Libor Market Models: the reasons behind the success

A focus on calibration

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Introduction Market models have become a standard in the bank industry.

This success is attested by the number of publications on the subject (not to mention the conferences…)

The repeated efforts to transpose this methodology to other underlyings (credit, inflation) is another remarkable sign.

Standard arguments cannot explain this phenomenon. The ability of these models to capture rate curve dynamics is more than questionable. Their implementation is demanding and their computational cost is high (even though

optimization techniques have been developed).

Exploring the reasons behind this success is very enlightening. As we will see, tractability, readability, flexibility are the keywords for understanding

the popularity of the LMM framework But some reasons may not be as bright as one could expect… Success also has a lot to do with calibration, but new classes of derivatives rise new

challenges which may prove difficult to address in this framework..

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Objectives

Our intention in this brief presentation is twofold.

Firstly, clarify the reasons why this modeling framework has reached such a success in the industry despite its strong limitations.

Naturally, there are many solutions to circumvent the limitations, but exposing these techniques will not be our intention here.

Instead, we would like to detail in an honest and practical way the reasons for this remarkable success - including the most questionable reasons.

Secondly, explore calibration as a key to this success.

Again, exploring advanced numerical techniques will not be our purpose.

Instead, we would like to expose some practical issues on calibration, with an evocation of the new challenges LMM are now confronted with.

The underlying objective is to provide some insight on how a model is used in the derivatives department of a bank.

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Outline

1. LMM: the reasons behind the success

Libor Market Models…

… have a tremendous success…

… the reasons of which need to be explained in details.

2. Calibration: practical issues and new challenges

The calibration process raises many delicate questions…

… and LMM offer a good control…

… but this flexibility may reach its limits with new-generation products.

The reasons behind the success

1. The libor market model framework

2. The marks of success

3. Exploring the reasons…

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The framework: what are we talking about ?

LMM dynamics

LMM parameters

The model is entirely characterized by volatility functions:

It is sometimes represented with scalar notations:

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What are we talking about ?(continued)

In preparation for the forthcoming sections, we propose an “HJM-biased” presentation of BGM :

This is a trivial observation, but it will be useful to understand what is new and what is not when switching from classical models to market models .

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This framework has become a standard.

They are commonly used for pricing most exotic interest rate derivatives

And there are more interesting signs of popularity…

First sign: when dealing with Bermudan options, the industry has preferred to explore new numerical techniques rather than change the model

Bermudan MC techniques

(estimation of continuation value: )

Markovian approximations for PDE implementation

(estimation of drift : )

Second sign: when facing the limitations of the model, the industry has preferred to extend the model rather than totally change the framework.

Stochastic-volatility and local-volatility extensions

Multiple-currency extension

A natural question arises: what is the rationale behind this success-story?

The marks of success

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What the reasons could be… but are not

Does LMM properly capture interest rates dynamics ?

Nobody sincerely believes in the lognormal dynamics of the forwards curve.

Statistical observation suggest the presence of jumps, regimes, etc

Importantly, implicit volatilities exhibit non-trivial smiles.

More globally, the deterministic volatility/correlation is very restrictive.

This assumption yields little control on joint moves of the parts of the curve.

Eventually, using Brownian motions is questionable.

Does the model allow an easy implementation ?

LMM are not strictly speaking Markovian

Even though satisfying Markovian approximations are attainable, the natural tool for implementing this model is a rather heavy Monte-Carlo simulation.

A naïve implementation consequently is very costly

A naïve implementation precludes backward valuation (PDE schemes)

Markovian approximations require the estimation of intricate conditional expectations

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What the reasons should be… and are indeed

Tractability : an overemphasized argument?

It is the most frequent argument and it is a strong one indeed: dealing with market quantities is very convenient.

But in our opinion, this argument is a bit overemphasized: most traditional models can be rewritten in more convenient form.

