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  • 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.

  • So it's fundamental financial restriction.