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.

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.

From the beginning, the series has been story driven - I began with a story idea - but research feeds it.

I don't write drafts. I write from the beginning to the end, and when it's finished, it's done.

Since the beginning, I always loved the game. When you grow up in Montreal, one day you want to be a professional hockey player. When I was six or seven, I knew that was what I wanted.

Before beginning, plan carefully.

Every new beginning comes from some other beginning's end.

The people who voted for President Obama are just beginning to wake up to exactly what they brought in. The 'change' they envisioned is not the 'change' they have gotten.

Religion has caused more harm than any other idea since the beginning of time. There's nothing good I can say about it. People use it as a crutch.

And that's how I start myself. I usually go back a couple of pages, maybe to the beginning of the chapter, and I start reading. And as I'm reading, I'm tweaking - putting in a different word, changing the syntax, putting that clause over there, you know that sort of thing.

In the beginning it was all black and white.

A palindrome is a word or pattern that instead of developing in different directions it folds in on itself so that the beginning and end mirror each other, that they are the same.

Learning is the beginning of wealth. Learning is the beginning of health. Learning is the beginning of spirituality. Searching and learning is where the miracle process all begins.

At the beginning of 1955 only about 60 percent of American homes had TVs.

Now this is not the end. It is not even the beginning of the end. But it is, perhaps, the end of the beginning.

Most musicians count at the beginning, and never count and talk to their musicians after that. They only talk to them at the end of the song. But I would count with them and talk.

What you really remember at the beginning was that you have to throw a budget together. We made some terrible mistakes at the beginning in my own budget that took us at least a year to catch up on.

There is a spirit and a need and a man at the beginning of every great human advance. Every one of these must be right for that particular moment of history, or nothing happens.

I am pleased to see that information campaigns, such as the America's WETLAND effort, are getting the message out, and people are beginning to realize that wetlands loss in Louisiana affects us all.

I started working on trying to sound like June from the very beginning.

The end comes when we no longer talk with ourselves. It is the end of genuine thinking and the beginning of the final loneliness.

Even though the future seems far away, it is actually beginning right now.

It is a curious sensation: the sort of pain that goes mercifully beyond our powers of feeling. When your heart is broken, your boats are burned: nothing matters any more. It is the end of happiness and the beginning of peace.

The American Revolution was a beginning, not a consummation.

All great deeds and all great thoughts have a ridiculous beginning. Great works are often born on a street corner or in a restaurant's revolving door.

The beginning of 1856 found me teaching in the family of a planter named Bryan, residing in Prince George County, Md., some fifteen or twenty miles from Washington.

I did all the work at the beginning up until the point where I couldn't handle the increasingly heavy art production burden alone. I needed, and got, assistance.

Art at its most significant is a Distant Early Warning System that can always be relied on to tell the old culture what is beginning to happen to it.

You know, in the beginning when your first payroll comes up and you have to borrow money to meet the payroll, you lose sleep the night before, and you say to yourself real fast, 'Well, maybe I should keep working a couple more years. It's sobering.

I think I realized it was an art form at the beginning, but it took me a really long time before I was able to view what I was performing myself as an art form.

In the beginning I had a real work problem. Every time I had job I had to convince the immigration authorities I was the only man for that job and get a special work permit until I went under contract to MGM.