Title: Equity and Credit Correlation Products and Copula Functions
1Equity and Credit Correlation Products and Copula
Functions
- Umberto Cherubini
- Matemates University of Bologna
- Birbeck College, London 04/02/2009
2Outline
- Motivation structured finance products and risk
management issues - Copula functions main concepts
- Radial symmetry and non-exchangeability
- Estimation and calibration
- Copula pricing cross-section dependence
- Copula pricing temporal dependence
- A general dynamic model for equity markets
- A general dynamic model for credit products
- Risk management applications
3Motivation
- Structured Finance Products and Risk Management
Issues
4Barrier Altiplano
- Assume a set of coupon periods, k 1,2,P.
- In each period k a set of dates j 1, 2, ,mk
- A basket of i 1, 2, , n assets
- Coupon paid
- ck if Si(tj)gt Ki for all i and all j
- 0 otherwise
5European Altiplano
- Investment period March 2000 - March 2005
- Principal repaid at maturity
- Coupon paid on march 15th every year.
- Coupon determination
- Coupon 10 if (i 1,2,3,4,5)
- Nikkei (15/3/200i) gt Nikkei (15/3/2000) e
- Nasdaq 100 (15/3/200i) gt Nasdaq 100 (15/3/2000)
- Coupon 0 otherwise
- Digital note ZCB bivariate digital call
options -
6Barrier Products
- Consider a product that at time tm pays 1 unit of
cash if the value of the underlying asset X(ti)
remains above, a given barrier B level on a set
of dates t1, t2,, tm. - This is a no-touch option, which is also called
digital barrier option, or uni-variate Altiplano. - Like the European digital option represents the
pricing kernel of European options, the barrier
digital product is the pricing kernel of barrier
options.
7First-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))
8Synthetic CDO
Senior Tranche
Originator
Junior 1 Tranche
Special Purpose Vehicle SPV
Protection Sale
Junior 2 Tranche
CDS Premia
Interest
Tranche
Investment
Collateral AAA
Equity Tranche
9Standard synthetic CDOs
- iTraxx (Europe) and CDX (US) are standardized CDO
deals. - The underlying portfolio of credit exposures is a
set of 125 CDS deals on primary names, same
nominal exposure, same maturity. - The tranches of the standard CDO are 5, 7 and 10
year CDS to buy/sell protection on the losses on
the underlying portfolio higher than a given
level (attachment) up to another level
(detachment) on a nominal value equal to the
difference between the two levels.
10Term structure of CDX
11Risks
- Products like these are made to provide exposure
to specific equity or credit risk sources, with
particular reference to correlation among them
(correlation products). - We will refer to this correlation as dependence
or association (more general terms) and we will
denote it cross-section dependence (that is
dependence among a set of assets or risk factors
evaluated at the same time).
12Cross-section dependence
- Any pricing strategy for these products requires
to select specific joint distributions for the
risk-factors or assets. - Notice that a natural requirement one would like
to impose on the multivariate distributions would
be consistency with the price of the uni-variate
products observed in the market (digital options
for multivariate equity and CDS for multivariate
credit) - In order to calibrate the joint distribution to
the marginal ones one will be naturally led to
use of copula functions.
13Temporal dependence
- Barrier Altiplanos the value of a barrier
Altiplano depends on the dependence structure
between the value of underlying assets at
different times. Should this dependence increase,
the price of the product will be affected. - CDX consider selling protection on a 5 or on a
10 year tranche 0-3. Should we charge more or
less for selling protection of the same tranche
on a 10 year 0-3 tranches? Of course, we will
charge more, and how much more will depend on the
losses that will be expected to occur in the
second 5 year period.
14Hybrids
- 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.
15Risk management applications
- Cross-section aggregation of risk measures (risk
measure integration). The problem is to compute
the joint distribution of losses due to different
risk factors (typically, market, credit,
operational) - Temporal aggregation of risk measures. The
problem is to compute the distribution of losses
over different time horizons. - Temporal aggregation is often a prerequisite to
cross-section aggregation since risk measures for
market risk are often computed at different
horizons with respect to those for credit and
operational risk.
