Equity and Credit Correlation Products and Copula Functions

1
Equity and Credit Correlation Products and Copula
Functions

• Umberto Cherubini
• Matemates University of Bologna
• Birbeck College, London 04/02/2009

2
Outline

• Motivation structured finance products and risk
  management issues
• Copula functions main concepts
• Radial symmetry and non-exchangeability
• Estimation and calibration
• Copula pricing cross-section dependence
• Copula pricing temporal dependence
• A general dynamic model for equity markets
• A general dynamic model for credit products
• Risk management applications

3
Motivation

• Structured Finance Products and Risk Management
  Issues

4
Barrier Altiplano

• Assume a set of coupon periods, k 1,2,P.
• In each period k a set of dates j 1, 2, ,mk
• A basket of i 1, 2, , n assets
• Coupon paid
• ck if Si(tj)gt Ki for all i and all j
• 0 otherwise

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European Altiplano

• Investment period March 2000 - March 2005
• Principal repaid at maturity
• Coupon paid on march 15th every year.
• Coupon determination
• Coupon 10 if (i 1,2,3,4,5)
• Nikkei (15/3/200i) gt Nikkei (15/3/2000) e
• Nasdaq 100 (15/3/200i) gt Nasdaq 100 (15/3/2000)
• Coupon 0 otherwise
• Digital note ZCB bivariate digital call
  options

6
Barrier Products

• Consider a product that at time tm pays 1 unit of
  cash if the value of the underlying asset X(ti)
  remains above, a given barrier B level on a set
  of dates t1, t2,, tm.
• This is a no-touch option, which is also called
  digital barrier option, or uni-variate Altiplano.
• Like the European digital option represents the
  pricing kernel of European options, the barrier
  digital product is the pricing kernel of barrier
  options.

7
First-to-default derivatives

• Consider a credit derivative product that pays
  protection, to keep things simple in a fixed sum
  L, the first time that a company in a reference
  basket of credit risks gets into default. The
  reference credit risks included in the basket are
  called names in the structured finance jargon.
• Again to keep things simple, let us assume that
  payment occurs at expiration date T of the
  derivative contract
• If Q(?1 gt T, ?2 gt T) denotes the joint survival
  probability of all the names in the basket, it is
  straightforward to check that the value of the
  derivative contract, named first-to-default
  turns out to be
• First to default LP(t,T)(1 Q(?1 gt T, ?2 gt
  T))

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Synthetic CDO
Senior Tranche
Originator
Junior 1 Tranche
Special Purpose Vehicle SPV
Protection Sale
Junior 2 Tranche
CDS Premia
Interest
Tranche
Investment
Collateral AAA
Equity Tranche
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Standard synthetic CDOs

• iTraxx (Europe) and CDX (US) are standardized CDO
  deals.
• The underlying portfolio of credit exposures is a
  set of 125 CDS deals on primary names, same
  nominal exposure, same maturity.
• The tranches of the standard CDO are 5, 7 and 10
  year CDS to buy/sell protection on the losses on
  the underlying portfolio higher than a given
  level (attachment) up to another level
  (detachment) on a nominal value equal to the
  difference between the two levels.

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Term structure of CDX
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Risks

• Products like these are made to provide exposure
  to specific equity or credit risk sources, with
  particular reference to correlation among them
  (correlation products).
• We will refer to this correlation as dependence
  or association (more general terms) and we will
  denote it cross-section dependence (that is
  dependence among a set of assets or risk factors
  evaluated at the same time).

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Cross-section dependence

• Any pricing strategy for these products requires
  to select specific joint distributions for the
  risk-factors or assets.
• Notice that a natural requirement one would like
  to impose on the multivariate distributions would
  be consistency with the price of the uni-variate
  products observed in the market (digital options
  for multivariate equity and CDS for multivariate
  credit)
• In order to calibrate the joint distribution to
  the marginal ones one will be naturally led to
  use of copula functions.

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Temporal dependence

• Barrier Altiplanos the value of a barrier
  Altiplano depends on the dependence structure
  between the value of underlying assets at
  different times. Should this dependence increase,
  the price of the product will be affected.
• CDX consider selling protection on a 5 or on a
  10 year tranche 0-3. Should we charge more or
  less for selling protection of the same tranche
  on a 10 year 0-3 tranches? Of course, we will
  charge more, and how much more will depend on the
  losses that will be expected to occur in the
  second 5 year period.

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Hybrids

• Correlation among different risk factors is the
  main risk in hybrids, or risky swaps as are
  called in the market.
• These are contracts between two parties, A and B,
  indexed to interest rates, currencies, or credit
  and contingent on survival of a third
  counterparty C.

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Risk management applications

• Cross-section aggregation of risk measures (risk
  measure integration). The problem is to compute
  the joint distribution of losses due to different
  risk factors (typically, market, credit,
  operational)
• Temporal aggregation of risk measures. The
  problem is to compute the distribution of losses
  over different time horizons.
• Temporal aggregation is often a prerequisite to
  cross-section aggregation since risk measures for
  market risk are often computed at different
  horizons with respect to those for credit and
  operational risk.

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Risk management of funds

• Temporal aggregation is a key feature of risk
  measurement of returns on managed funds. You are
  not interest in a 10 day risk measure if you are
  evaluating investment in a mutual fund. You are
  even less interested if you are planning to
  invest in a closed-end fund or a hedge fund.
• The issue is to compute VaR or other risk
  measures over different time horizons, an issue
  called long term VaR.
• The state of the art is i) to include a drift in
  the specification of the dynamics and ii) to
  apply the square-root rule, that is to multiply
  the risk measure by T1/2.
• The square root law is obviously subject to
  very strict assumptions. The process must have
  i.i.d. gaussian innovations.

