Yes, our tree has an interesting shape. The center branches reflect the shape of the zero curve. When extreme parts of the tree are reached the branching pattern changes to accommodate the mean reversion.

We concluded that you cannot rely on delta hedging alone. It sounds simplistic to say that now, but back then, this was the sort of thing people were only just beginning to realize.

Yes, our tree has an interesting shape. The center branches reflect the shape of the zero curve. When extreme parts of the tree are reached the branching pattern changes to accommodate the mean reversion.

Our starting point then was trying to find a way to incorporate mean reversion into the HoLee model.

Briefly speaking, our conclusion is that stochastic volatility does not make a huge difference as far as the pricing is concerned if you get the average volatility right. It makes a big difference as far as hedging is concerned.

The HoLee model was the first term structure model. I remember reading their paper soon after it was published and as it was fairly different from many of the other papers that I had read, I had to read it quite a few times. I realized that it was a really important paper.

I didn't become interested in derivatives until 1982, 1983.

I think VAR is a very healthy development within the industry.

Alan White and I spent the next two or three years working together on this. We developed what is known a stochastic volatility model. This is a model where the volatility as well as the underlying asset price moves around in an unpredictable way.

The real challenge was to model all the interest rates simultaneously, so you could value something that depended not only on the three-month interest rate, but on other interest rates as well.

Our research led on to other things, such as the fact that exchange rates are not lognormally distributed.

In the interest rate area, traders have for a long time used a version of what is known as Black's model for European bond options; another version of the same model for caps and floors; and yet another version of the same model for European swap options.

If each of your time steps is one week long, you are not modeling the stock price terribly well over a one-week time period, because you are saying that there are only two possible outcomes.

I guess any simple idea that is really good will catch on quickly.

We started giving presentations at practitioner conferences in 1986, and since then all of our derivatives research has been stimulated by contact with practitioners.

The problem with interest rates are that you are not modeling a single number, you are modeling a whole term structure, so it is a sort of different type of problem.

There are challenges in terms of the measurement of VAR for what are known as nonlinear derivatives, where things like gamma and vega are important dimensions of the risk.

Our tree is actually a tree of the short-term interest rate. The average direction in which the short-term interest rate moves depends on the level of the rate. When the rate is very high, that direction is downward; when the rate is very low, it is upward.

One important measurement issue concerns the fat tails problem that I mentioned earlier. VAR is concerned with extreme outcomes. If the tails of the probability distributions we are using are too thin, our VAR measures are likely to be too low.

When interest rates are high you want the average direction in which interest rates are moving to be downward; when interest rates are low you want the average direction to be upward.

I would rather go to any extreme than suffer anything that is unworthy of my reputation, or of that of my crown.

It is the tendency of the social burdens to crush out the middle class, and to force society into an organization of only two classes, one at each social extreme.

Almost the only persons who may be said to comprehend even approximately the significance, principles, and purposes of Socialism are the chief leaders of the extreme wings of the Socialistic forces, and perhaps a few of the money kings themselves.

Not being a comic book fan, being thrown into that and seeing the extreme - it's taken very seriously. So I tried to do as much learning as I could about it so I wasn't mean or anything.

Virtue is defined to be mediocrity, of which either extreme is vice.

I think unleashing 3,000 smart bombs against the city of Baghdad in the first several days of the war... to me, if those were unleashed against the San Francisco Bay Area, I would call that an act of extreme terrorism.

By the time this concert ends this evening, 30,000 Africans will have died because of extreme poverty. By this time tomorrow evening, another 30,000. This does not make sense.

The whole drive of western culture, the part of it which is serious, is towards an extreme objectification. It's carried to the point where the human subject is treated almost as if it's dirt in the works of a watch.

Characters are an extreme form in Shakespeare's theater.

Assassination is the extreme form of censorship.

I like photographing the people I love, the people I admire, the famous, and especially the infamous. My last infamous subject was the extreme right wing French politician Jean-Marie Le Pen.

It may be an extreme example brought about by abnormal circumstances - but the criteria of human rights kick in, surely, precisely when the conditions are extreme and the situation is abnormal.

I should point out, however, that at first some difficulty was experienced in observing the phenomena predicted by the theory, owing to the extreme smallness of the variations in the period of oscillation.

I love extreme sports, I like snowboarding and motorcross and rollerblading and hockey.

What is going on in America is extreme. The youth cult, they worship youth so much it's almost paranoid. And LA is the Mecca of it all; they're taking it to the hilt.

If there is anything worse than international warfare, is civil warfare, and the United States was destined to experience it in the extreme of bitterness.

I am not a tree-hugger and I don't think mine is an extreme point of view.

I don't get over the wonder of it, and 'The Last Samurai' was an extreme example of that. Every day when I went to the set, I couldn't believe what I was seeing.

Texas has arguably the most extreme separation between the well off and everyday people in the United States.

Our role is to develop techniques that allow us to provide emergency life-saving procedures to injured patients in an extreme, remote environment without the presence of a physician.

I'm a bit of a worrier, to an extreme. I'll crack a joke, then worry if I've offended someone - even when they're laughing. I have a guilt complex, always worrying.

I have never seen such extreme partisanship, such bitter partisanship, and such forgetfulness of the fate of our fathers and of the Constitution.

English policy may not yet have made the definite decision to attack us; but it doubtless wishes, by all and every means, even the most extreme, to hinder every further expansion of German international influence and of German maritime power.

Yes, our tree has an interesting shape. The center branches reflect the shape of the zero curve. When extreme parts of the tree are reached the branching pattern changes to accommodate the mean reversion.

Infancy is what is eternal, and the rest, all the rest, is brevity, extreme brevity.

Ladies and gentlemen, I take office at a time in which the world is living in extreme contradictions.

The constraints of melodrama can be a great blessing, because they demand that all the characters involved - as absurd and extreme as they may initially seem - must stay utterly rooted in their own reality, or the whole project collapses.

We'll engage in pretty extreme violence in the world but, you know, the one thing that comes to humans as easily as eating or breathing or sleeping, is sex.

To the American people I say, awaken to what is happening. It is the duty of each citizen to be vigilant, to protect liberty, to speak out, left and right and disagree lest be trampled underfoot by misguided zealotry and extreme partisanship.