Correlation Products and Copula Functions

1
Correlation Products and Copula Functions

• Umberto Cherubini
• Matemates University of Bologna
• Birbeck College, London 14/02/2007

2
Outline

• Motivation structured finance products
• Copula functions main concepts
• Radial symmetry and non-exchangeability
• Estimation and calibration
• Copula pricing methods
• Altiplanos, Everest co
• Counterparty risk

3
Motivation
4
Correlation product example

• Investment period March 2000 - March 2005
• Principal repaid at maturity
• Coupon paid on march 15th every year.
• Coupon determination
• Coupon 10 if (i 1,2,3,4,5)
• 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

5
Questions

• In order to price and hedge this product both the
  investors and the issuer have to address three
  questions
• Is the product long or short in the underlying
  assets?
• Is the product long or short in the volatility of
  the underlying assets?
• Is the product long or short in the correlation
  between the underlying asset?

6
Risks

• A product like this is constructed to provide
  exposure to equity risk, with particular
  reference to correlation between Nasdaq and
  Nikkey (correlation products).
• The product is also exposed to other, ancillary
  risks in particular, credit risk of the issuer
  and possible correlation with the underlying
  (counterparty risk)

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))

8
Risks

• The first-to-default swap is constructed to
  provide exposure to default risk, with particular
  reference to correlation among the credit events
  (correlation products).
• The product is also exposed to other, ancillary
  risks in particular, the risk of term structure
  movements and credit risk of the issuer and
  possible correlation with the underlying
  (counterparty risk)

9
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.

10
Copula functions in finance

• Given the prices of single-bet financial
  contracts, what is the price of a multiple-bet
  contract? There is no unique answer to that
  question (Bernard Dumas)
• The fortune of copula applications to finance is
  due to the massive increase in supply and demand
  of multiple-bet products.

11
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.

12
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.

13
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)

14
Copula functions
15
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) .

16
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 the copula C
• Examples
• C(u,v) uv
• C(u,v) min(u,v)
• C(u,v) max(u v 1,0)

17
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.

18
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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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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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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Conditional probability I

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

22
Right/left tail in/decreasing

• Take two variables S1 and S2.
• S1 is said to be left tail decreasing in S2 if
  Pr(S1 ? x S2 ? y)
• is decreasing in S2.
• S1 is said to be right tail incrasing in S2 if
  Pr(S1 gt x S2 gt y)
• is increasing in S2.

23
Conditional probability II

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

24
Stochastic increasing

• Take two variables S1 and S2.
• S1 is said to be stochastic increasing or
  decreasing in S2 if
• Pr(S1 ? x S2 y)
• is increasing or decreasing in S2.

25
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.

26
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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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(u,v) 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(u,v) 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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Examples of copula functionsArchimedean copulas

• Archimedean copulas are build from a suitable
  generating function ? from which we compute
• C(u,v) ? 1 ?(u)?(v)
• An example is Clayton copula. Setting
• ?(t) t a 1/a
• we obtain
• C(u,v) maxu av a 1,0 1/a

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Radial symmetry and exchangeability
32
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)

33
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

34
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).

35
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 ?)

36
Conditioning bias

• Correlation figures measured on conditional
  samples are different from the unconditional
  figure (conditioning bias). Ahn and Chen computed
  the bias for the gaussian distribution in closed
  form, and proposed a measure of average
  exceedance correlation.
• The measure, H, is a weighted square difference
  of a set of empirical exceedance correlations
  ?(?) with respect to the theoretical figure
  ?(?).

37
Exceedance rank-correlation

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

38
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.

39
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 , assume
• Prob of a decrease in Euro wrt Dollar 50
• Prob of a decrease in Pound wrt Dollar 50
• Prob of a 4 drop in Euro wrt Dollar 1
• Prob of a 3 drop in Pound wrt Dollar 1
• Say the joint probability of a decrease in Euro
  and a major drop in Pound is higher than the
  joint probability of a decrease in Pound and a
  major drop in Euro.
• Dominance the destiny of Pound sterling is
  linked to that of Euro, more than the other way
  round

42
Clayton copula
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Clayton non-exchangeable
44
C(u,v) C(v,u)
45
C(0.5,u) vs C(v,0.5)
46
Estimation and simulation
47
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.

48
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.

49
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.

50
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.

51
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.

52
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)

53
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))

54
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.

55
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.

56
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.
  