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Copula functions

• Main Concepts

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Compatibility problems

• In statistics compatibility refers to the
  relationship between joint distributions (say of
  dimension n) and marginal distributions (for all
  dimensions k lt n).
• In finance compatibility means that the price of
  multi-variate derivatives, namely the value of
  contingent claims written on a set of events have
  to be consistent with the values of derivatives
  written on subsets of the events.

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Single bets

• Assume you get 1000 if 3 months from now the US
  stock market is at least 2 lower than today and
  zero otherwise. How much are you willing to pay
  for this bet? Say 200 . This price is linked to
  a 20 probability of success.
• Assume you get 1000 if 3 months from now the
  Canadian stock market is at least 3 lower than
  today and zero otherwise. How much are you
  willing to pay for this bet? Say 200 . This
  price is linked to a 20 probability of success.

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Multiple bets (Altiplano)

• Assume you get 1000 iff
• 3 months from now the US stock market is at least
  2 lower than today
• AND the Canadian market is at least 3 lower than
  today.
• How much are you willing to pay for this bet? Say
  66.14 . This price is linked to a 6.614
  probability of success. Of course
• 6.614 v(t,T)C(20,20)

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Copula functions

• Copula functions are based on the principle of
  integral probability transformation.
• Given a random variable X with probability
  distribution FX(X). Then u FX(X) is uniformly
  distributed in 0,1. Likewise, we have v FY(Y)
  uniformly distributed.
• The joint distribution of X and Y can be written
• H(X,Y) H(FX 1(u), FY 1(v)) C(u,v)
• Which properties must the function C(u,v) have in
  order to represent the joint function H(X,Y) .

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Copula function Mathematics

• A copula function z C(u,v) is defined as
• 1. z, u and v in the unit interval
• 2. C(0,v) C(u,0) 0, C(1,v) v and C(u,1) u
• 3. For every u1 gt u2 and v1 gt v2 we have
• VC(u,v) ?
• C(u1,v1) C (u1,v2) C (u2,v1) C(u2,v2) ? 0
• VC(u,v) is called the volume of copula C

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Copula functions Statistics

• Sklar theorem each joint distribution H(X,Y) can
  be written as a copula function C(FX,FY) taking
  the marginal distributions as arguments, and vice
  versa, every copula function taking univariate
  distributions as arguments yields a joint
  distribution.

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Copula functions and dependence structure in risks

• Copula functions represent a tool to separate the
  specification of marginal distributions and the
  dependence structure.
• Say two risks A and B have joint probability
  H(X,Y) and marginal probabilities FX and FY. We
  have that H(X,Y) C(FX , FY), and C is a copula
  function.
• Examples
• C(u,v) uv, independence
• C(u,v) min(u,v), perfect positive dependence
• C(u,v) max (u v - 1,0) perfect negative
  dependence
• The perfect dependence cases are called Fréchet
  bounds.

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Positive orthant dependence

• Copula functions are clearly linked to
  dependence.
• The first measure of dependence we could think of
  refers to the sign.
• Positive (negative) orthant dependency determines
  whether variables co-move in the same direction
  or in opposite directions. In the previous
  example
• C(20,20) 6,614 gt 0.20.2 4

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Copula function and dependence structure

• Copula functions are linked to non-parametric
  dependence statistics, as in example Kendalls ?
  or Spearmans ?S
• Notice that differently from non-parametric
  estimators, the linear correlation ? depends on
  the marginal distributions and may not cover the
  whole range from 1 to 1, making the
  assessment of the relative degree of dependence
  involved.

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Dualities among copulas

• Consider a copula corresponding to the
  probability of the event A and B, Pr(A,B)
  C(Ha,Hb). Define the marginal probability of the
  complements Ac, Bc as Ha1 Ha and Hb1 Hb.
• The following duality relationships hold among
  copulas
• Pr(A,B) C(Ha,Hb)
• Pr(Ac,B) Hb C(Ha,Hb) Ca(Ha, Hb)
• Pr(A,Bc) Ha C(Ha,Hb) Cb(Ha,Hb)
• Pr(Ac,Bc) 1 Ha Hb C(Ha,Hb) C(Ha, Hb)
• Survival copula
• Notice. This property of copulas is paramount to
  ensure put-call parity relationships in option
  pricing applications.

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Conditional probability I

• The dualities above may be used to recover the
  conditional probability of the events.

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Conditional probability II

• The conditional probability of X given Y y can
  be expressed using the partial derivative of a
  copula function.

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Tail dependence in crashes

• Copula functions may be used to compute an index
  of tail dependence assessing the evidence of
  simultaneous booms and crashes on different
  markets
• In the case of crashes

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and in booms

• In the case of booms, we have instead
• It is easy to check that C(u,v) uv leads to
  lower and upper tail dependence equal to zero.
  C(u,v) min(u,v) yields instead tail indexes
  equal to 1.

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Examples of copula functions The Fréchet family

• C(x,y) bCmin (1 a b)Cind aCmax , a,b
  ?0,1
• Cmin max (x y 1,0), Cind xy, Cmax
  min(x,y)
• The parameters a,b are linked to non-parametric
  dependence measures by particularly simple
  analytical formulas. For example
• ?S a - b
• Mixture copulas (Li, 2000) are a particular case
  in which copula is a linear combination of Cmax
  and Cind for positive dependent risks (agt0, b
  0), Cmin and Cind for the negative dependent
  (bgt0, a 0).

