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

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

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

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

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

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

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

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

Â technology because, when you deal with fixing com productions.

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

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

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

Â field actually much more amendable to mathematical modelling than equities.

Â Okay.

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

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

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

Â income options desk that were selling options on Treasury bonds.

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

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

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

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

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

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

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

Â anymore. And so what they were doing was, they

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

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

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

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

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

Â trying to build models to understand these things.

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

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

Â will shortly describe, that I set about modifying.

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

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

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

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

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

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

Â much less mathematically sophisticated than they are now.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â If I can gesticulate to explain what I mean.

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

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

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

Â process. Meaning, it's a bridge between its

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

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

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

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

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

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

Â Brownian bridge. So for example Ravid came up with this

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

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

Â variable. Making the yield to maturity log normal

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

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

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

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

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

Â reasons. the predominant one being that, like

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

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

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

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

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

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

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

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

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

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

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

Â passes and options whose expiration changes as time passes.

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

Â You cannot model one bond as an independent entity.

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

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

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

Â riskless rate. But the riskess itself is a reflection of

Â the bond price. And so implicity are already modelling

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

Â separate instruments. And in fact, Ravi taking some interesting

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

Â Model that treated yield as an independent variable.

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

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

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

Â so, this was the way people did things.

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

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

Â >> Yes, so we were actually.

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

Â Fisher was located. Boltoy worked for him.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Â rates as reflecting expectation of future short rates.

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

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

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

Â discounting all future possible shortrates and expectations.

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

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

Â not average perameters but average prices.

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

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

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

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

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

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

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

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

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

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

Â mathematical education as some of them do now.

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

Â draw diagrams that showed what rates were doing.

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

Â a way that traders were very comfortable with.

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

Â sophisticated. We decided to do everything binomially,

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

Â way of doing business.

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

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

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

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

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

Â How did you decide on that approach.

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

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

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

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

Â was logged normally. And logged normal interest rates were

Â more realistic than normal interest rates which can go negative.

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

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

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

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

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

Â correctly describes the behavior of[UNKNOWN].

Â And they were theoreticians. And we were actually partitioners working

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

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

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

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

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

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

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

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

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

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

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

Â yield curve. On our model they had to reproduce the

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

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

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

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

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

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

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

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

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

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

Â bond. And we targeted the volatility of bonds

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

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

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

Â distribution calibrated to reproduce these long yields and long volatilities.

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

Â models. You know your model is not strictly

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

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

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

Â >> Over the years you've shared many

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

Â modeling, what was his overall philosophy.

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

Â >> I came from a physics background, and

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

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

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

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

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

Â rates with stochastic process. And consistently price every instrument

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

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

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

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

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

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

Â to that point sort of six or seven years later.

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

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

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

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

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

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

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

Â model. That's the end of the quote.

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

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

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

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

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

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

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

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

Â parts of it. And you calibrate a lot of different

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

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

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

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

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

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

Â introspection. I particularly like the appeals to

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

Â science of the way people behave and and an art.

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

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

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

Â conventional tests of statistical signifigance seem almost worthless.

Â I particularly like that, because when people new.

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

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

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

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

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

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

Â period. And if you try calibrating one model to

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

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

Â understood that. He was also a big believer in

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

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

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

Â tried to make money because you want to encourage them.

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

Â >> Thank you very much.

Â