字幕表 動画を再生する 英語字幕をプリント The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. DENIS GOROKHOV: So I work at Morgan Stanley. I run corporate treasury strategies at Morgan Stanley. So corporate treasury is the business unit that is responsible for issuing and risk management of Morgan Stanley debt. I also run desk strategies own the New York inflation desk. That's the business which is a part of the global interest rate business, which is responsible for trading derivatives linked to inflation. And today, I'm going to talk about the HJM model. So HJM model-- the abbreviation stands for Heath-Jarrow-Morton, these three individuals who discovered this framework in the beginning of 1990s. And this is a very general framework for pricing derivatives to interest rates and to credit. So on Wall Street, big banks make a substantial amount of money by trading all kinds of exotic products, exotic derivatives. And big banks like Morgan Stanley, like Goldman, JP Morgan-- trades thousands and thousands of different types of exotic derivatives. So a typical problem which the business faces is that new types of derivatives arrive all the time. So you need to be able to respond quickly to the demand from the clients. And you need to be able not just to tell the price of derivative. You need to be able also to risk manage this derivative. Because let's say if you sold an option, you've got some premium, if something goes not in your favor, you need to pay in the end. So you need to be able to hedge. And you can think about the HJM model, like this kind of framework, as something which is similar to theoretical physics in a way, right? So you get beautiful models-- it's like a solvable model. For example, let's say the hydrogen atom in quantum mechanics. So it's relatively straightforward to solve it, right? So we have an equation, which can be exactly solved. And we can find energy levels and understand this fairly quickly. But if you start going into more complex problems-- for example, you add one more electron and you have a helium atom-- it's already much more complicated. And then if you have complicated atoms or even molecules, it's unclear what to do. So people came up with approximate kind of methods, which allow nevertheless solve everything very accurately numerically. And HJM is a similar framework. So you can-- it allows to price all kinds of [INAUDIBLE] derivatives. And so it's very general. It's very flexible to incorporate new payoffs, all kinds of correlation between products and so on, so forth. And this HJM model-- [INAUDIBLE] natural [INAUDIBLE] more general framework like Monte Carlo simulation. And before actually going into details of pricing exotic interest rates and credit derivatives, let me just first explain how this framework appears in the most common type of derivatives, basically equity-linked product. So like a very, very simple example, right? So let's say if we have a derivatives desk at some firm, and they sell all kinds of products. Of course, ideally, let's say there's a client who wants to buy something from you. Of course, the easiest approach would be to find the client and do an opposite transaction with him, so that you're market neutral, at least in theory. So if you don't take into account counterparties and so on. However, it's rather difficult in general, so the portfolios are very complicated. And there's always some residual risk. So this is the cause of dynamic hedging. So for this example, very simple example, a dealer just sold a call option on a stock. And if you do this, then in principle, the amount of money which you can lose is unlimited. So you need to be able to hedge dynamically by trading underlying, for example, in this case. So just a brief illustration of the stock markets, you see how random it has been for the last 20 years or so. So first of all, this year, some kind of-- from beginnings of the 1990s to around 2000, we see really very sharp increase. And then we have dot-com bubble, and then we have the bank [INAUDIBLE] of 2008. And if you trade derivatives whose payoff depends, for example, on the FTSE 100 index, you should be very careful. All right? Because market can drop, and you need to be hedged. So you need to be able to come with some kind of good models which can recalibrate to the markets and which can truly risk manage your position. So the so the general idea of pricing derivatives is that one starts from some stochastic process. So in this example here, it's probably like the simplest possible-- nevertheless a very instructive-- model, which is essentially like these [INAUDIBLE] Black-Scholes formalism, which is where we have the stock, which follows the log-normal dynamics. I have a question. Do you have a pointer somewhere, or not? It's just easier-- OK, OK. PROFESSOR: Let's see. There's also a pen here, where you can use this. DENIS GOROKHOV: Oh, I see. PROFESSOR: Have you used this before? You press the color here that you want to use, say, and then you can draw. You press on the screen. DENIS GOROKHOV: Oh, I see. Excellent. That's even better. OK, so it seems like the