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Fréchet Family C(0.3,0,3)
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Fréchet Family C(0.3,0.7)
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Examples of copula functionsEllictical copulas

• Ellictal multivariate distributions, such as
  multivariate normal or Student t, can be used as
  copula functions.
• Normal copulas are obtained
• C(u1, un )
• N(N 1 (u1 ), N 1 (u2 ), , N 1 (uN )
  ?)
• and extreme events are indipendent.
• For Student t copula functions with v degrees of
  freedom C (u1, un )
• T(T 1 (u1 ), T 1 (u2 ), , T 1 (uN ) ?,
  v)
• extreme events are dependent, and the tail
  dependence index is a function of v.

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Gaussian copula vs mixture
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Examples of copula functionsArchimedean copulas

• Archimedean copulas are build from a suitable
  generating function ? from which we compute
• C(u,v) ? 1 ?(u)?(v)
• The function ?(x) must have precise properties.
  Obviously, it must be ?(1) 0. Furthermore, it
  must be decreasing and convex. As for ?(0), if it
  is infinite the generator is said strict.
• In n dimension a simple rule is to select the
  inverse of the generator as a completely monotone
  function (infinitely differentiable and with
  derivatives alternate in sign). This identifies
  the class of Laplace transform.

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Examples Frank copula

• Take
• ?(t) log(1 exp( ?t)/(1 exp( ?)
• such that the inverse is
• ? 1(s) ? 1 log(1 (1 exp( ?)exp( s)
  
• the Laplace transform of the logarithmic series.
  Then, the copula function
• C(u1,, un) ? 1 ?(u1)?(un)
• is called Frank copula. It is symmetric and does
  not have tail dependence (either lower or upper).

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Frank copula vs gaussian
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Examples Clayton copula

• Take
• ?(t) t ? 1/ ?
• such that the inverse is
• ? 1(s) (1 ?s) 1/ ?
• the Laplace transform of the gamma distribution.
• Then, the copula function
• C(u1,, un) ? 1 ?(u1)?(un)
• is called Clayton copula. It is not symmetric
  and has lower tail dependence (no upper tail
  dependence).

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Clayton copula vs Frank copula
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Examples Gumbel copula

• Take
• ?(t) (log t)?
• such that the inverse is
• ? 1(s) exp( s 1/? )
• the Laplace transform of the positive stable
  distribution.
• Then, the copula function
• C(u1,, un) ? 1 ?(u1)?(un)
• is called Gumbel copula. It is not symmetric and
  has upper tail dependence (no lower tail
  dependence).

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Gumbel copula vs Frank copula
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Kendall function

• For the class of Archimedean copulas, there is a
  multivariate version of the probability integral
  transfomation theorem.
• The probability t C(u,v) is distributed
  according to the distribution
• KC (t) t ?(t)/ ?(t)
• where ?(t) is the derivative of the generating
  function. There exist extensions of the Kendall
  function to n dimensions.
• Constructing the empirical version of the Kendall
  function enables to test the goodness of fit of a
  copula function (Genest and McKay, 1986).

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Kendall function Clayton copula
46
Hedge Funds Market Neutral
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Hierarchical copulas

• Consider a set of copula functions
• C(u1,, unm)
• ?21 1 ?21 ??11 1(?11(u1)?11(un))
• ?21 ??12
  1(?12(u1)?12(um)
• where ?21 , ?11 and ?12 are generators.
• This is a copula iff the dependence of the ?21
  generator is lower or equal to the dependence of
  the ?11 and ?12 generators
• This is called Archimedean hierarchical copula
  and allows to extend Archimedean copula to
  arbitrary dimensions with different bivariate
  dependence

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Radial symmetry and exchangeability
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Radial symmetry

• Take a copula function C(u,v) and its survival
  version
• C(1 u, 1 v) 1 v u C( u, v)
• A copula is said to be endowed with the radial
  symmetry (reflection symmetry) property if
• C(u,v) C(u, v)

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Radial symmetry example

• Take u v 20. Take the gaussian copula and
  compute N(u,v 0,3) 0,06614
• Verify that
• N(1 u, 1 v 0,3) 0,66614
• 1 u v N(u,v 0,3)
• Try now the Clayton copula and compute Clayton(u,
  v 0,2792) 0,06614 and verify that
• Clayton(1 u, 1 v 0,2792) 0,6484 ? 0,66614

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Radial symmetry economics

• In economics and econometrics, radial symmetry
  has led to discover phenomena of correlation
  asymmetry.
• Empirical evidence have been found that
  correlation is higher for downward moves of the
  stock market than for upward moves (Longin and
  Solnik, Ang and Chen among others).

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Exceedance correlation

• Longin and Solnik have first proposed the concept
  of exceedance correlation correlation measured
  on data sampled in the tails.
• Step 1. Standardize data si (Si ?) /?
• Step 2. Select sub-samples si gt ?, si lt ?
• Step 3. Corr (si gt ?, sj gt ?),Corr (si lt ?, sj
  lt ?)
• For (radial) symmetric distributions
• Corr (si gt ?, sj gt ?) Corr (si lt ?, sj lt ?)

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Exceedance rank-correlation

• Schmid and Schmidt propose a similar concept of
  conditional rank-correlation.

