The World According to John Hull
John Hull is a rarity in the derivatives businessa downtoearth academic who is is as comfortable in the trading room as he is in the classroom.
He is best known as the coauthor, 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, AJ 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 mid1980s.
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
threemonth interest rate," you would assume that the threemonth 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 threemonth 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 meanreversion 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 oneweek 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 shortterm interest rate. The average direction in which the shortterm 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 nonrecombining tree.
To explain what I mean by this, consider first the wellknown 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 nonrecombining 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 twofactor model. In a onefactor model all rates move
in the same direction over any short period of time. For example, if the
3month rate moves up, the 5year rate also moves up. In practice things
are not always this simple. Sometimes the 5year rate and the 3month rate
move in opposite directions. A twofactor model allows this to happen. We
have done some work extending our original onefactor 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 modelerto 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 twofactor 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 valueatrisk 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. Valueatrisk 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.
