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The World According to John Hull

John Hull is a rarity in the derivatives business-a down-to-earth academic who is is as comfortable in the trading room as he is in the classroom. He is best known as the co-author, with Alan White, of the Hull­White interest rate model, which earned both men wide acknowledgment as seminal thinkers in this field. He moves seamlessly between his role as a professor of finance at the University of Toronto and head of his thriving consulting business, A-J Financial Systems. The firm's work on swap option pricing is embedded in Renaissance Software's Opus system, and the firm has also been active in training and consulting activities.

Hull's second textbook on derivatives, Options, Futures, and Other Derivatives, has just been released in its third edition by Prentice Hall. The book's ambition is to propose "a unifying approach to the valuation of all derivatives." Hull was interviewed in September by editor Joe Kolman.

Derivatives Strategy: When did you start getting interested in derivatives?

John Hull: I didn't become interested in derivatives until 1982, 1983. As a result of an article I had written on plain vanilla foreign currency options, I got a call from a bank in Toronto asking me to address a group of foreign currency dealers. I suggested to Alan White that he come along. We had a very interesting day. We took the traders through the Black­Scholes pricing model and we did some Monte Carlo simulations to show how delta hedging works.

One of the participants pointed out that delta hedging does not work very well in practice because volatility bounces around. We suggested that maybe it would work better if you used the latest information about volatility when you are calculating the hedge parameters. He was adamant that it does not work well at all.

As a result of that interaction, we agreed we would do some further Monte Carlo simulations with volatility moving around. And the participant was absolutely right, of course. Delta hedging does not work very well when volatility moves around because there is a whole source of uncertainty that you are not hedging against. 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. We looked at both the pricing of options and hedging of options when our model is assumed.

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.

DS: When you started looking into the limitations of the Black­Scholes model, was it like discovering the Emperor had no clothes?

JH: That is much to strong a statement. The Black­Scholes model is a robust model that has stood the test of time. Our stochastic volatility research revealed more of a hedging problem than a pricing problem. 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.

Our research led on to other things, such as the fact that exchange rates are not lognormally distributed. There is a "fat tails" effect, so extreme outcomes are more likely than the lognormal distribution would suggest. Traders were beginning to realize this in the mid-1980s.

We started giving presentations at practitioner conferences in 1986, and since then all of our derivatives research has been stimulated by contact with practitioners. For academic researchers, this is perhaps unusual. Academics tend to read what other academics have written and say, "Well I could do a little bit more on this," and then do a little bit more. Virtually all our ideas have come from talking to practitioners about the sort of problems that they are having.

DS: So, much of your career has been dedicated to improving on the Black­Scholes model.

JH: Certainly the stochastic volatility work falls into that category. As Black­Scholes doesn't take into account the possibility that the counterparty may default, I guess we could say that our credit risk research also does so. In a very general sense, nearly all the research in this field is about improving Black­Scholes, because if you just sat back and said that Black­Scholes does everything, there would not be anything more to do.

DS: Can you describe what led up to what is now referred to as the Ho­Lee model?

JH: We got interested in interest rate research back in 1987, when there was a sense that as far as stock options, foreign currency options, futures options and index options were concerned, the Black­Scholes model covered it. There were small bits of tweaking going on such as our stochastic volatility research, but basically the Black­Scholes model is a pretty robust model.

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. As I was considering this problem, I read the Ho­Lee paper, which was the first attempt to model a term structure and was published in 1986.

DS: Before that, what did people do?

JH: Before that, people could only model one interest rate at a time. So if you said, "I've got a derivative that depends on the three-month interest rate," you would assume that the three-month rate was lognormal, and you would use the Black­Scholes model. But you would not be saying anything about any other interest rates. 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.

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. So I reworked the paper to make sure that I thoroughly understood it. I noticed that Ho­Lee did not have any mean-reversion in their model. This worried me a little bit, because I think that most of us would agree that when interest rates become really high, there are economic forces pulling them back down again. And when interest rates become very low there are economic forces pulling them back up.

Our starting point then was trying to find a way to incorporate mean reversion into the Ho­Lee model. The first thing that we did was to develop all the mathematics associated with an extension of the Ho­Lee model incorporating mean reversion. Later on, we developed some numerical procedures for implementing the model. These involved the use of a trinomial tree.

DS: Would you explain just what a trinomial tree is and why it's more accurate than other trees?