16Risk management of funds
- Temporal aggregation is a key feature of risk
measurement of returns on managed funds. You are
not interest in a 10 day risk measure if you are
evaluating investment in a mutual fund. You are
even less interested if you are planning to
invest in a closed-end fund or a hedge fund. - The issue is to compute VaR or other risk
measures over different time horizons, an issue
called long term VaR. - The state of the art is i) to include a drift in
the specification of the dynamics and ii) to
apply the square-root rule, that is to multiply
the risk measure by T1/2. - The square root law is obviously subject to
very strict assumptions. The process must have
i.i.d. gaussian innovations.
17Copula functions
18Compatibility 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.
19Single 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.
20Multiple 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)
21Copula 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) .
22Copula function Mathematics
- A copula function z C(u,v) is defined as
- 1. z, u and v in the unit interval
- 2. C(0,v) C(u,0) 0, C(1,v) v and C(u,1) u
- 3. For every u1 gt u2 and v1 gt v2 we have
- VC(u,v) ?
- C(u1,v1) C (u1,v2) C (u2,v1) C(u2,v2) ? 0
- VC(u,v) is called the volume of copula C
23Copula 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.
24Copula 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.
25Positive 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
26Copula 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.
27Dualities 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.
28Conditional probability I
- The dualities above may be used to recover the
conditional probability of the events.
29Conditional probability II
- The conditional probability of X given Y y can
be expressed using the partial derivative of a
copula function.
30 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
31and 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.
32Examples 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).
33Fréchet Family C(0.3,0,3)
34Fréchet Family C(0.3,0.7)
35Examples of copula functionsEllictical copulas
- Ellictal multivariate distributions, such as
multivariate normal or Student t, can be used as
copula functions. - Normal copulas are obtained
- C(u1, un )
- N(N 1 (u1 ), N 1 (u2 ), , N 1 (uN )
?) - and extreme events are indipendent.
- For Student t copula functions with v degrees of
freedom C (u1, un ) - T(T 1 (u1 ), T 1 (u2 ), , T 1 (uN ) ?,
v) - extreme events are dependent, and the tail
dependence index is a function of v.
36Gaussian copula vs mixture
37Examples of copula functionsArchimedean copulas
- Archimedean copulas are build from a suitable
generating function ? from which we compute - C(u,v) ? 1 ?(u)?(v)
- The function ?(x) must have precise properties.
Obviously, it must be ?(1) 0. Furthermore, it
must be decreasing and convex. As for ?(0), if it
is infinite the generator is said strict. - In n dimension a simple rule is to select the
inverse of the generator as a completely monotone
function (infinitely differentiable and with
derivatives alternate in sign). This identifies
the class of Laplace transform.
38Examples Frank copula
- Take
- ?(t) log(1 exp( ?t)/(1 exp( ?)
- such that the inverse is
- ? 1(s) ? 1 log(1 (1 exp( ?)exp( s)
- the Laplace transform of the logarithmic series.
Then, the copula function - C(u1,, un) ? 1 ?(u1)?(un)
- is called Frank copula. It is symmetric and does
not have tail dependence (either lower or upper).
39Frank copula vs gaussian
40Examples Clayton copula
- Take
- ?(t) t ? 1/ ?
- such that the inverse is
- ? 1(s) (1 ?s) 1/ ?
- the Laplace transform of the gamma distribution.
- Then, the copula function
- C(u1,, un) ? 1 ?(u1)?(un)
- is called Clayton copula. It is not symmetric
and has lower tail dependence (no upper tail
dependence).
41Clayton copula vs Frank copula
42Examples Gumbel copula
- Take
- ?(t) (log t)?
- such that the inverse is
- ? 1(s) exp( s 1/? )
- the Laplace transform of the positive stable
distribution. - Then, the copula function
- C(u1,, un) ? 1 ?(u1)?(un)
- is called Gumbel copula. It is not symmetric and
has upper tail dependence (no lower tail
dependence).
43Gumbel copula vs Frank copula
44Kendall function
- For the class of Archimedean copulas, there is a
multivariate version of the probability integral
transfomation theorem. - The probability t C(u,v) is distributed
according to the distribution - KC (t) t ?(t)/ ?(t)
- where ?(t) is the derivative of the generating
function. There exist extensions of the Kendall
function to n dimensions. - Constructing the empirical version of the Kendall
function enables to test the goodness of fit of a
copula function (Genest and McKay, 1986).
