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.

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 want to thank all the fans for their support over the years. Keep supporting the Knicks, we're moving in the right direction here and we will be exciting to watch and competing against the best in the NBA. Come back to the Garden for those exciting games.

Then I decided I couldn't just crawl in the corner and die, so I started putting pen to paper and wrote some songs. I had no idea what for or who I was going to work with. I tried to find my way and direction.

The laws were not made so much for the direction of good men, as to circumscribe the bad.

The mind moves in the direction of our currently dominant thoughts.

Life is an operation which is done in a forward direction. One lives toward the future, because to live consists inexorably in doing, in each individual life making itself.

On one hand, to be able to go from one direction in the sky to study such an object to another direction to study another object, and on the other hand to be able to maintain accurately the position in space.

The more one listens to ordinary conversations the more apparent it becomes that the reasoning faculties of the brain take little part in the direction of the vocal organs.

Backward is just not a natural direction for Americans to look - historical ignorance remains a national characteristic.

I have thought about Happy Days made into a movie. As far as the original cast not being a part of it, wow, I don't know who could be who!I just don't see it going in that direction. I can see the original cast doing the movie very easily though.

But I try not to become preoccupied with that because with whatever direction I follow, with whatever advice I've followed or not followed, It's landed me in New York, in a very beautiful hotel, talking to people about something that I love. So I ain't that far off.

No one could understand the bond between me and my brother. I struggled to understand the forces that drove his soul in one direction and mine in another.

I am always more interested in performance and character depiction, and my direction says as much.

You've got to think about big things while you're doing small things, so that all the small things go in the right direction.

People, like sheep, tend to follow a leader - occasionally in the right direction.

The general knowledge of time on the island depends, curiously enough, on the direction of the wind.

I didn't know Albert back then - I just learned to play that way. He and I were the only guys that played left-handed. Then left-handed people came from every direction.

As the unity of the modern world becomes increasingly a technological rather than a social affair, the techniques of the arts provide the most valuable means of insight into the real direction of our own collective purposes.

In group lesson number six I think we learned how to turn backwards and then just kind of wiggle. That wasn't really skating backward, but I guess I was going in the right direction.

I think it was probably down to the fact that we weren't together personally as a band. We weren't pulling in the same direction. I always feel if you're having a good time in the studio it actually comes across on the tape and that was a bit of a miserable album for us.

It takes little talent to see what lies under one's nose, a good deal to know in what direction to point that organ.

I mean, George Bush is a man of prayer. He talks to the lord. He tries to get his direction from the lord.

Children in a family are like flowers in a bouquet: there's always one determined to face in an opposite direction from the way the arranger desires.

Give immediate instruction to all your posts in said territory, under your direction, at no time and on no pretence to hoist, or suffer be hoisted, the English flag.

If you cry 'forward', you must without fail make plain in what direction to go.

There are a lot of times that if a detail in a scene or a beat, feels unnatural, they'll allow me to explore another direction to go until we're all comfortable with what we are doing.

It's better to hang out with people better than you. Pick out associates whose behavior is better than yours and you'll drift in that direction.

Things went in a direction that I didn't want to go. I started doing bit parts and things that are pretty much laid out for people in my position. The parts were pretty generic.

There is no great sport in having bullets flying about one in every direction, but I find they have less horror when among them than when in anticipation.

Stupidity, outrage, vanity, cruelty, iniquity, bad faith, falsehood - we fail to see the whole array when it is facing in the same direction as we.