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Fixed Income Derivatives Pricing in Practice

Course video 31 of 53

Calibration of term-structure models; the Black-Derman-Toy and Ho-Lee models. Limitations of term-structure models and derivatives pricing models in general. Introduction to credit-default swaps (CDS) and the pricing of CDS and defaultable bonds.

Financial Engineering is a multidisciplinary field drawing from finance and economics, mathematics, statistics, engineering and computational methods. The emphasis of FE & RM Part I will be on the use of simple stochastic models to price derivative securities in various asset classes including equities, fixed income, credit and mortgage-backed securities. We will also consider the role that some of these asset classes played during the financial crisis. A notable feature of this course will be an interview module with Emanuel Derman, the renowned ``quant'' and best-selling author of "My Life as a Quant". We hope that students who complete the course will begin to understand the "rocket science" behind financial engineering but perhaps more importantly, we hope they will also understand the limitations of this theory in practice and why financial models should always be treated with a healthy degree of skepticism. The follow-on course FE & RM Part II will continue to develop derivatives pricing models but it will also focus on asset allocation and portfolio optimization as well as other applications of financial engineering such as real options, commodity and energy derivatives and algorithmic trading.

Skills You'll Learn

Pricing, Financial Modeling, Financial Risk, Financial Engineering

Taught By

Martin Haugh
Co-Director, Center for Financial Engineering
Garud Iyengar
Professor

Transcript

In this module which is our last module on fixed income derivatives pricing we're

going to talk about the practice of fixed income derivatives pricing.

We'll explain how it is really not much more than using observable liquid Security

prices to extrapolate and compute the prices of non-observable security prices.

We're going to discuss, again, the problem of calibration and

how it is a difficult problem to solve in practice.

So, in practice, more complex models than binomial models are used to price fixed

income derivatives today.

In fact, the models that are used today are a lot more complicated than

the binomial models we've seen so far.

However, the pricing philosophy is still very much the same.

It works as follows.

We specify a model under the risk mutual dynamics which we are calling q of theta

and where theta is a vector of parameters that you need to choose.

So, for example, the theta's could be the ai's and

bi's that we have seen in the Ho-Lee and BDT models.

So, we fixed our model.

There is some unknown parameters theta.

What we will do is we will price all securities with our new model using risk

neutral pricing.

And if you recall this is our risk neutral pricing framework.

So, what we do is we say that Z t over B t where Z t is the price of the security in

question, B t is the value of the cash account So that is equal to the expected

value, conditional in time t information, under the risk neutral probabilities

of the expected value of the security at time t + s divided by the cash account,

plus all the intermediate coupons or cash flows that you receive between times t + 1

and t + s, again divided by the cash account value.

This is our risk neutral pricing framework,

we know that there cannot be any in the model.

When we price all of our securities like this.

So, that's step number two.

Step number three is then to choose data so that the market prices of

appropriate liquid securities agree with model prices of those securities.

This is the model calibration procedure.

And we've seen in the last module, some examples of how to do model calibration in

the context of the Black–Derman–Toy model.

And when our calibration securities [INAUDIBLE] coupon bond prices.

The calibration problem typically requires minimizing a sum of squares of

the following form.

So remember, the Theta are the vector of parameters we want to choose in our model

These could be the a i's or b i's in the black [INAUDIBLE] your [INAUDIBLE] model

so the goal is to choose the thetas in such a way as to minimize the following.

So we have a bunch of securities which are indexing by i here and

we have a market price of the i security which is pi market and

then we have our model price of the i security, pi model and

this model price is calculated using this equation here, equation 14.

This is our risk mutual pricing that we use in our model.

Doing this ensures that our model prices are arbitrage free.

So in our model, we can compute this price.

We see this in the marketplace.

And omega i is a weight, it's a positive weight reflecting

the importance of the ith security in our calibration.

Maybe omega i is large for some values or for some securities,

and omega i is small for other values.

Omega i could reflect, maybe, how certain we are about the market price that we see.

Maybe we think this market price is stale, maybe it's a few minutes old, or

maybe there's a wide bid ask on this market price, so

we're not as confident of the true value of the security.

Either way the omega i's can be used to reflect the importance of the i security

in the calibration.

We also have this additional term, which I won't spend too much time on.

But, theta prev stands for theta of the previous calibration.

So, the previous model calibration might have occurred the previous day or

the previous hour or the previous half hour.

What we want to do typically is make sure that our calibrated parameters don't move

around too much.

There are various reasons for this, but

it helps to keep the model somewhat stable through time if we penalize

Theta being too far away from the previously calibrated vector theta.

And lambda will be a parameter that's greater than or

equal to 0 which reflects the importance of this term to the overall calibration.

Once this model has been calibrated we can use it to hedge and

price more exotic or illiquid securities.

One problem,

however, is that this calibration problem is often very difficult to solve.

This is typically a known convex optimization problem.

Well, if you don't know what I mean by nonconvex basically the upshot of that

is that there will be many local minima.

And it would be very hard to find the true solution to this optimization problem.

As market conditions change we often find that we need to recalibrate frequently,

often many times a day.

If the model wasn't right then actually you would only need to calibrate once.

So, in fact your model typically in practice is not right it's never even

close to being right.

We always need to recalibrate it.

So, this is very different to the situation on the physical sciences

for example.

Or typically if you have the right model then you don't need to

recalibrate it very often.

In finance unfortunately you do.

So, derivatives pricing in practice then is little more than

using observable market prices to interpolate and

extrapolate to price non-observable security prices.

But risk-neutral pricing at the model level at least implies that we can

extrapolate and interpolate in an arbitrage-free manner.

And in case you're wondering,

we haven't said anything about the true probabilities in this section.

We've discussed fixing con derivatives up to to now, and

we've completely ignored true probabilities.

We've only discussed risk-neutral probabilities, Q and one minus Q,

and in fact in the binomial model, we set them equal to a half.

So, does that mean that the derivative prices and

practise don't depend on true probabilities?

Well that is not true.

Of course they depend on practise and

the dependence actually enters into the calibration process because the true

probabilities will enter the market's perception.

Of market conditions, economic conditions, political conditions and so on.

And so, the markets perception of the two probabilities will certainly enter in here

into the market prices of these securities.

So, implicitly when we do this calibration we're actually Including

the market's views, the market's probabilities,

the market's level of risk aversion, into the entire derivative's pricing exercise.

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