Libor Market Models: the reasons behind the success

1
Libor Market Models the reasons behind the
success

• A focus on calibration

2
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..

3
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.

4
Outline

• LMM the reasons behind the success
• Libor Market Models
• have a tremendous success
• the reasons of which need to be explained in
  details.
• 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.

5
The reasons behind the success

• The libor market model framework
• The marks of success
• Exploring the reasons

6
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

7
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 .

8
The marks of success

• 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?

9
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

10
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

11
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 / qi
• Volatility defined as
  and dimension 1

12
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).

13
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.

14
Calibration

• The questions behind calibration
• LMM and calibration the perfect match?
• New products, new challenges...

15
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

16
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.

17
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

18
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.

19
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.

20
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

21
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.

22
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.

23
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

24
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).

25
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)

26
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 i2
• This entirely defines the first column, of the
  matrix
• Then one can proceed recursively in the same way
  for the other columns

27
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...

28
An alternative formula for calibration
29
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

30
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

31
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)

32
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

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Questions Answers
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