Readability : the “Gaussian process view”

A model can now be characterized through the forward covariance cube (T -1 = 0):

Linear combinations of such elements can be interpreted as swaption variances, caplet variances, spot or forward. For example:

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What the reasons should be… and are indeed(continued) Simplicity: the flexibility of HJM with an intuitive parameterization

Naturally LMM are a particular case of HJM (hopefully) HJM is based on a full volatility surface BGM works with a vector of volatilities

In this perspective, market models fill the conceptual gap between: The classical, simple models - too restrictive The HJM framework - too general

Note that it is often stated that the major break is log-normality of rates Log-normality is definitely an essential feature of (the first version of) LMM But it is useful to understand that the major change attached to LMM is a reduction in the

complexity of parameterization

In this perspective, it is crucial to see the “continuity” with classical models For example, the Hull-White model can be presented as a (displaced) BGM

Displacement 1 / i

Volatility defined as: and dimension 1:

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What the reasons should not be… and are anyway

Familiarity with Black-Scholes

LMM framework allows to think of the curve as a (highly correlated) basket.

Each libor follows a BS-type diffusion under its martingale measure

Familiarity with BS in terms of possible extensions, robustness, etc, can thus be transposed to the interest rates world.

Familiarity with Gaussian Calculus

Correlation as a characterization interdependence is poor but convenient

Same observation for the variance as a characterization of dispersion.

Simplicity of MC schemes

LMM are more naturally suited to MC schemes (even though it is not compulsory)

The industry is very prone to implement generic solutions and such solutions are more rapidly attained with a simulation approach (it can be delegated to non-specialists).

To a some extent, it is the simplest choice (from an organizational point of view).

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Conclusion (of part 1)

Does the success of LMM result from an educational bias in the quants/traders community? To some extent, the answer is yes. But it is not shocking: pricing models are meant to serve as decision-making tools and

should be adapted to their users. And there is more to it than that…

To fully appreciate this success, one has to understand the very role of a model in a trading room.

Actually, it is a rather modest role. Interpolate available information (pricing) Connect risks from different sources (hedging)

But for this role, calibration is critical. The calibration set can be thought of as a choice of interpolation points. The model and its parameterization can be thought of as a choice of interpolation

method. This « interpolation » analogy is not very convincing but helps understanding why one

should not expect too much from a model.

Calibration

1. The questions behind calibration

2. LMM and calibration: the perfect match?

3. New products, new challenges...

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Calibration in practice

The steps for calibration

Model parameterization

Determination of constraints (target instruments)

Choice of calibration mode (cascade vs. global)

Numerical methods (inversion / minimization)

These steps express specific views on risk management issues

Curve and volatility dynamics

Product risk factor analysis

Risk diversification of trading portfolios

Computation time capacity

Structure of the market in terms of products & risks

Again, our intention is to expose the beliefs hidden in the calibration process before exploring the virtues of LMM

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Parameterization

It is an expression of a view (or an intention) on curve dynamics

How does one expect the curve and, as importantly, its dispersion structure to evolve?

Using the interpolation analogy, this question reads: what information should one retrieve from the interpolated points?

A quick review on volatility

Notations: t time of observation T fixing date of underlying rate

Function of t : time-dependent structures (like short-rate models)

Function of T: underlying-dependent structures (equity-like model)

Function of T-t : stationary structures (non low-dimension Markovian)

Mixture of such forms are commonly used (example: stationary with scaling)

Example: callable products

Forward volatility is one of the key risk factors.

Consequently, using non-stationary is dangerous

But it may be a choice when one knows the bias of the model.

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Parameterization (continued)

Naturally the same kind of taxonomy holds for correlation.

Choosing the rank of the model is the first delicate question

Empirical evidence suggests not to exceed three, but some nice parametric forms impose full rank correlation matrices

Then all the questions regarding stationary, time-dependence, etc need to be addressed.

This choice should be dictated by volatility structure (in practice, only covariance really matters so using different choices is dangerous)

Extensions of LMM require additional parameterization

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Calibration targets

It is an expression of a view on products’ risk factors What points in the market should be considered as relevant for the pricing of the

structured product? Using the interpolation analogy, this question reads: which points should we

interpolated from (apart from the curve itself) ?

Example: Bermudan swaption It is natural to calibrate on underlying swaptions However, the main exotic risk factor is forward volatility Some may think that the spread between caps and swaptions volatilities says

something about forward volatility (under correlation assumptions) In this perspective, using caplets makes sense for calibration.