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Exchangeable copulas

• Most of the copula functions used in finance are
  symmetric or exchangeable, meaning
• C(u,v) C(v,u)
• In a recent paper, Nelsen proposes a measure of
  non-exchangeability
• 0 ? 3 sup C(u,v) C(v,u) ? 1
• Nelsen also identifies a class of maximum
  non-exchangeable copulas.

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Non exchangeable copulas

• A way to extend copula functions to account for
  non-exchangeability was suggested by Khoudraji
  (Phd dissertation, 1995).
• Take copula functions C(.,.) and C(.,.), and 0 lt
  ?, ? lt 1 and define
• C?, ?(u,v) C(u1 ?, v1 ?) C(u ?, v ?)
• The copula function obtained is in general
  non-exchangeable. In particular, this was used by
  Genest, Ghoudi and Rivest (1998) taking C(.,.)
  the product copula and C(.,.) the Gumbel copula
• C?, ?(u,v) u1 ?v1 ?C(u ?, v ?)

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Non-exchangeable copulas example

• Take u 0,2 and v 0,7 and the Gaussian copula.
  Verify that
• N(u, v 30) N (v, u 30) 16,726
• Compute now
• C(u,v) u0,5 N(u0,5, v 30) 15,511
• and
• C(v,u) v0,5 N(v0,5, u 30) 15,844

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Non-Exchangeability economics

• To have an idea of the ecnomic meaning of
  exchangeability, consider the following case
• Lehmann Bros provides capital insurance to life
  insurance policies to Mediolanum, an Italian
  bank-insurance company. Assume that in case of
  default of Lehmann, Mediolanum takes over the
  cost of insurance (as it actually happened after
  September 15 2008)
• Consider now the joint probability of default of
  Lehmann and Mediolanum. If the marginal
  probability of default of Lehmann is high, the
  joint distribution will also be high. But if the
  marginal probability of Mediolanum is high, the
  joint distribution might not be as high as in the
  previous case.

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C(u,v) C(v,u)
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Partial exchangeability

• Notice that the Archimedean hierarchical copula
  allows for some non-exchangeability among
  variables in different groups.
• For example, assume a hierarchical representation
  in which assets are grouped by sector at the
  lower level and between sector at the higher
  level. Of course, the dependence structure within
  the group is exchangeable, but between the groups
  it is not. This case is called partial
  exchangeability.
• Econometricians would recognize similarities with
  the within and between estimators.

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Estimation and simulation
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Copula function calibration

• A first straightforward way of determining the
  copula function representing the dependence
  structure between two variables was proposed by
  Genest and McKay, 1986.
• The algorithm is particularly simple
• Change the set of variables in ranks
• Measure the association between the ranks
• Determine the parameter of the copula (a family
  of copula has to be chosen) in order to obtain
  the same association measure.
• Plot the Kendall function of each copula and
  select the copula which is closest to the
  empirical Kendall function.

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Nasdaq 100 vs Nikkey 225
63
Gumbel fit
64
Clayton fit
65
Copula density

• The cross derivative of a copula function is its
  density.
• The copula density times the marginal density
  yields the joint density
• The density is also called the canonical
  representation of a copula.

66
Copula function likelihood

• Using the canonical representation of copulas one
  can write the log-likelihood of a set of data.
• Notice that the likelihood may be partitioned in
  two parts one only depends on the copula density
  and the other only on the marginal densities.

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Maximum Likelihood Estimation

• Maximum Likelihood Estimation (MLE). The
  Likelihood is written and maximised with respect
  to both the parameters of the marginal
  distributions and those of the copula function
  simultaneously
• Inference from the margin (IFM). The likelihood
  is maximised in two stages by first estimating
  the parameters of the marginal distributions and
  then maximizing it with respect to the copula
  function parameter
• Canonical Maximum Likelihood (CML). The marginal
  distribution is not estimated but the data are
  transformed in uniform variates.

68
Conditional copula functions

• A problem with specification of the copula
  function is that both dependence parameters and
  the marginal distributions can change as time
  elapses
• The conditional copula proposal (Patton) is a
  solution to this problem
• The key feature is that Sklars theorem can be
  extended to conditional distribution if both the
  margins and the copula function are function of
  the same information set.

69
Conditional copula estimation.

• The conditional copula is
• C(H(S1tIt), H(S2tIt) ? It)
• Step 1. Estimate Garch models for variables S1
  and S2
• Step 2. Apply the probability integral transform
  to both S1 and S2 and test the specification
• Step 3. Estimate the dynamics of the dependence
  parameter in a ARMA model
• ?t ?(f(?t - 1,u t - 1,u t p , v t - 1,,v t
  p ))
• with ? ?? (0, 1)

70
Dynamic copula functions

• An alternative, proposed by Van der Goorberg,
  Genest Verker is based on the estimation, for
  Archimedean copulas, of the dependence
  non-parametric statistic as a function of
  marginal conditional volatilities
• In particular, they specify Kendalls ? as
• ?t ?0 ?1 log (max(h1t,h2t))

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Monte Carlo simulationGaussian Copula

• Cholesky decomposition A of the correlation
  matrix R
• Simulate a set of n independent random variables
  z (z1,..., zn) from N(0,1), with N standard
  normal
• Set x Az
• Determine ui N(xi) with i 1,2,...,n
• (y1,...,yn) F1-1(u1),...,Fn-1(un) where Fi
  denotes the i-th marginal distribution.

