It is a great pleasure to have Professor Emanuel Derman for a guest lecture today. Professor Derman is co-head of risk at Prisma Capital Partners. He's also the director of Financial Engineering Program at Columbia University. He's the author of two books, he's latest book, Models Behaving Badly: Why Confusing Illusion With Reality Can Lead to Disaster, on Wallstreet and in life, Was one of Business Week's top 10 books for 2011. His other book, My Life as a Quant, was one of Business Week's top ten books in 2004. Way back in 1996, he wrote an article titled Model Risk, where he pointed out the dangers that inevitably accompany the use of models. Among his many awards and honors, he was named the Sungard IAFE Financial Engineer of the year, in the year 2000. Professor Derman has Phd in theoretical physics from Columbia. In the 1980s he worked at AT&T Bell Labs, on developing programming languages for business modelling. From 1985 to 2002 he worked on Wall Street, running quantitative strategies, research groups, and fixed income equities and risk management. He was appointed as managing director at Goldman, Sachs and company in 1997. The financial models that he developed there, in particular, the Black-Derman-Toy interest model and the Derman Kani Luke volatility model have become widely used in the industry. Today, we'll be speaking with Professor Derman about models, how they are developed in Wallstreet, using the Black-Derman-Toy interest rate model as the main context. You moved from Bell Labs to Goldman in 1985. Can you describe what the state of the art was in terms of the products being used, the models being used and how you made the transition from Bell Labs to Goldman. >> Yeah. I came from Bell Labs to Goldman in 1985. And, it was actually a real pleasure for me at the time. 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.