45Kendall function Clayton copula
46Hedge Funds Market Neutral
47Hierarchical copulas
- Consider a set of copula functions
- C(u1,, unm)
- ?21 1 ?21 ??11 1(?11(u1)?11(un))
- ?21 ??12
1(?12(u1)?12(um) - where ?21 , ?11 and ?12 are generators.
- This is a copula iff the dependence of the ?21
generator is lower or equal to the dependence of
the ?11 and ?12 generators - This is called Archimedean hierarchical copula
and allows to extend Archimedean copula to
arbitrary dimensions with different bivariate
dependence
48Radial symmetry and exchangeability
49Radial 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)
50Radial 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
51Radial 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).
52Exceedance 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 ?)
53Exceedance rank-correlation
- Schmid and Schmidt propose a similar concept of
conditional rank-correlation.
54Exchangeable 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.
55Non 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 ?)
56Non-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
57Non-Exchangeability economics
- To have an idea of the ecnomic meaning of
exchangeability, consider the following case - Lehmann Bros provides capital insurance to life
insurance policies to Mediolanum, an Italian
bank-insurance company. Assume that in case of
default of Lehmann, Mediolanum takes over the
cost of insurance (as it actually happened after
September 15 2008) - Consider now the joint probability of default of
Lehmann and Mediolanum. If the marginal
probability of default of Lehmann is high, the
joint distribution will also be high. But if the
marginal probability of Mediolanum is high, the
joint distribution might not be as high as in the
previous case.
58C(u,v) C(v,u)
59Partial exchangeability
- Notice that the Archimedean hierarchical copula
allows for some non-exchangeability among
variables in different groups. - For example, assume a hierarchical representation
in which assets are grouped by sector at the
lower level and between sector at the higher
level. Of course, the dependence structure within
the group is exchangeable, but between the groups
it is not. This case is called partial
exchangeability. - Econometricians would recognize similarities with
the within and between estimators.
60Estimation and simulation
61Copula function calibration
- A first straightforward way of determining the
copula function representing the dependence
structure between two variables was proposed by
Genest and McKay, 1986. - The algorithm is particularly simple
- Change the set of variables in ranks
- Measure the association between the ranks
- Determine the parameter of the copula (a family
of copula has to be chosen) in order to obtain
the same association measure. - Plot the Kendall function of each copula and
select the copula which is closest to the
empirical Kendall function.
62Nasdaq 100 vs Nikkey 225
63Gumbel fit
64Clayton fit
65Copula 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.
66Copula 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.
67Maximum 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.
68Conditional 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.
69Conditional 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)
70Dynamic 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))
71Monte 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.
72Monte 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.
73Other 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.
74Copula pricing cross-section dependence
75Digital 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 -
76Coupon determination
77Super-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.
78Copula 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
79a bearish coupon
80Bivariate 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.
81Radial 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
82Pricing strategies
- The pricing of call and put options whose pay-off
is dependent on more than one event may be
obtained by - Integrating the the value of the pay-off with
respect to the copula density times the marginal
density - Integrating the conditional probability
distribution times the marginal distribution of a
risk factor - Integrating the joint probability distribution
83From 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)
84Joint probability distribution approach
- Assume a product with pay-off
- Maxf(S1(T), S2(T)) K, 0
- The price can be computed as
85Conditional probability distribution approach
- Assume a product with pay-off
- Maxf(S1(T), S2(T)) K, 0
- The price can be computed as
86AND/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.
87Equity-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.
88Everest
- 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)
89Exercises
- 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
90Copula pricingtemporal dependence
91Copula product
- The product of a copula has been defined (Darsow,
Nguyen and Olsen, 1992) as - AB(u,v) ?
- and it may be proved that it is also a copula.
92Markov processes and copulas
- Darsow, Nguyen and Olsen, 1992 prove that 1st
order Markov processes (see Ibragimov, 2005 for
extensions to k order processes) can be
represented by the ? operator (similar to the
product) -
- A (u1, u2,, un) ?B(un,un1,, unk1) ?