In more sophisticated products, the choice is highly non-trivial. For instance, callable cms-spread products have at least 3 obvious risks This sometimes push for a more global approach.

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Calibration mode : global vs. cascade calibration

It is an expression of organizational choices

It is very closely related to the determination of the calibration set.

But it may also be very related to the structure of the business and to the level of sophistication of the persons in charge of quotation.

Principles

Global calibration consists in using an arbitrary set of vanilla instruments as calibration targets (typically a whole set of caps and swaptions)

Cascade calibration consists in solving a series of one-dimension problems (based on a specific parameterization of the model)

Implications

A global calibration is well suited to an organization where a high level of accuracy is not required for each price but where a large number of quotations are addressed.

In this case, once calibrated, the model may be shared for distinct quotations

A local calibration is typically more adapted when transparent risk reports and high accuracy are mandatory.

In this case, the model will typically be recalibrated at each quotation.

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Calibration mode : global vs. cascade calibration(continued) Pros and cons of global calibration

It avoids the complex questions regarding risk factor analysis (is it a good thing ?)

It allows using a unique model for wide range of products, ensuring some consistency in risk analysis reports

But it is computationally costly (global minimization schemes)

It is sometimes numerically unstable (due to the existence of local minima)

Risk reports (deltas, vegas, etc) may prove difficult to decipher.

Pros and cons of cascade calibration

It is easy, fast and robust (because of dimension one)

It often implies that models are product-dependent

It requires a thorough analysis of product risk factors

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Calibration algorithm: numerical choices

It is an expression of skills but of the environment as well. Obviously, numerical methods depend on quantitative talents inside the institution. But in many situations, it also reflects the structure of the market. And it is naturally related to technological constraints inside the institution.

Naturally algorithmic choices depend on previous steps (parameterization, constraints, calibration mode, etc) Valuation formula for the target instruments Root-finding or minimization methods

But the market (and technological) environment may be determining: In a market where strong risk diversification is allowed an institution may prefer to

resort to a global approach, using a global model with a heavy calibration procedure (where a unique model can be shared for many quotations)

However, structured products markets are often one-way markets (clients always trade the same side for a given exotic risk), which rather pushes for product-adapted (“on the fly”) models. In this case, fast and unbiased formulae are required.

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LMM and calibration

We rapidly exposed the successive steps in the calibration process:

Parameterization

Selection of constraints

Selection of a methodology (cascade or global)

Selection of formulae and numerical schemes

Now, it is interesting to explore why does the LMM outmatch other models in this process

What particularities does LMM bring into the process ?

What makes LMM so easy to use ?

For this purpose, we briefly review each step in the process.

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LMM and calibration: parameterization

LMM offer a clear view on volatility

The model directly characterize the volatility functions of libors, which are the direct underlyings of vanilla caplets

Consequently volatility forms have a clear interpretation in terms of the evolution of caplet implicit volatilities

LMM offer a clear distinction between volatility and correlation risks

Most calibration constraints can be thought of as basket option problems.

Even spread options are easy to handle in this framework

In particular, the distinction between caps and swaptions can thought as the combination of a question of time repartition of volatility and a question of correlation

In practice…

Simple is beautiful: stepwise constant functions are ok.

Readability is critical: starting with something reasonably stationary is wise

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LMM and calibration: constraints

Naturally, the model has little to do with this stage

Ideally, calibration constraints are a question of product risks, not model properties

it is indeed dangerous to have some preconceptions regarding the model.

LMM do not impose as many limitations as classical models do

Their flexibility allows considering many constraints without loosing too much in accuracy

Besides, the built-in calibration of caplets is remarkable (as long as structured libor swap legs are involved, this is a very nice feature)

In practice…

If a global minimization scheme is used, the whole caps/swaptions matrix is used

Otherwise, depending on the product, it might be a column of caplets, a column of swaptions, a diagonal of swaptions, etc (typically a combination of these).

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LMM and calibration: calibration mode

Whether global or local, LMM calibration proves very adaptable

Global calibration can be expressed in simple terms

Swaptions and caps imposes constraints on the covariance cube.

