Goldman was, and the whole of Wall Street was experiencing a burgeoning in the

application of financial models and quantitative financial models to the, to

fixed income in particular. And people, Black Shoals had appeared in

1973 and people were now busy extending Black Shoals and the methodology of Black

Shoals to other sectors. And the disk I would thought a Goldman

was a fixed income options treasury trading disk, and the big battle at that

time was to try to extend Black Scholes to work for options on treasury bonds.

And fixed income was a very good area for applying Black Scholes and quantitative

technology because, when you deal with fixing com productions.

For example Treasury Bonds, they have payments every six months and the

principle back in 30 years. And so there are a lot of fixed points

that you could describe very accurately, mathematically, and it makes the whole

field actually much more amendable to mathematical modelling than equities.

Okay.

>> How are people modelling these products and how are they using the exisiting

models to price and trade these products? Okay, so when I arrived I was set to

work, I was in a quantitative strategies area but I was set to work other fixed

income options desk that were selling options on Treasury bonds.

Options on Treasury bonds were kind of the analog of credit default swaps being

a hot product in the, in the 2000s, that was the hot product of 1985.

the economic reason was that interest rates had been coming down since the

Carter era in the late 70s. And treasury rates, long term Treasury

bond yields that sort of hit 16 or 17% and they were now coming down to 10 or

11%. And a lot of the investors in the world

who invested in fixed income were used to earning 15, 16% and couldn't earn that

anymore. And so what they were doing was, they

were selling call options on the particular Treasury bonds that they owned

to try to get a little extra premium. In exchange for giving away the upside as

they went up, the idea being to enhance their yield.

And so this was a big a big area of application for Treasury bond trading

desks and option trading desks. And for quantitative people who are

trying to build models to understand these things.

When I first came to work in the fixed income strategies area, I worked for a

guy called Ravi Dattatreya, who had actually built the first model, that I

will shortly describe, that I set about modifying.

And he had a very pragmatic attitude towards things, and one day he said to

me, you don't really need to much skill over here, all you need is addition,

subtraction, multiplication and division. And most of the time you don't need

division. And he kind of had a valid point, which I

think isn't valid anymore now. I mean that people on Wall Street, both

at the quantitative level and at the trading level are much less numerate and

much less mathematically sophisticated than they are now.

And you could get away with a lot less advanced skill and in a way before I

launch into describing what I worked on. It was kind of amateur heaven in a sense

that nobody, there weren't a lot of option trading books, there weren't a lot

of option valuation books. And if you were reasonably smart and came

in and had a good background, you were expected to pick this stuff up from a few

papers and start working. the product that I started to work on was

pricing options on Treasury bonds. And the natural way which the guy I

worked for of he had first modeled this and this was a common practice on Wall

Street, was to treat a Treasury bond as a kind of stock.

And just apply Black-Scholes to it by moving the model across, by treating the

bond as the under-liar instead of the stock price.

And so you treated a bond as a stochastic variable whose price could vary in the

future, and you treated the, the coupon unpaid somewhat as a dividend and you

modeled it's future evolution. there was one problem with this approach,

and that's that, if you look at a stock, thirty years from now a stock can take on

any uncertain price you can imagine. Whereas a bond, if there's no credit

risk, will always pay you back its principal in thirty years.

people in the business call this a pull to par in the model.

That one price has to go back to its initial principle value.

And so if you use Black Scholes to model the stochastic evolution of a bond price,

it's good for the first few months because you can't tell that the bond is

eventually going to revert to par. But if you go out to 3 or 4 years, the

fact that it's got a finite time to maturity starts to make a big difference.

If I can gesticulate to explain what I mean.

a stock price evolves out into the future with more and more uncertainty.

The bond price goes up, but then it has to come back to par after 30 years, and

the technical term for that is that, it's a Brownian Bridge for the stochastic

process. Meaning, it's a bridge between its

current price, and its terminal price at expiration, which you know.

And all the uncertainty is somewhere in the middle, or the maximum uncertainty's

somewhere in the middle. And, nevertheless it wasn't a bad model

to use for, for short term options on Treasury bonds.

For longer term options, people came up with all kinds of, I want to say kluges

or adjustments freely pragmatic ones to try to make the price process be a

Brownian bridge. So for example Ravid came up with this

idea that, instead of modeling the bond price as the sarcastic variable, you

model the bond price as a yield to maturity as a log normal sarcastic

variable. Making the yield to maturity log normal

means it never goes below zero, it's always positive and it can get very large

but since it's a yield to maturity, it doesn't affect the price that much.

When you get close to expiration, even a yield of 10,000% has no effect on the

price one day before expiration, so the price does behave like Brownian bridge.

so there is an improvement, it wasn't really satisfactory for a bunch of

reasons. the predominant one being that, like

Scholes when you, when you do When you do stock options on two different underlying

stocks. Say you do Apple and Google, an option on

Apple and an option on Google pretty much have nothing to do with each other.

You're free to model them independently. You only have to worry about the relation

between Apple and Google, and the correlation and the covariance, if you

did an option on the combined index that involved Apple and Google.

