My Life as a Quant (38 page)

Iraj and I developed an alternate view of the future tree of index levels. We pictured the usual constant-volatility binomial tree redrawn on a flexible sheet of rubber, which we could then stretch and distort so as to resemble the tree of
Figure 14.7
. On this deformed tree, the size of the moves in the index at each node of the tree could differ, representing a varying volatility whose value differed from node to node. In theorists' jargon, the index would have a varying
local volatility
;by “local volatility” we meant the short-term volatility of the index at a particular future level and time. The constant or global volatility of
Figure 14.6
was inconsistent with the market's tentlike implied volatility surface of
Figure 14.2
. There must be, we figured, an
implied binomial tree
whose local volatility could be chosen to match the market's implied volatility surface. We expected it to look like the tree in
Figure 14.7
, in which the local volatility of the index rises as the index falls and vice versa, in order to reflect the surface's variation with strike.

Figure 14.7
At left, an implied binomial tree of future index moves. Each future percentage move has a local volatility that rises as the index level drops. Can one deduce the shape of the implied binomial tree from the implied volatility surface at right?

It was easy to imagine such a tree. It was even easy to build such a tree by literally making up a rule for how local volatility varied within the tree and then constructing it. Given such a tree, you could use it to calculate the prices of many different options and then plot their implied volatility surface. We could see that it was possible to pick a local volatility whose variation produced a realistic-looking volatility surface. But the ultimate problem we faced was the inverse of what we were doing. We needed to start with the implied volatility surface the market presented and deduce from it the unique local volatilities that reflected it. The implied volatility surface was the primary object, and the whole procedure we envisaged would only constitute a true theory if you could extract from it a unique implied binomial tree.

Throughout 1993, as the QS group continued to build the more elaborate risk-management systems that occupied most of our time, we continued to ruminate over the smile. In spare moments we tinkered with implied trees, still uncertain as to whether there was a truly unique relationship between the volatility surface in
Figure 14.7
and the tree we hoped it implied. We knew we could go from the tree to the surface, but what was the incontrovertible way to go from the surface to the tree? We discussed it with Dave Rogers and his traders, who, because of our sheet-of-rubber analogy, always called it the
flexible tree
. We built versions of it and used them to price and hedge varieties of options, but, too busy with supporting the software needs of the desk, we avoided an all-out attempt on the question of uniqueness.

The relationship between surface and tree reminded me of the lecture by Mark Kaç I had heard as a graduate student at Columbia thirty years earlier, when he solved the question of hearing the shape of a drum. Physicists call this an inverse-scattering problem because, whereas most models in physics proceed from the physical law to the results, inverse problems work backwards. Newton's theory of gravitation, for example, commences with the law of gravitational attraction between the sun and the planets, and deduces the planetary orbits. Inverse scattering problems go in reverse—given the observations, they ask, what law would produce them? Imagine, for example, that astronomers observed some strange perturbation in the orbit of the earth. What change in the law of gravitational attraction would account for it?

Our search for a method to extract a unique implied tree from the volatility surface was an inverse scattering problem. This approach is more typical of financial modeling than it is of physics. In physics, the beauty and elegance of a theory's laws, and the intuition that led to them, is often compelling, and provides a natural starting point from which to proceed to phenomena. In finance, more of a social than a natural science, there are few beautiful theories and virtually no compelling ones, and so we have no choice but to take the phenomenological approach. There, much more often, one begins with the market's data and calibrates the model's laws to fit it. This calibration is a kind of inverse-scattering approach, too, and it was what we were trying to do in our attempt to construct implied trees.

