tree fits today's smile. It says nothing convincing about how the smile will evolve. Its predicted hedge ratios are therefore systematically biased in the same way Dupire's are, and for the same reason: volatility in the model is a rigid function of price and time, not a genuinely random thing.
• The tails remain guesswork. Deep out-of-the-money options are illiquid or unquoted. The distribution's tails, which are exactly what a risk manager cares about, are being determined mostly by the smoothing prior rather than by data. That should be stated more often than it is.
• It is a snapshot with no theory. The tree tells you what the market believes today. It has no view on whether that belief is reasonable, no memory of yesterday, and no ability to say the crash premium looks expensive.

The one-line takeaway

Rubinstein rebuilt his own binomial tree from the outside in, starting from the market's implied ending distribution and growing the tree backwards, and by choosing the smoothest distribution consistent with noisy prices rather than forcing an exact fit, he produced an implied tree that reproduces the volatility smile, handles early exercise, and, unlike its contemporaries, cannot generate negative probabilities.