market is very random. We need to be able to come up with some kind of dynamics. And it turns out that the log-normal dynamics is a very reasonable first approximation for the actual dynamics. So in this example, we have stochastic differential equation for the stock price. And it consists-- it's the sum of two terms. This is a drift, it's some kind of deterministic part of the stock price dynamics. And here, also, we have diffusion. So here, dB is the Brownian motion driving the stock, and S is the price of the stock here. Mu is the drift. And sigma is the volatility of the stock. Particularly, it shows the randomness. And it's the randomness impact on the stock price. And using this model, one can derive the Black-Scholes formula. And the Black-Scholes formula shows how to price derivatives whose payoff depends on the price of the stock. So here, if you look at this differential equation, then you can answer the question. Let's say if you started from some initial value for the stock at time t. And then we started the clock. Which are now to be at time capital T. And given that time T, then stock price is S_T. So what's the probability distribution for the stock at time T? So this kind of equation can be very easily solved. And one can obtain analytically the probability distribution function at any [INAUDIBLE] moment of time. So I mean, I just think I'll write a few equations, because it's very important to understand this. So I'm sure you probably have seen something like this already, but let me just show you the main ideas beyond this formula. So if you have a random process-- let's say A is some process, stochastic process, which is normal. So it follows some drift. Plus some volatility term. Right? So the difference between this equation is that I don't multiply by A here and A here. Especially, it's much simpler to solve. So the solution for this equation is very straightforward. So at any moment of time T, if you start at moment 0, the solution of the equation would be something like this. Drift-- right? I'm simply integrating. Plus-- and I assume that B of t is standard Brownian motion, so at time 0, it's 0. And then it's very easy to see now that... is equal to the Brownian motion. But this is nothing else. It's some random number, which is normally distributed, times square root of time. So epsilon is proportional to it. OK, so basically, this means that this is normally distributed. And its-- and probability distribution for this quantity is equal to-- we know it's exactly, right, because this is like a standard Gaussian distribution. And if you simply substitute A into here, then you will obtain the probability distribution for the actual quantity. And I'll just write it for the completeness. So basically, we obtain probability distribution for the standard variable. So this is straightforward. So the only difference between the case I'm doing here is that the dynamics is assumed to be log-normal. Right? And the interpretation is very simple. If it's normal, then the price of the stock can become negative. Which is just a financial nonsense. So the [INAUDIBLE] log-normal dynamics basically is a good first approximation. And in this case, what helps as a result is just known as Ito's lemma. So I just first of all write it, and then I will explain how you can obtain it. And if you look at this equation-- let me write it once again-- which is basically the drift plus-- then it turns out that, of course, since--- it-- intuitively it's clear that the dynamics of logarithm of S is-- dynamic of logarithm of S is normal. So essentially, we obtain something like this. So if you now substitute this into this, you locked in a very simple formula. OK, so here, I used the result, which is known as Ito's lemma, which I'm going to explain right now. Like how it was obtained-- basically, it tells us that when we differentiate the function of a stochastic variable. Then besides the trivial term, which is basically the first derivative times dS, there's an additional term, which is proportional to the second derivative. And it's non-stochastic, so I'll explain why it's this. But if you do it-- if you look at this equation, then you see essentially this formula. It's very, very similar to this formula. The only difference now is that alpha is just mu minus one half of sigma squared. So that's a how, if you iteratively use this solution, and simply substitute A by log S, you will come to this equation. So this is very important. So it's a very important effect, like-- yes? AUDIENCE: The fact that it can't be negative, does that exclude certain possibilities? When there's a normal Gaussian, can go negative or positive? DENIS GOROKHOV: Yes, but stock-- from a financial point of view, stock cannot be a liability. Right? You buy a stock. This means basically, you pay some money. And you have basically some sort of, say, option on the profit of the company. So they can't charge you by default. So it can't go negative for the stock. Also, in principle, there might be derivatives, which can be both positive or negative payoff, but not the stock.