JH: The usual tree is the binomial tree developed by Cox, Ross and Rubinstein in the 1970s for stock prices. In that tree, the stock price starts at a certain level and then in each small time interval, there is either an up movement or a down movement.

Now of course, in reality, the stock price can exhibit a whole spectrum of changes. So this model is a simplification. 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. But when you put many time periods together, it turns out that you have got a pretty accurate representation of all the things the stock price can do.

This binomial tree was very well established; everybody used it from the late 1970s onward. The reason it does not work very well for interest rates is because of mean reversion. 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.

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. You need an extra degree of freedom in a tree to incorporate this mean reversion. That is where the trinomial tree idea comes from.

DS: That sort of allowed you, in your mind, to make the models much more accurate, because you took into account that mean reversion was in fact happening in the real world.

JH: 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.

DS: And what was the reaction to this model when it was first published?

JH: I think it has always been very well received. We have given presentations at practitioner conferences explaining it. Now the model has been refined and extended. One of the things underlying the Ho­Lee model and the first version of our model was that the interest rates had to be normally distributed. We have now developed a version of the model that accommodates a wide range of probability distributions.

DS: At this point, there's no shortage of models competing with yours.

JH: One alternative is the Heath, Jarrow, Morton [HJM] model, which is more general than ours. It gives the user much more freedom in the choice of volatility assumptions. The problem with the model is that it leads to a non-recombining tree.

To explain what I mean by this, consider first the well-known Cox, Ross and Rubinstein binomial tree. This is a recombining tree because an up branch followed by a down branch leads to the same node as a down branch followed by an up branch. After 1 step there are 2 nodes; after 2 steps there are 3 nodes; after 3 steps there are 4 nodes; and so on. After n steps there are n+1 nodes.

An HJM tree is non-recombining because an up branch followed by a down branch does not in general lead to the same node as a down branch followed by an up branch. This means that after 1 step there are 2 nodes; after 2 steps there are 4 nodes; after 3 steps there are 8 nodes; and so on. After 30 steps there are about 1 billion nodes. This is difficult to handle.

HJM propose a two-factor model. In a one-factor model all rates move in the same direction over any short period of time. For example, if the 3-month rate moves up, the 5-year rate also moves up. In practice things are not always this simple. Sometimes the 5-year rate and the 3-month rate move in opposite directions. A two-factor model allows this to happen. We have done some work extending our original one-factor model and our trinomial tree technique to accommodate two factors.

DS: So the thrust of your recent work is adding sophistication and flexibility to the existing models.

JH: That is the general aim of a modeler-to become closer to the way the real world actually works. For interest rate models, one factor captures about 75­80 percent of what we actually observe. The second factor captures perhaps another 15 percent of movements. Once everyone is comfortable with two factors it is natural to consider a third factor to capture some of that last 5­10 percent.

DS: Do you think we are at that stage, that we have captured 90 percent of the real world?

JH: I guess we are in the 90­95 percent range with a two-factor model But let us not forget that an important objective for the researcher is to produce a model that traders feel comfortable with as well as one that captures reality as closely as possible. 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. Practitioners are so comfortable with these models that they are unlikely to switch from them. Only recently have researchers fully recognized this and developed ways of extending the models so they become complete models for how the yield curve can evolve.

I think this is fascinating research. Alan and I have been working on some of the implementation issues involved in it. I think it is an area where there will be many exciting developments over the next few years.

DS: Were you surprised by the sudden burst of interest in value-at-risk two years ago?

JH: I guess any simple idea that is really good will catch on quickly. It is natural to ask a question such as, "How much could we lose over the next ten days?" It is something everyone can relate to. You've got to be a quant to understand what it means to say, "Our gamma exposure is 372.5," but a statement such as, "I am 99 percent certain that we will not lose more than $3 million over the next ten days" is relatively easy to understand. Value-at-risk is a simple composite risk measure. Previously we needed a series of Greek letters such as delta, gamma, vega, and so on to describe different aspects of risk.

DS: The VAR measure is simple enough so it can be adopted by a much wider group than any model involving the Greeks.

JH: I agree. I think VAR is a very healthy development within the industry. 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. I think that we are in relatively early days as far as VAR research is concerned. Alan White and I have not as yet written anything on VAR but we have done a number of consulting projects in this area and expect to do more.

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. Another important measurement issue concerns how variables move together. Often an extreme movement in one market variable will precipitate extreme movements in other market variables. These are both areas Alan and I are currently researching.