57
Simulating non-exchangeable copulas

• To simulate the copula
• C?, ?(u,v) C(u1 ?, v1 ?) C(u ?, v ?)
• draw
• u1 and v1 from C(u,v)
• u2 and v2 from C(u,v)
• and determine
• u max(u11/?,u21/(1 ?))
• v max(v11/?,v21/(1 ?))

58
Copula pricing methods
59
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

60
Coupon determination
61
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.

62
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

63
a bearish coupon
64
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.

65
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

66
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
• Integrating the joint probability distribution

67
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)

68
Joint probability distribution approach

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

69
Conditional probability distribution approach

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

70
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.

71
Altiplanos, Everest co
72
Altiplanos

• Assume a coupon is set and paid at time tj (reset
  date).
• Assume a basket of n 1,2 assets, with prices
  Sn(tj).
• Call Sn(t0) the reference prices, typically
  recorded at the beginning of the contract and
  used as strikes.
• Call Ij the characteristic function taking value
  1 if Sn(tj)/Sn(t0) gt B and 0 otherwhise for both
  the assets.
• Coupon is a bivariate option, for coupon rate c

73
Altiplanos with memory

• Assume a coupon is set and paid at time tj (reset
  date).
• Assume a basket of n 1,2 titoli, with prices
  Sn(tj).
• Call Sn(t0) the reference prices, typically
  recorded at the beginning of the contract and
  used as strikes.
• Call Ij the characteristic function taking value
  1 if Sn(tj)/Sn(t0) gt B and 0 otherwhise for both
  the assets.
• Each time the characteristic function yields 1,
  coupons c are paid for that year and for all the
  previous years in which the trigger event had not
  taken place (memory feature).

74
Altiplano
75
Altiplano with memory B 70
76
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.

77
Everest

• 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 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)

78
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

79
An Altiplano with barrier bet

• Bet 1000 iff both the Canadian and the US
  markets do not lose more the 20 in a year from
  today.
• The underlying assets of the contract
• Running minimum of the Canadian market (must be
  greater than 80 of todays value)
• Running minimum of the US market (must be greater
  than 80 of todays value)

80
The Altiplano barrier problem

• Assume we observe a sequence of two viariables
• X X(t1), X(t2),, X(tm) Y Y(t1), Y(t2),,
  Y(tm)
• whose dependence is described by
• Q(X(tm) gt K, Y(tm) gt H) Cm(QXm (K), QYm (H))
• Which is the copula corresponding to the joint
  distribution
• Q(minj?mX(tj) gt K, minj?m Y(tj) gt H) ?

81
The idea
82
Dependence structure

• If event A(i) occurs (the running minimum of Y at
  time m is greater than H) then B(i) should take
  place as well (the level of Y at time m is
  greater than H) but not vice versa
• A(i) ? B(i)
• This implies
• C(Q(B(i) )?Q(A(i) )) min(Q(B(i) )?Q(A(i) ))
  Q(A(i) )

83
Cross-section compatibility

• Assume Q(X(tm) gt K, Y(tm) gt H)
• C(Q(X(tm) gt K),Q(Y(tm) gt H))), then
• Q(minj?m X(tj) gt K, minj?m Y(ti) gt H)
• C(Q(minj?m X(tj) gt K),Q(minj?m Y(tj) gt H)))
• Proof
• Q(X(tm) gt K, Y(tm) gt H) C (Q(B(1)),Q(B(2)))
• Q(A(1)? B(1)?A(2) ?B(2)
• C(min(Q(A(1)), Q(B(1)), min(Q(A(2)), Q(B(2)))
• Q(A(1)?A(2))
• Q(minj?m X(ti) gt K, minj?m Y(tj) gt H)

84
Altiplano

• Assume an Altiplano product that pays one unit of
  cash if all the assets Si are above a given
  barrier Bi at a future date tm.
• Denote Qmi(Bi) Pr((Si (tm) gt Bi), the
  probability that asset Si be above the barrier at
  time tm under the risk neutral measure.
• Denote C(Qm1(B1), Qm2(B2), Qmk(Bk)) the
  dependence structure among the assets
• The price of the Altiplano is
• Altiplano v(t,tm)C(Qm1(B1), Qm2(B2), Qmk(Bk))

85
Altiplano with barrier

• Assume an Altiplano product that pays one unit of
  cash if all the assets Si are above a given
  barrier Bi by a future date tm (on a set of
  monitoring dates t1, t2,, tm).
• The price of the Altiplano is
• Altiplano
• v(t,tm)C(Qm1(A1), Qm2(A2), Qmk(Ak))