72
Monte Carlo simulationStudent t Copula

• Cholesky decomposition A of the correlation
  matrix R
• Simulate a set of n independent random variables
  z (z1,..., zn) from N(0,1), with N standard
  normal
• Simulate a random variable s from ?2? indipendent
  from z
• Set x Az
• Set x (?/s)1/2y
• Determine ui Tv(xi) with Tv the Student t
  distribution
• (y1,...,yn) F1-1(u1),...,Fn-1(un) where Fi
  denotes the i-th marginal distribution.

73
Other simulation techniques

• Conditional sampling the first marginal is
  obtained by generating a random variable from the
  uniform distribution. The others are obtained by
  generating other uniform random variables and
  using the inverse of the conditional
  distribution.
• Marshall-Olkin Laplace transforms and their
  inverse are used to generate the joint variables.
  

74
Copula pricing cross-section dependence
75
Digital Binary Note Example

• Investment period March 2000 - March 2005
• Principal repaid at maturity
• Coupon paid on march 15th every year.
• Coupon determination
• Coupon 10 if (i 1,2,3,4)
• Nikkei (15/3/200i) gt Nikkei (15/3/2000) e
• Nasdaq 100 (15/3/200i) gt Nasdaq 100 (15/3/2000)
• Coupon 0 otherwise
• Digital note ZCB bivariate digital call
  options

76
Coupon determination
77
Super-replication

• It is immediate to check that
• MaxDCNky DCNsd v(t,T),0 Coupon
• and
• Coupon MinDCNky,DCNsd
• otherwise it will be possible to exploit
  arbitrage profits.
• Fréchet bounds provide super-replication prices
  and hedges, corresponding to perfect dependence
  scenarios.

78
Copula pricing

• It may be easily proved that in order to rule out
  arbitrage opportunities the price of the coupon
  must be
• Coupon v(t,T)C(DCNky/v(t,T),DCNsd /v(t,T))
• where C(u,v) is a survival copula representing
  dependence between the Nikkei and the Nasdaq
  markets.
• Intuition.Under the risk neutral probability
  framework, the risk neutral probability of the
  joint event is written in terms of copula, thanks
  to Sklar theorem,the arguments of the copula
  being marginal risk neutral probabilities,
  corresponding to the forward value of univariate
  digital options.
• Notice however that the result can be prooved
  directly by ruling out arbitrage opportunities on
  the market. The bivariate price has to be
  consistent with the specification of the
  univariate prices and the dependence structure.
  Again by arbitrage we can easily price

79
a bearish coupon
80
Bivariate digital put options

• No-arbitrage requires that the bivariate digital
  put option, DP with the same strikes as the
  digital call DC be priced as
• DP v(t,T) DCNky DCNsd DC
• v(t,T)1 DCNky /v(t,T) DCNsd /v(t,T)
• C(DCNky /v(t,T),DCNsd /v(t,T))
• v(t,T)C(1 DCNky /v(t,T),1 DCNsd /v(t,T))
• v(t,T)C(DPNky /v(t,T),DPNsd /v(t,T))
• where C is the copula function corresponding to
  the survival copula C, DPNky and DPNsd are the
  univariate put digital options.
• Notice that the no-arbitrage relationship is
  enforced by the duality relationship among
  copulas described above.

81
Radial symmetry finance

• Now consider the following pricing problem.
• Bivariate digital call on Nikkei and Nasdaq with
  marginal probability of exercise equal to u and v
  respectively.
• Bivariate digital put on Nikkei and Nasdaq with
  marginal probability of exercise equal to u and v
  respectively.
• Radial symmetry means that
• C(u,v) C(u,v)
• so that
• DP(u,v) DC(u,v)
• Imagine to recover implied correlation from call
  and put prices you would recover a symmetric
  correlation smile

82
Pricing strategies

• The pricing of call and put options whose pay-off
  is dependent on more than one event may be
  obtained by
• Integrating the the value of the pay-off with
  respect to the copula density times the marginal
  density
• Integrating the conditional probability
  distribution times the marginal distribution of a
  risk factor
• Integrating the joint probability distribution

83
From the pricing kernel to options

• The idea relies on Breeden and Litzenberger
  (1978)
• By integrating the pricing kernel (i.e. the
  cumulative or decumulative risk neutral
  distribution) we may recover put and call prices
• From simple digital call and put options we can
  recover call and put prices simply set Pr(S(T)
  u) Q(u)

84
Joint probability distribution approach

• Assume a product with pay-off
• Maxf(S1(T), S2(T)) K, 0
• The price can be computed as

85
Conditional probability distribution approach

• Assume a product with pay-off
• Maxf(S1(T), S2(T)) K, 0
• The price can be computed as

86
AND/OR operators

• Copula theory also features more tools, which are
  seldom mentioned in financial applications.
• Example
• Co-copula 1 C(u,v)
• Dual of a Copula u v C(u,v)
• Meaning while copula functions represent the AND
  operator, the functions above correspond to the
  OR operator.

87
Equity-linked bonds

• Assume a coupon which is defined and paid at time
  T.
• Assume a basket of n 1,2,N assets, whose
  prices are Sn(T).
• Denote Sn(t0) the reference prices, typically
  registered at the origin of the contract, and
  used as strike prices.
• The coupon of a basket option is
• maxAverage(Sn(T)/Sn(0),1k
• (1 k) maxAverage(Sn(T)/Sn(0) (1k),0
• with n 1,2,,N and a minimum guaranteed return
  equal to k.

