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Correlation Products and Copula Functions

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

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

Motivation

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

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?

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)

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 ))

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)

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.

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.

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.

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.

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)

Copula functions

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) .

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)

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.

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.

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.

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

Conditional probability I

- The dualities above may be used to recover the

conditional probability of the events.

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.

Conditional probability II

- The conditional probability of X given Y y can

be expressed using the partial derivative of a

copula function.

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.

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.

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

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.

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).

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.

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

Radial symmetry and exchangeability

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)

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

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).

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 ?)

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

?(?).

Exceedance rank-correlation

- Schmid and Schmidt propose a similar concept of

conditional rank-correlation.

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.

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 ?)

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

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

Clayton copula

Clayton non-exchangeable

C(u,v) C(v,u)

C(0.5,u) vs C(v,0.5)

Estimation and simulation

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.

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.

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.

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.

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.

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)

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))

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.

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.

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.

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 ?))

Copula pricing methods

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

Coupon determination

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.

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

a bearish coupon

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.

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

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

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)

Joint probability distribution approach

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

Conditional probability distribution approach

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

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.

Altiplanos, Everest co

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

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).

Altiplano

Altiplano with memory B 70

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.

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)

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

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)

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) ?

The idea

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) )

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)

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))

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))

Counterparty risk

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

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

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

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.

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.

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

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.

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.

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)

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

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

Vulnerable call and put options

Vulnerable put-call parity

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

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

Credit risk for the fixed payer

Credit risk for the fixed receiver

Counterparty risk spread ratings

Counterparty risk spread ratings

The effects of dependence

Swap credit risk Aaa

Swap credit risk Aa

Swap credit risk A

Swap credit risk Baa

Swap credit risk Ba

Swap credit risk B

Swap credit risk Caa

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.