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## Correlation Products and Copula Functions

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Title: 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
• 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.

19
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.

20
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

21
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

27
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.

28
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).

29
Examples of copula functions Ellictical 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.

30
Examples of copula functions Archimedean 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

31
32
• 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
• 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
• 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 ?)
• 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 ?)

40
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

41
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
43
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 simulation Gaussian 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 simulation Student 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
• 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.
• 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
• 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
105
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.