But when you come to bonds, if you want to do a five year option on a 30

year bond, and you also want to do a three year option on a 30 year bond.

You can't actually pretend that those two options are independent because the, the

five year option will be a three year option two years from now.

And so there's a relation between bonds that get shorter in maturity as time

passes and options whose expiration changes as time passes.

And so you actually implicitly involved in modeling the whole yield curve.

You cannot model one bond as an independent entity.

Another way to see this is that in Black Scholes, you have to discount the

expected payoffs at the riskless. And when you Apply Black Shoals to bonds,

you have to discount the expected value of the option on the bond, at the

riskless rate. But the riskess itself is a reflection of

the bond price. And so implicity are already modelling

two bonds, when you treat the riskless rate, and the underlying bond has

separate instruments. And in fact, Ravi taking some interesting

approaches to trying to do this, he tried to embed in the crude Brownian Bridge

Model that treated yield as an independent variable.

He tried to make the short discount rate move parallel to the long term yield to

maturity, to reflect the fact in a crude way that yields always tend to move up or

down together, or they're not really one for one.

so, this was the way people did things.

>> Can you describe how you, Fisher Black, and Boltoy got into a collaboration?

And how this idea of modeling short rates came to be?

>> Yes, so we were actually.

Fisher Black and Boltoy were actually in the equities division, which is were

Fisher was located. Boltoy worked for him.

And I was in fixed income, working with the bond option desk.

And I spent my first two or three months rewriting the bond option model, fixing

some technical errors. Trying to build a calculator, meaning a

front end, there were no calculations in those days, and I built a front end for

people to use the model. I had a lot of experience at Bell Labs

building front ends in units. And our Editor user interface that

actually make it, made it very easy for salespeople to talk to clients, model a

deal, save a price, talk to them the next day modify it a little bit.

And it was kind of interesting that, fixing up the model helped their business

but I would almost argue at the beginning that adding a good user interface.

And good ergonomics help the business much more than actually improving the

model to some extent. And when I finished that they had

meanwhile involved Fischer who was obviously the world expert on options in

trying to model the whole yield curve. Because we all understood pretty clearly

that if you want to rebuild a model for options and bonds, you actually had to

model the yield curve consistently. And so they sent me to interview with

Fisher, and I joined the collaboration with him and Bill Toy up in equities to

try to build a better model that had no arbitrage violations.

And would model the whole yield curve and all options and fix income instruments

derivative on the yield curve. It was pretty clear to us that we had to

start with a one factor model. Although, late, later we actually tried

to extend it to two factors. Because, Black Scholes is a one factor

model and, we all had a sort of a pragmatic idea that, you start simple and

add complexity later. So, if you are going to model the whole

yield curve, and you are only going to use only one factor, the natural thing is

to use the short rate, because. In an intuitive way you can think of long

rates as reflecting expectation of future short rates.

And so if you model the short rates as stochastic process, you can then try to

make sure that long rates come out as the right expected value of short rates, so a

long bond prices to be more precise. Come out as the expected value of, of

discounting all future possible shortrates and expectations.

One of the things I actually learned at the time which I always try to tell

students now, is that you quickly learn that what you have to do is finance is

not average perameters but average prices.

Because of convexity, and so You shouldn't average short rates to get long

term prices, you should average bun prices to get bun prices.

We started by modeling the short rate, and figured you could model long rates as

the expected value, in some sense of future short rates.

We also adopted a binomial model approach for a variety of reasons.

The first was, it was very simple to picture, and we were all very familiar

with the binomial model. The second is, one of the things, even

these days I think, in trying to persuade sales people and traders in particular to

rely on a new model for business, is they have to understand it.

And traders in the 1980s were not as numerate and didn't have advanced

mathematical education as some of them do now.

And so it was kind of important to us to use a binomial model because you could

draw diagrams that showed what rates were doing.

You could show the nodes, you could show the discounting from period to period in

a way that traders were very comfortable with.

So for both PR reasons and because we didn't like to be too mathematically

sophisticated. We decided to do everything binomially,

and build the computer program to do it so that we could deliver it to them as a

way of doing business.

>> So, it appears that in developing the BDT model there were lots of approximations

made, there was a single factor model, you put in a second factor later on.

there was also this philosophical idea that you had to somehow calibrate models

to bond prices. That it wasn't a model which was going to

give you all the details. was that a conscious decision?

How did you decide on that approach.

>> Yeah, that, that's an interesting question because, although BDT.

There, there were two models that came out around the time we wrote our model.

There was Ho-Lee, which was kind of similar in a normal framework and we had

BDT which was a little bit more advanced, and allowed you to vary volatilities but

was logged normally. And logged normal interest rates were

more realistic than normal interest rates which can go negative.

There had actually been ten years earlier a bunch of, we had a different attitude.

There had actually been ten years earlier a bunch of continuous time extensions of

black shelves sort of fixed income world. The first one was[UNKNOWN] check which

was really a[UNKNOWN] model. And the second one was by[UNKNOWN] but

their aim was different. They were trying to build a model that

correctly describes the behavior of[UNKNOWN].

