Building a Tree That Already Knows the Smile: Derman and Kani

In the same month that Bruno Dupire published his local volatility formula in Risk, Emanuel Derman and Iraj Kani published an article in the same magazine that reached essentially the same destination by a completely different road. Dupire's route was continuous mathematics: partial differential equations, forward Kolmogorov, calculus. Derman and Kani's route was a tree, and their version is the one most practitioners actually understand.

The two papers together mark the moment the industry stopped treating the volatility smile as an embarrassment and started treating it as information.

The problem: the tree assumes away the thing we can see

The Cox-Ross-Rubinstein binomial tree is built from a single volatility number. You choose a volatility, that determines how far the stock moves up and down at each step, and the entire lattice follows. Every node has the same volatility. Every path is drawn from the same distribution.

The market disagrees, loudly. Look at real index options after 1987 and out-of-the-money puts are priced with far higher implied volatilities than at-the-money calls. A one-volatility tree cannot reproduce that, and therefore cannot price a barrier option or an American option in a way that is consistent with the vanilla prices used to hedge it.

Derman and Kani, both at Goldman Sachs, asked the question from the desk's point of view. Can we build a binomial tree that is not given a volatility, but which is instead constructed so that it reproduces every option price we can see in the market?

The key idea via analogy: growing a tree from its fruit

A normal tree is built forwards. You pick the parameters, you plant the seed, and the branches follow.

An implied tree is built the other way. You start from the fruit, the observed option prices, and you work out what tree must have grown them.

The construction is iterative and, once you see it, quite intuitive.

Start at today's stock price, the root. Move out one time step. In a standard binomial tree, you would need to know the volatility to place the up-node and the down-node. In an implied tree, you instead ask: what up-node, down-node and transition probability would exactly reproduce the market prices of the options that expire at this time step? That is a small system of equations. Solve it, and the nodes are placed, not by assumption, but by the market.

Now move out another step. You know where you are, and you know the prices of options expiring at the next date, so you can determine the next layer of nodes and probabilities in the same way. Repeat. The tree grows outwards, one layer at a time, each layer calibrated to the options expiring at that maturity.

What emerges is a lattice in which each node has its own local volatility. Down in the lower-left region of the tree, where the stock has fallen, the up and down moves are large: volatility is high. Up in the top-right, where the stock has rallied, the moves are gentle: volatility is low. The tree has automatically encoded the market's belief that a falling market is a volatile market.

That is precisely Dupire's local volatility, arrived at discretely.

Why the discrete version earned its place

You might ask why anyone needs the tree version when Dupire's formula is more general and more elegant. Two reasons, and they are practical ones.

First, trees handle early exercise. Most listed equity options are American, exercisable at any time. Pricing them requires the backward-induction step of comparing "hold" versus "exercise" at every node, which is trivial in a tree and awkward in a PDE. Derman and Kani's implied tree can price American options while fitting the smile, which was exactly what the desk needed.

Second, it is comprehensible. A trader can look at an implied tree and see, node by node, what the model believes. That transparency matters enormously in a business where the person taking the risk needs to understand the machine. Dupire's formula is a beautiful abstraction. Derman and Kani's tree is a picture you can point at.

Derman, Kani and later Neil Chriss extended the idea to implied trinomial trees, which give three branches instead of two and turn out to be considerably more stable numerically, because the extra degree of freedom lets you fix the node positions in advance and solve only for probabilities.

Why it mattered

• It brought local volatility onto trading floors. Dupire proved it; Derman and Kani made it usable. The implied tree was, for years, the practical mechanism by which the smile got into exotics models at real banks.
• It made the smile mean something. Before this, "the smile" was a description of a market quirk. After, it was a statement about future volatility conditional on the price level, and you could read that statement directly off the tree.
• It fits American options and the smile at the same time. No other technique of the era could do both.
• It made local volatility teachable. Almost every exposition of local volatility begins with a picture of an implied tree, because it is the only version of the idea you can draw.

The honest limitations

• It is numerically delicate, and it can fail outright. This is the model's notorious weakness. The iterative construction solves a small system of equations at every node, and nothing guarantees the solution is sensible. Poorly behaved market data, or the ordinary noise in quoted prices, can produce negative transition probabilities, which is arithmetic nonsense: the model is telling you an outcome has less than zero chance of happening. Practitioners developed a range of overrides and smoothing techniques to keep the tree well-behaved, and much of the subsequent literature is about repairing exactly this. The trinomial version was introduced partly for this reason.
• The problem is worst in the tails. The far reaches of the tree, deep in the money and far out, are calibrated to option prices that are illiquid, wide or simply not quoted, so the tree is being fitted to data that barely exists. Unsurprisingly, that is where it breaks.
• It inherits every conceptual flaw of local volatility. The tree fits today's smile exactly and predicts, wrongly, how the smile will move tomorrow. Its hedge ratios are therefore systematically biased, and its forward-starting volatility dynamics are unrealistic. Fitting today's prices is not the same as describing tomorrow's world.
• It is a snapshot, not a dynamic. The tree is rebuilt every day from that day's prices. It has no memory and no theory of why the smile is the shape it is, so it cannot tell you whether today's smile is cheap or expensive.
• It scales badly. Trees are fine for one underlying. They become impossible for several.

The one-line takeaway

Derman and Kani inverted the binomial tree, growing it outwards from observed option prices so that each node acquires its own local volatility and the tree reproduces the whole smile by construction, delivering local volatility in a form that could price American options and that a trader could actually look at, at the cost of a construction that is fragile enough to produce negative probabilities when the market data is anything less than pristine.