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.

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.

Roosevelt's magic lay in one facet of his personality: He knew how to take the risk. No other man in public life I knew could so readily take the challenge of the new.

To do the Ozzfest again would be great. I'd like to finish with a final Sabbath album. You always feel that it is still a challenge.

People are not perfect... very often the relationships that are strongest are those where people have worked through big crises, but they've had to work through them. So the challenge to us is to work through that.

Every single book is a challenge.

Critics are sentinels in the grand army of letters, stationed at the corners of newspapers and reviews, to challenge every new author.

The lopsided attitudes of college professors pose a serious challenge to learning because students are so susceptible to becoming lopsided sheep.

I really enjoy comedy. It's a real challenge.

I had a hard time with that hockey. I hadn't grown up skating, so that was my biggest challenge. We worked on it and worked on it. But then when we first shot it, it was so hard for me.

The challenge is to practice politics as the art of making what appears to be impossible, possible.

Change can either challenge or threaten us. Your beliefs pave your way to success or block you.

My greatest challenge has been to change the mindset of people. Mindsets play strange tricks on us. We see things the way our minds have instructed our eyes to see.

I'd say that it's important for music to be there that gives you a challenge, that rearranges things in your head.

I like good stories. Quality products and character are what's important. Even if the script isn't that strong, if I challenge myself with a great character, I'll go for it.

I tend to arrive in the rehearsal process with very strongly developed ideas about what I want to do. But I don't like those ideas to be things that are not subject to change, or subject to development, or subject to challenge.

I'm not looking to set a standard... but, I believe I have offered a challenge to others with my work.

If you live a life of make-believe, your life isn't worth anything until you do something that does challenge your reality. And to me, sailing the open ocean is a real challenge, because it's life or death.

And, I'd never done Tennessee Williams, and I had done Broadway musicals, so it was a challenge.

When you remove the risk, you remove the challenge. When you remove the challenge, you wither on the vine.

After Land I wanted to continue exploring the theme but I needed a new challenge so turned to colour. I explored Bradford and produced a series of urban landscapes that I liked, but because Land had made such an impact on the general public my colour work wasn't reviewed.

The worlds I paint leave a lot to engage the imagination by hinting at what lies beyond the four edges of the painting. I think getting beyond the four edges of an opportunity or challenge is one of the basic skills you need in business.

For me, it's a bigger challenge, it's much harder to do and much more rewarding to do well, then just to think up stuff of your own, hit or miss, because you've got to see to it that you don't torpedo any of his punch lines.

This same habitual blindness to spiritual, substantive dimensions of every significant challenge continues to handicap Hollywood.

Every artist has to grow, and has to challenge themselves with a new form of expression.

The next challenge for Christianity is to remind Europeans that we are called to seek the truth.

Give yourself an even greater challenge than the one you are trying to master and you will develop the powers necessary to overcome the original difficulty.

The biggest challenge has been simulating a tornado with wind machines and dirt and debris. Right when you walk on the set, you feel the energy of a tornado. But the hardest thing is trying to get dialogue out in all of that.

I look at raising funds for The Perlman Music Program as a challenge and as a way to provide opportunities for people who care about the future of classical music.

My father instilled in me the attitude of prevailing. If there's a challenge, go for it. If there's a wall to break down, break it down.

The challenge in working in politics, particularly if you're working for a political party, is that everyone's a messenger.