Using standard approximations, these constraints have a quadratic form

In the end, the problem can be expressed in terms of semi-definite programming, for which abundant literature can be found

But cascade calibration is more interesting…

A simple example is caplet column calibration with stationary volatility

Assume stationarity

Write the constraints:

Then solving the problem is a trivial bootstrapping.

This exactly is where LMM is strong: a fully stationary volatility column can be obtained without effort (and regardless of constraints on correlation parameters)

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An example of cascade calibration

LMM allows full calibration of the vanilla matrix. Here we consider the problem: Targets: full vanilla matrix (forget about smile here) Calibrated parameters: volatilities (in the strict sense) Fixed parameters: correlations:

It is useful to have a nice representation of forward volatility structure:

Armed with this representation, the calibration process is straightforward The first element is given by the caplet of exercise date T0 Then the other elements in the first column are recursively calculated from swaptions of exercise

date T0 and maturity Ti for all i≥2 This entirely defines the first column, of the matrix Then one can proceed recursively in the same way for the other columns

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LMM and calibration: numerical issues

The calibration of LMM requires using simple and efficient formulae

The standard market formula for swaptions consists in three steps:

Write the swap rate as a function of libors

Write the dynamics of the swap rate

Simplify the expression assuming deterministic weights (freeze expression at forward rates) and log-normality

This is a very practical and intuitive approach, but

It has a limited scope: swaptions only

It does not allow computing convexity adjustments (for CMS options calibration)

A slightly more general approach may thus prove useful for a more ambitious calibration...

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An alternative formula for calibration

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New products, new challenges

We have explored the advantages of LMM for calibration

Intuitive parameters, especially in terms of caplet implied volatility.

Flexible parameterization, with a quadratic expression of constraints

Feasibility of powerful cascade calibration

Existence of simple and accurate formulae

This positive image would be deceptive if we ignored the challenges imposed by new classes of products.

Structured swaps with multiple underlyings (Libors, CMS)

Popularity of products with an exposure to the slope of the curve (CMS-spreads)

We will use an example: “lock-up on CMS spread”

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New products, new challenges (continued) Product description

The product is a structured leg (embedded in a structured swap) Each quarter, the client receives the spread cms10y – cms2y (with a leverage and an shift)

floored at some strike: MAX(ADDITOR + LEVERAGE * CMS-SPREAD, STRIKE) When the CMS-SPREAD exceeds some LIMIT, the coupon becomes fixed at a predetermined

level until maturity

Risk factor analysis Naturally the implied volatility of CMS-spread is essential But the trigger mechanism implies a binary risk (end of the structured leg), triggered by a

spread. The magnitude of binary risk is determined by the mark-to-market of the residual leg. Consequently the exotic risk is the forward volatility and correlation, and the inter-temporal

correlation between CMS-spreads.

Challenges in terms of parameterization Provide a parameterization with some control on this “second-order” correlation Once this has been achieved, understand how classical model extensions (to account for

smile) affect or does not affect the conclusions

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Conclusion

LMM are not perfect, but who needs a perfect model ?

Models are important (heavy decisions at stake)…

… but not that important (to some extent, models work as interpolation tools)

Above all, there is no such thing as the “perfect” model (believing in one is dangerous)

LMM are not trivial, but who needs a trivial model ?

Products are sophisticated and sophisticated models are required to price and hedge them

Computation cost is not as critical as it used to be.

Traders have a high level of sophistication (most are ex-quants)

So what do we need exactly ?

A model that is naturally adapted to the human, organizational, and technical environment

A model that allows flexible calibration (flexible enough to keep up the pace of the evolution of the market in terms of payoff sophistication and risk complexity)

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Conclusion (continued)

This is where the value of LMM lies: they are well adapted…

…to the people (educational bias on BS, taste for simplicity)

…to the market (in terms of information to calibrate to and in terms of products to price)

…to the technology (computational capacity increases rapidly)

What is critical is calibration, and LMM do more than well on this side

LMM bring the sophistication HJM to the reach of non-specialists

It allows flexible, accurate and rich calibration while keeping everything intuitive and simple

Intuitive enough? As far as new generation products are concerned, the heralded “interpretability” of LMM may soon reach its limits…

Questions & Answers