86
Counterparty risk
87
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

88
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

89
Credit risk in a nutshell

• Expected loss EL
• It is the discount required to account for losses
  due to default of the issuer or the counterparty
• Default probability DP
• It is the probability of default of the issuer or
  the counterparty
• Loss given default Lgd
• It is the percentage of value to be lost in case
  of default of the issuer or the counterparty
  (alternatively, the recovery rate RR is the
  amount recovered)
• EL DP X Lgd

90
A baseline model

• Assume a forward contract stipulated at time 0
  with for delivery time T, and assume default may
  occur only at T. It is easy to check that the
  value of the contract in this case is
• CF(t) EQv(t,T)Lgdi1imax?(S(T) F(0)),0
• with ? 1, - 1 for long and short positions,
  v(t,T) the discount factor, Lgdi and 1i the
  loss-given-default and default indicator function
  for counterparty i. CF(t) is instead the
  default-free value of the contract.
• Counterparty risk is a short position in options,
  of the call type for long positions and of the
  put type for short ones.

91
Sources of risk

• Counterparty risk is evaluated as
• EQv(t,T)Lgdi1imax?(S(T) F(0)),0
• and is made up by four sources of risk
• Discount factor risk (interest rate risk)
• Underlying asset risk
• Counterparty default risk
• Recovery risk
• All of these sources of risk may be correlated.

92
Default at maturity

• Under the further assumption of orthogonality
  between counterparty risk and the underlying
  asset we have
• v(t,T)EQLgdi1i EQmax?(S(T) F(0)),0
• where Q(T) is time T forward martingale measure.
  Notice that in case the pay-off of the product is
  a constant the formula yields the price of a
  defaultable zero-coupon-bond, namely
• Di(t,T) v(t,T) v(t,T)EQ(T)Lgdi1i,
• or else
• Di(t,T) v(t,T) v(t,T)ELi,
• with ELi EQLgdi1i the expected loss.
• Under the independence assumption above we have
  then
• v(t,T) Di(t,T) EQ(T)max?(S(T) F(0)),0

93
Default before maturity

• Let us now relax the hypothesis of default at
  maturity. To keep things simple, partition time
  in a grid of dates t1,t2,tn
• Denote by Gj(ti) the survival probability of
  counterparty j A, B beyond ti.
• Compute
• GB(ti -1) GB(ti) Call(S(ti), ti v(ti
  ,T)F(0), ti )
• GA(ti -1) GA(ti) Put(S(ti), ti v(ti
  ,T)F(0), ti )
• respectively for long and short positions.
• Sum these values across all the exercise dates.

94
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.

95
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)

96
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

97
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

98
Vulnerable call and put options
99
Vulnerable put-call parity
100
Swap credit risk

• In a swap contract, both the parties are exposed
  to counterparty risk. Credit risk should account
  for the joint event of the counterparty
  defaulting over the life on the contract, and the
  contract being in the money for the other
• If counterparty A pays fixed the risk for her is
• while for B

The approach due to Sorensen and Bollier, 1994
suggests to represent swap credit risk as a
portfolio of payer or receiver swaptions. Their
approach rests on the hypothesis of independence
of the swap rate term structure and credit risk
of the counterparties. Strike rate k of the
swaptions equals the swap rate at inception or
depends on the collateral where applicable
101
Pricing swap credit risk with copulas

• Denoting GB(tj) the survival probability of
  counterparty B beyond time tj. Then her default
  probability between time tj-1 and tj is GB(tj-1)
  GB(tj). Furthermore, assume that, under the
  appropriate swap measure Q(u) Pr(sr(tj) u)
• Swap credit risk for the fixed payer can be
  evaluated as

102
Credit risk for the fixed payer
103
Credit risk for the fixed receiver
104
Counterparty risk spread ratings
105
Counterparty risk spread ratings
106
The effects of dependence
107
Swap credit risk Aaa
108
Swap credit risk Aa
109
Swap credit risk A
110
Swap credit risk Baa
111
Swap credit risk Ba
112
Swap credit risk B
113
Swap credit risk Caa
114
Reference Bibliography

• Nelsen R. (2006) Introduction to copulas, 2nd
  Edition, Springer Verlag
• Joe H. (1997) Multivariate Models and Dependence
  Concepts, Chapman Hall
• Cherubini U., E. Luciano and W. Vecchiato (2004)
  Copula Methods in Finance, John Wiley Finance
  Series.