88
Everest

• Assume a basket of n 1,2,N assets, whose
  prices are Sn(T).
• Denote Sn(t0) the reference prices, typically
  registered at the origin of the contract, and
  used as strike prices.
• The coupon of an Everest note is
• maxmin(Sn(T)/Sn(0),1k
• (1 k) maxmin(Sn(T)/Sn(0) (1k),0
• with n 1,2,,N and a minimum guaranteed return
  equal to k.
• The replicating portfolio is
• Everest note ZCB 2-colour rainbow (call on
  minimum)

89
Exercises

• Verify that a product giving a call on the
  maximum of a basket is short correlation
• Hint 1 ask whether the pay-off includes a AND or
  OR operator
• Hint 2 verify the result writing the replicating
  portfolio in a bivariate setting
• Verify that a long position in a first to
  default swap is short correlation

90
Copula pricingtemporal dependence
91
Copula product

• The product of a copula has been defined (Darsow,
  Nguyen and Olsen, 1992) as
• AB(u,v) ?
• and it may be proved that it is also a copula.

92
Markov processes and copulas

• Darsow, Nguyen and Olsen, 1992 prove that 1st
  order Markov processes (see Ibragimov, 2005 for
  extensions to k order processes) can be
  represented by the ? operator (similar to the
  product)
• A (u1, u2,, un) ?B(un,un1,, unk1) ?
• i

93
Properties of ? products

• Say A, B and C are copulas, for simplicity
  bivariate, A survival copula of A, B survival
  copula of B, set M min(u,v) and ? u v
• (A ? B) ? C A ? (B ? C) (Darsow et al. 1992)
• A? M A, B? M B (Darsow et al. 1992)
• A? ? B? ? ? (Darsow et al. 1992)
• A ? B A ? B (Cherubini Romagnoli, 2008)

94
Symmetric Markov processes

• Definition. A Markov process is symmetric if
• Marginal distributions are symmetric
• The ? product
• T1,2(u1, u2) ? T2,3(u2,u3) ? Tj 1,j(uj 1 ,
  uj)
• is radially symmetric
• Theorem. A ? B is radially simmetric if either i)
  A and B are radially symmetric, or ii) A ? B A
  ? A with A exchangeable and A survival copula of
  A.

95
Example Brownian Copula

• Among other examples, Darsow, Nguyen and Olsen
  give the brownian copula
• If the marginal distributions are standard
  normal this yields a standard browian motion. We
  can however use different marginals preserving
  brownian dynamics.

96
Time Changed Brownian Copulas

• Set h(t,?) an increasing function of time t,
  given state ?. The copula
• is called Time Changed Brownian Motion copula
  (Schmidz, 2003).
• The function h(t,?) is the stochastic clock. If
  h(t,?) h(t) the clock is deterministic (notice,
  h(t,?) t gives standard Brownian motion).
  Furthermore, as h(t,?) tends to infinity the
  copula tends to uv, while as h(s,?) tends to
  h(t,?) the copula tends to min(u,v)

97
Copula martingale processes

• A problem for pricing applications is to impose
  the martingale restriction in the Markov process
  representation.
• Cherubini, Mulinacci and Romagnoli, 2008 propose
  a particular strategy to use copula functions to
  build martingale process.
• The novelty of the idea is to use copula
  functions to model the dependence structure
  between the increment of a stochastic process and
  its level before the increment.

98
CheMuRo Model

• Take three continuous distributions F, G and H.
  Denote C(u,v) the copula function linking levels
  and increments of the process and D1C(u,v) its
  partial derivative. Then the function
• is a copula iff

99
A special class of processes

• F represents the probability distribution of
  increments of the process, H represents the
  distribution of the level of the process before
  the increment and G represents the level of the
  process after the increment.
• Distribution G is obtained by an operation that
  we denote C-convolution of F and H.
• Lévy processes are obtained as a class in which
• C(u.v) uv, the operator is the convolution.
• F G H increments are stationary

100
Dependent increment with gaussian marginals

• As an example, consider the increments to be
  standard normal F ?. We have
• Notice that marginals are gaussian, since they
  refer to the sum of gaussian variables The
  dependence structure is instead given by the
  copula function C(u,v).

101
Temporal dependence and scaling law

• Notice that given a marginal distribution and a
  copula describing the link between increment and
  the level before it, we simultaneously derive the
  marginal distribution of the level following the
  increment and the dependence structure with the
  level before.
• Notice that once distribution of the increments
  and dependence with the levels have been
  selected, the dynamics of the process is
  completely specified. So, there is a relationship
  between dependence of the increments and scaling
  law of the process.

102
Clayton dependence and gaussian marginals 5 perc
103
Dependent increments? SP 500
104
Dependent increments? USD/Euro
105
HFIndex convertible arbitrage
106
HFIndex Dedicated short bias
107
HFIndex Emerging Markets
108
Copula based dynamics

• Given the evidence above the class of model that
  seems more appropriate to describe the dynamics
  of assets seems to be
• with

109
Martingale restrictions

• In the model with independent increments the
  martingale requirement is very easy to implement.
  In fact, it suffices to choose zero mean
  distributions for increments.
• Notice that independent increments are not a
  necessary requirement. In fact, the martingale
  requirement may be ensured for symmetric
  distributions of increments given specific
  dependence structures.

110
A general dynamic model for equity markets
111
The model of the market

• Our task is to model jointly cross-section and
  time series dependence.
• Setting of the model
• A set of ?S1, S2, ,Sm? assets conditional
  distribution
• A set of ?t0, t1, t2, ,tn? dates.
• We want to model the joint dynamics for any time
  tj, j 1,2,,n.
• We assume to sit at time t0, all analysis is made
  conditional on information available at that
  time. We face a calibration problem, meaning we
  would like to make the model as close as possible
  to prices in the market.