And they were theoreticians. And we were actually partitioners working

with a trading disk and our job Wasn't to model the yield curve.

Our job was to model options on the yield curve and give realistic prices for them

for traders. And so we had to take the yield curve as

a given, in the same way as option, option traders, option pricers take the

underlying stock price as a given, they don't try to decide whether it's right or

wrong. In the same way, we had take the yield

curve as that's the way it is, now price an option on it.

So the whole question of calibration became a big issue, and maybe that's the

first time in modern history of building these models that it became a name.

So the idea was we'll build a model of short rates, but we had to make sure that

when you price all fixed income on zero coupon bonds or treasury bonds on the

yield curve. On our model they had to reproduce the

price of treasury bonds at the instant that the option was priced.

Because when you price an option you want to make sure that you at least price the

end line correctly, so this was a question of calibration.

We chose a log-normal distribution cross sectionally of short rates and each

log-normal distribution of short rates had a mean, standard deviation or

volatility and we calibrated those to fit the price of a bun with that majority.

By pricing the bun but this cutting all the way down to trees.

So it's kind of[UNKNOWN], you price the tree of bond, you fixed everything then

you went to three and a half years. And you added another layer to the tree

with the right mean, and the right standard deviation to place the four year

bond. And we targeted the volatility of bonds

which the traders gave us because that was important for the option.

And we targetted the yield of bonds so the price of bonds because that was given

to you by the treasury bond market... And the idea was to choose your sort rate

distribution calibrated to reproduce these long yields and long volatilities.

Since then I would say that has become a pretty standard method of running all

models. You know your model is not strictly

correct, but you want it to reproduce the price of liquid instruments that are

underliers, and you calibrate the model everyday if you have to.

Since the financial[UNKNOWN] to make it fit the prices of underliers.

>> Over the years you've shared many

interesting quotes of Fischer Black with me, about modeling, about how he approach

modeling, what was his overall philosophy.

It would be great if you could share some of those with our students.

>> I came from a physics background, and

actually I came to Wall Street as I said in late 1985.

And I got very excited about it. Getting a shot in the arm about applying

physics and math techniques to new area that I have not done before.

And, I think I like a lot of people have this illusion that you could sort of

build a grand unified theory of finance, in which you would model all fixed income

rates with stochastic process. And consistently price every instrument

in the world and look for arbitrage opportunities, and BDT was an

arbitrage-free model in its, in its own limited one factor way.

And Fisher was actually much more pragmatic about all of this.

He was quite happy to live with an imperfect financial market and have

different models that weren't consistent with each other in different areas.

And didn't have this overarching desire to unify everything and I only really got

to that point sort of six or seven years later.

And I, I have a couple of nice quotes that he wrote in the late 80s and early

90s which I think are reflective of his understanding of the way models work.

So I have a couple of quotes from papers that he wrote.

One he says, it's better to quote estimate a model than to test it.

I take quote unquote calibration to be a form of estimation.

So I'm sympathetic to it. So long as we don't take seriously the

structure of a model we calibrate. Best of all though is to quote, explore a

model. That's the end of the quote.

I think what he's saying there is fighting against the people who don't

like calibration. There are people who say you're taking a

wrong model and fitting it to the data with wrong paramiters.

And his argument was I don't think this is gospel truth.

I think I'm just trying to get a handle on how things will behave in this model.

And I sort of come around to the idea that you should think of all of these

models as imaginary worlds that you're trying to construct.

Which don't reflect the real world in all its details but may Be consistent with

parts of it. And you calibrate a lot of different

models to the same data and see how, why the range of prices you get, when you,

when you pass the same instrument calibrated to the same underliers under

different stochastic models. he's got another quote which I like too,

even better. He says, my job I believe is to persuade

others that my conclusions are sound. I will use an array of devices to do this

theory, stylized affects, time series data surveys and appeals to

introspection. I particularly like the appeals to

introspection because he's making clear that finance isn't just a science, it's a

science of the way people behave and and an art.

And he's looking inside himself to try to get an idea of, what's a sensible way

that people would try to come at prices, and then model that.

the last quote I wanted to say, is, He says, in the world of real research,

conventional tests of statistical signifigance seem almost worthless.

I particularly like that, because when people new.

Either students or even people who are economists come to Wall Street.

In my experience, they always. Have greta expectations for models and

think they're going to be used in a, in a way that explains the truth.

And think you should test them very carefully to calibrate them, find the

best model and then use that. And the truth is, the financial world

goes through regimes of change, and the same models don't work in the same

period. And if you try calibrating one model to

30 years. It doesn't work because you really have

to use different, different models at different times, and I think he kind of

understood that. He was also a big believer in

rationality, Fisher, in that he once wrote an internal article at Goldmann

that said, you should pay traders not for the results that they get but for the

stories that they tell. About why they made money or why they

tried to make money because you want to encourage them.

To think, rather than to simply reward them for luck.

>> Thank you very much.