Sometime in late 1993 I went to visit our trading desk in London, where I also gave a talk at a
Risk
magazine conference on exotic options. Between conference sessions, I met Graham Cooper, the new editor of
Risk
, and also ran into John Hull. During our conversations I told them what Iraj and I had been exploring. Graham and John told me that they had heard that Bruno Dupire at Paribas Capital Markets in London and Mark Rubinstein, the Berkeley finance professor who was one of the codevelopers of the original constant-volatility binomial tree model, had been tackling the same problem. Worried about giving away proprietary information to our competition, I called Dave Rogers in New York and quickly got his approval to allude in public to what Iraj and I had done. I hurried back to my hotel room and quickly appended a few transparencies to my presentation to describe our implied tree approach. After my talk, Graham invited me to submit an article on our work to
Risk
, while John, hearing me describe our trees sometimes as “flexible” and sometimes as “implied,” nudged me towards the use of “implied.”

With competition at our heels, Iraj and I anxiously rededicated ourselves to proving the uniqueness of our tree. Most of our day was typically spent enhancing the desk's trading models, responding to requests for pricing new structures, and building trading software. Whenever we had spare time away from desk support, we tried again to define a scheme for uniquely extracting the local volatility at each future node of the implied binomial tree.

We began with the market's implied volatility surface on a given day, as illustrated in
Figure 14.2
. We then constructed a binomial tree like the one in
Figure 14.8
. Each shaded triangle in this tree, at each index level and future time, evolves with a different local volatility whose magnitude is represented by the degree of shading. Higher index levels correspond to lower (paler) volatilities; lower index levels correspond to higher (darker) volatilities. How pale or how dark must you choose them to match the initial implied volatility surface of
Figure 14.2
? That was the question.

Figure 14.8
An implied binomial tree with varying local volatilities represented by the shaded triangles in the tree.

The
local
volatility in
Figure 14.8
is a local quality of the tree, the microscopically viewed volatility within each single small internal triangle. In contrast, the implied volatility in
Figure 14.2
is a
global
quality, a wide-angle overview of all the internal triangles seen from 30,000 feet. We viewed the implied volatility of an option as the average
1
of all the local volatilities that the index will experience during the life of that option.

Consider the option whose expiration and strike correspond to the time and index level at the location of the small flashlight in the next-to-last row of the tree in
Figure 14.9
. The value of its implied volatility depends upon the values of the local volatilities in the shaded right-striped triangles; those are the local volatility regions that the index can traverse in moving towards the strike during the life of the option. It's useful to think of the option expiring at the flashlight as an X-ray source that illuminates all the local volatilities in the internal right-striped triangles in the tree.

Figure 14.9
The implied volatility of the option whose expiration and strike lie at the circle illuminates the local volatilities in the right-striped region of the tree.

Similarly, the option whose strike lies at the location of the lantern in the tree of
Figure 14.10
illuminates the local volatilities in the shaded left-striped triangles.

Figure 14.10
The implied volatility of the option whose expiration and strike lie at the location of the lantern illuminates the local volatilities in the left-striped region of the tree.

The option with a strike at the flashlight illuminates one part of the tree, while the option whose strike lies at the lantern illuminates another part. But no single option, neither the one struck at the lantern nor the one struck at the flashlight, sheds light on just one triangle in the tree, the single node whose volatility was the obscure object of our desire.

We kept struggling. We wanted a set of options that illuminated the volatilities at just one internal node. But each scheme we tried failed—there seemed to be no recipe for the local volatility at a single node.

Then, one day, as we played around with a five-row toy version of our tree on a spreadsheet, we found something miraculous happened, something so strange that for a few minutes we thought it was due to a programming error in the spreadsheet. We noticed, almost by accident, that if we used
three
distinct option strikes to illuminate the interior of the tree, two of them with adjacent earlier strikes and one with a strike one period later—if, so to speak, we directed X-rays into the tree from three different angles—the illumination canceled everywhere except at the single node where they intersected. This is illustrated in
Figure 14.11
. It was astonishing: We had found an algorithm that determined the local volatility at a single node in terms of the market implied volatilities of the options with strikes at the three surrounding nodes.

Figure 14.11
How three options can illuminate one node. (a) Two options with adjacent strikes illuminate the triangles to their left. (b) A single option expiring one period later illuminates all the previous triangles plus one more. (c) Subtracting the earlier expirations from the later leaves just one internal triangle illuminated.