112
Assumptions

• Assumption 1. Risk Neutral Marginal Distributions
  The marginal distributions of prices Si(tj)
  conditional on the set of information available
  at time t0 are Qi j
• Assumption 2. Markov Property. Each asset is
  generated by a first order Markov process.
  Dependence of the price Si(tj -1) and asset
  Si(tj) from time tj-1 to time tj is represented
  by a copula function Tij 1,j(u,v)
• Assumption 3. No Granger Causality. The future
  price of every asset only depends on his current
  value, and not on the current value of other
  assets. This implies that the m x n copula
  function admits the hierarchical decomposition
• C(G1 (Q11, Q12, Q1n), Gm(Qm1, Qm2, Qmn))

113
No-Granger Causality

• The no-Granger causality assumption, namely
• P(Si(tj)? S1(tj 1),, Sm(tj 1)) P(Si(tj)?
  Si(tj 1))
• enables the extension of the martingale
  restriction to the multivariate setting.
• In fact, we assuming Si(t) are martingales with
  respect to the filtration generated by their
  natural filtrations, we have that
• E(Si(tj)?S1(tj 1),, Sm(tj 1))
• E(Si(tj)?Si(tj 1)) S(t0)
• Notice that under Granger causality it is not
  correct to calibrate every marginal distribution
  separately.

114
Running maxima and minima

• Due to the first order Markov assumption the
  dynamics of each asset can be represented as a
  function of the bivariate copulas Tij 1,j(u,v)
• Running Maximum Gij (u1, u2, uj)
• Ti1,2(u1, u2) ? Ti2,3(u2,u3) ? Ti j 1,j(uj
  1 , uj)
• Running Minimum Gij (u1, u2, uj)
• Ti1,2(u1, u2) ? Ti2,3(u2,u3) ? Ti j 1,j(uj
  1 , uj)
• Notice that the operator implies recursions like
• Gij (u1, u2, uj)
• Gij 1(u1, u2, uj 1) ? Ti j 1,j(uj 1 ,
  uj)

115
Univariate barrier Altiplanos

• Risk neutral valuation implies compatibility
  restrictions between barrier and European
  options. In fact, set
• DC(K, tk) forward price of the European digital
  option paying one unit of cash iff S(tk) gt K.
• NT(K, tk) forward price of the barrier digital
  option paying one unit of cash iff S(tp) gt K, for
  all p 1,2,,k.
• Price compatibility requires then
• NT(K, tk) NT(K,tk1) ? Ti k 1,k(DC(K, tk1) ,
  DC(K, tk))
• Notice that the Markov property assumption
  implies this recursive structure (bootstrapping)

116
Barrier BootstrappingBrownian motion and O-U
clock
117
Cross-section compatibility

• Assume Q(Si(tm) gt H, Sj(tm) gt K)
• ?(Q(Si(tm) gt H),Q(Sj(tm) gt K)), then
• Q(mink?m Si(tk) gt H, mink?m Sj(tk) gt K)
• ?(Q(mink?m Si(tk) gt H),Q(mink?mSj(tk) gt K)))
• Sketch of proof for the symmetric case. Use the
  reflection principle Pr(mink?m Si(tk) gt K) 2
  Pr(Si(tm) gt K) 1 to prove
• Pr(mink?m Si(tk) gt H, mink?m Si(tk) gt K)
• C(Pr(mink?m Si(tk) gt H), Pr(mink?m Sj(tk) gt K)
• C(Pr(Ui gt ui), Pr(Uj gt uj))
• C(Pr(Ui 1)/2gt (ui 1)/2, Pr(Uj 1)/2 gt (uj
  1)/2
• Pr(Si(tm) gt H, Sj(tm) gt K) ?((ui 1)/2, (uj
  1)/2)

118
European and Barrier Altiplanos

• DCi(K, tm) forward price of the European
  digital option paying one unit of cash iff Si(tm)
  gt K.
• NTi(K, tm) forward price of the barrier digital
  option paying one unit of cash iff Si(tp) gt K,
  for all p 1,2,,m.
• The price of a European Altiplano is
• EA ?(DC1(K, tm), DC2(K, tm),, DCn(K, tm))
• The price of a barrier Altiplano is
• BA ?(NT1(K, tm), NT2(K, tm),, NTn(K, tm))

119
European and barrier AltiplanosTemporal
Dependence Shocks
120
European and barrier AltiplanosCross-Section
Dependence Shocks
121
Multivariate credit products
122
Application to credit market

• Assume the following data are given
• The cross-section distribution of losses in every
  time period ti 1,ti (Y(ti )). The
  distribution is Fi.
• A sequence of copula functions Ci(x,y)
  representing dependence between the cumulated
  losses at time ti 1 X(ti 1), and the losses
  Y(ti ).
• Then, the dynamics of cumulated losses is
  recovered by iteratively computing the
  convolution-like relationship

123
A temporal aggregation algorithm

• Denote X(ti 1) level of a variable at time ti
  1 and Hi 1 the corresponding distribution.
• Denote Y(ti ) the increment of the variable in
  the period ti 1,ti. The corresponding
  distribution is Fi.
• Start with the probability distribution of
  increments in the first period F1 and set F1
  H1.
• Numerically compute
• where z is now a grid of values of the variable
• 3. Go back to step 2, using F3 and H2 compute
  H3

124
Distribution of losses 10 y
125
Temporal dependence
126
Equity tranche term structure
127
Senior tranche term structure
128
Houston, we have a problem

• The application of the algorithm to credit leads
  to a problem. As the support of the amount of
  default is bounded, the algorithm must be
  modified accordingly, including constraints.
• Continuous distribution of losses
• D1C (w,FY(K FX1(w))) 1, ? w ? 0,1
• Discrete distribution of losses
• C(FX(j),FY(K j)) C(FX(j 1),FY(K j)) P(X
  j)
• j 0,1,,K
• These constraints define a recursive system that
  given the initial distribution of losses and the
  temporal dependence structure yields the
  distribution of losses in future periods.

