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

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.

At the end, the realization is that she had to get to a place in her life where she could drop her guard and make peace with the fact that whether she had a small amount of time, that she had to kind of live it completely through, instead of living by the rules.

President Johnson and I have a lot in common. We were both born in small towns and we're both fortunate in the fact that we think we married above ourselves.

The fact is I've been in Massachusetts for the last two weeks, and it seems over the last few days that the price is increasing by the hour at the pump, so there needs to be an aggressive investigation.

In fact, it is a farce to call any being virtuous whose virtues do not result from the exercise of its own reason.

Working-class Americans have waited too long, close to a decade in fact, for an increase in the minimum wage. This has been the second longest period without a pay raise since the Federal minimum wage law was first enacted in 1938.

The fact is, Latinos now own their own lives and I believe they can afford to look back without judgment.

The fact that it's science fiction gives you the license to do anything you want to do.

It has to do with the fact that Ford, for all his greatness, is an Irish egomaniac, as anyone who knows him will say.

I never knew a more presumptuous person than myself. The fact that I say that shows that what I say is true.

I think the fact that I'm so well known to be gay makes it very difficult to have a convincing relationship with a woman on screen. It wouldn't be at all difficult for me to kiss a woman - I'll kiss a frog if you like.

People expect us to be different, but we're not. We're very similar people, and it's because we're so similar and close to each other that we make each other laugh - in fact we make each other laugh more than we make anyone else laugh.

I can't tell you how hard I worked the last year. In fact, I worked so hard that I know I can't maintain that same work level in 2001, so I've got to quit something.

The miracle of light pours over the green and brown expanse of saw grass and of water, shining and slowly moving, the grass and water that is the meaning and the central fact of the Everglades. It is a river of grass.

The key to accepting responsibility for your life is to accept the fact that your choices, every one of them, are leading you inexorably to either success or failure, however you define those terms.

In fact I have nightmares about having children. I want to carry a baby and feel the life within me and in my dream, I do. But every time after it's born, there's this incredible fear, this pounding pulse of fear. It's a real bad nightmare.

All my songs were solo voices. Just me singing. In fact, that was the gimmick - no gimmick. Just singing straight with not too much background.

The fact is that young people are going to have sex whether you like it or not.

We have stabilized our economy. We took over a very sick economy, and we were hemorrhaging 750,000 jobs a month. We have stopped the hemorrhaging. In fact, we had 140,000 job growth last month. And that's what I call progress.

As a matter of fact they'd blacken us down. I guess there's a reason that according to what the Caucasian wanted us to look like. He wanted us to look-if we were Black, then he had his idea of what we look like.

I felt pretty comfortable with Westerns, apart from the fact I couldn't ride.

There are in fact two things, science and opinion; the former begets knowledge, the later ignorance.

Every day some new fact comes to light - some new obstacle which threatens the gravest obstruction. I suppose this is the reason which makes the game so well worth playing.

There is nothing as mysterious as a fact clearly described. I photograph to see what something will look like photographed.

In fact, words are well adapted for description and the arousing of emotion, but for many kinds of precise thought other symbols are much better.

It has long been known that the chemical atomic weight of hydrogen was greater than one-quarter of that of helium, but so long as fractional weights were general there was no particular need to explain this fact, nor could any definite conclusions be drawn from it.

That the apostolic office is temporary, is a plain historical fact.

I knew exactly what I was, and there was no hang-up with me. None whatsoever. The fact that the pigment of my skin maybe being lighter brown than other people of my race, maybe some of them, but you know our race has all colors.

Facts, according to my ideas, are merely the elements of truths, and not the truths themselves; of all matters there are none so utterly useless by themselves as your mere matters of fact.

Science is organized common sense where many a beautiful theory was killed by an ugly fact.