129
Risk management applications
130
Value-at-Risk Aggregation

• Assume you want to compute the Value at Risk of
  an investment on a hedge funds over different
  time horizons.
• Applying the square root law would imply
  independent increments, which is inconsistent
  with findings for hedge funds.
• One could estimate the dependence structure
  between increments and levels and apply the
  aggregation algorithm described above.

131
Value-at-Risk for Hedge Funds
132
Counterpart risk in derivatives

• Most of the derivative contracts, particularly
  options, forward and swaps, are traded on the OTC
  market, and so they are affected by credit risk
• Credit risk may have a relevant impact on the
  evaluation of these contracts, namely,
• The price and hedge policy may change
• Linear contracts can become non linear
• Dependence between the price of the underlying
  and counterparty default should be accounted for

133
The replicating portfolio approach

• The idea is to design a replicating portfolio to
  hedge and price counterparty derivatives
• Goes back to Sorensen and Bollier (1994)
  approach counterparty risk in swaps represented
  as a sequence of swaptions
• Copula functions may be used to extend the idea
  to dependence between counterparty risk and the
  underlying

134
Dependence structure

• A more general approach is to account for
  dependence between the two main events under
  consideration
• Exercise of the option
• Default of the counterparty
• Copula functions can be used to describe the
  dependence structure between the two events
  above.

135
Vulnerable digital call option

• Consider a vulnerable digital call (VDC) option
  paying 1 euro if S(T) gt K (event A). In this
  case, if the counterparty defaults (event B), the
  option pays the recovery rate RR.
• The payoff of this option is
• VDC v(t,T)H(A,Bc)RR H(A,B)
• v(t,T) Ha H(A,B)RR H(A,B)
• v(t,T)Ha (1 RR)H(A,B)
• DC v(t,T) Lgd C(Ha, Ha)

136
Vulnerable digital put option

• Consider a vulnerable digital put (VDP) option
  paying 1 euro if S(T) K (event Ac). In this
  case, if the counterparty defaults (event B), the
  option pays the recovery rate RR.
• The payoff of this option is
• VDP DP v(t,T)(1 RR)H(Ac,B)
• P(t,T)Ha v(t,T)(1 RR)H(Ac,B)
• P(t,T)Ha v(t,T)(1 RR)Hb C(Ha, Hb)
• v(t,T)(1 Ha) v(t,T) Lgd Hb
  C(Ha, Hb)
• v(t,T) VDC v(t,T) Lgd Hb

137
Vulnerable digital put call parity

• Define the expected loss EL Lgd Hb.
• If D(t,T) is a defaultable ZCB issued by the
  counterparty we have
• D(t,T) v(t,T)(1 EL)
• Notice that copula duality implies a clear
  no-arbitrage relationship
• VDC VDP v(t,T) v(t,T) EL D(t,T)
• Buying a vulnerable digital call and put option
  from the same counterparty is the same as buying
  a defaultable zero-coupon bond

138
Vulnerable call and put options
139
Vulnerable put-call parity
140
Example swap credit risk Counterparty BBB
141
Reference Bibliography I

• Nelsen R. (2006) Introduction to copulas, 2nd
  Edition, Springer Verlag
• Joe H. (1997) Multivariate Models and Dependence
  Concepts, Chapman Hall
• Cherubini U. E. Luciano W. Vecchiato (2004)
  Copula Methods in Finance, John Wiley Finance
  Series.
• Cherubini, U. (2004) Pricing Swap Credit Risk
  with Copulas, working paper
• Cherubini U. E. Luciano (2003) Pricing and
  Hedging Credit Derivatives with Copulas,
  Economic Notes, 32, 219-242.
• Cherubini U. E. Luciano (2002) Bivariate
  Option Pricing with Copulas, Applied
  Mathematical Finance, 9, 69-85
• Cherubini U. E. Luciano (2002) Copula
  Vulnerability, RISK, October, 83-86
• Cherubini U. E. Luciano (2001) Value-at-Risk
  Trade-Off and Capital Allocation with Copulas,
  Economic Notes, 30, 2, 235-256

142
Reference bibliography II

• Cherubini U. S. Mulinacci S. Romagnoli
  (2008) Copula Based Martingale Processes and
  Financial Prices Dynamics, working paper.
• Cherubini U. Mulinacci S. S. Romagnoli
  (2008) A Copula-Based Model of the Term
  Structure of CDO Tranches, in Hardle W.K., N.
  Hautsch and L. Overbeck (a cura di) Applied
  Quantitative Finance,,Springer Verlag, 69-81
• Cherubini U. S. Romagnoli (2008) The
  Dependence Structure of Running Maxima and
  Minima Results and Option Pricing Applications,
  Mathematical Finance, forthcoming
• Cherubini U. S. Romagnoli (2008) Computing
  Copula Volume in n Dimensions, Applied
  Mathematical Finance, forthcoming