- i
93Properties of ? products
- Say A, B and C are copulas, for simplicity
bivariate, A survival copula of A, B survival
copula of B, set M min(u,v) and ? u v - (A ? B) ? C A ? (B ? C) (Darsow et al. 1992)
- A? M A, B? M B (Darsow et al. 1992)
- A? ? B? ? ? (Darsow et al. 1992)
- A ? B A ? B (Cherubini Romagnoli, 2008)
94Symmetric Markov processes
- Definition. A Markov process is symmetric if
- Marginal distributions are symmetric
- The ? product
- T1,2(u1, u2) ? T2,3(u2,u3) ? Tj 1,j(uj 1 ,
uj) - is radially symmetric
- Theorem. A ? B is radially simmetric if either i)
A and B are radially symmetric, or ii) A ? B A
? A with A exchangeable and A survival copula of
A.
95Example Brownian Copula
- Among other examples, Darsow, Nguyen and Olsen
give the brownian copula - If the marginal distributions are standard
normal this yields a standard browian motion. We
can however use different marginals preserving
brownian dynamics.
96Time Changed Brownian Copulas
- Set h(t,?) an increasing function of time t,
given state ?. The copula - is called Time Changed Brownian Motion copula
(Schmidz, 2003). - The function h(t,?) is the stochastic clock. If
h(t,?) h(t) the clock is deterministic (notice,
h(t,?) t gives standard Brownian motion).
Furthermore, as h(t,?) tends to infinity the
copula tends to uv, while as h(s,?) tends to
h(t,?) the copula tends to min(u,v)
97Copula martingale processes
- A problem for pricing applications is to impose
the martingale restriction in the Markov process
representation. - Cherubini, Mulinacci and Romagnoli, 2008 propose
a particular strategy to use copula functions to
build martingale process. - The novelty of the idea is to use copula
functions to model the dependence structure
between the increment of a stochastic process and
its level before the increment.
98CheMuRo Model
- Take three continuous distributions F, G and H.
Denote C(u,v) the copula function linking levels
and increments of the process and D1C(u,v) its
partial derivative. Then the function - is a copula iff
-
99A special class of processes
- F represents the probability distribution of
increments of the process, H represents the
distribution of the level of the process before
the increment and G represents the level of the
process after the increment. - Distribution G is obtained by an operation that
we denote C-convolution of F and H. - Lévy processes are obtained as a class in which
- C(u.v) uv, the operator is the convolution.
- F G H increments are stationary
100Dependent increment with gaussian marginals
- As an example, consider the increments to be
standard normal F ?. We have - Notice that marginals are gaussian, since they
refer to the sum of gaussian variables The
dependence structure is instead given by the
copula function C(u,v).
101Temporal dependence and scaling law
- Notice that given a marginal distribution and a
copula describing the link between increment and
the level before it, we simultaneously derive the
marginal distribution of the level following the
increment and the dependence structure with the
level before. - Notice that once distribution of the increments
and dependence with the levels have been
selected, the dynamics of the process is
completely specified. So, there is a relationship
between dependence of the increments and scaling
law of the process.
102Clayton dependence and gaussian marginals 5 perc
103Dependent increments? SP 500
104Dependent increments? USD/Euro
105HFIndex convertible arbitrage
106HFIndex Dedicated short bias
107HFIndex Emerging Markets
108Copula based dynamics
- Given the evidence above the class of model that
seems more appropriate to describe the dynamics
of assets seems to be - with
109Martingale restrictions
- In the model with independent increments the
martingale requirement is very easy to implement.
In fact, it suffices to choose zero mean
distributions for increments. - Notice that independent increments are not a
necessary requirement. In fact, the martingale
requirement may be ensured for symmetric
distributions of increments given specific
dependence structures.
110A general dynamic model for equity markets
111The model of the market
- Our task is to model jointly cross-section and
time series dependence. - Setting of the model
- A set of ?S1, S2, ,Sm? assets conditional
distribution - A set of ?t0, t1, t2, ,tn? dates.
- We want to model the joint dynamics for any time
tj, j 1,2,,n. - We assume to sit at time t0, all analysis is made
conditional on information available at that
time. We face a calibration problem, meaning we
would like to make the model as close as possible
to prices in the market.
112Assumptions
- Assumption 1. Risk Neutral Marginal Distributions
The marginal distributions of prices Si(tj)
conditional on the set of information available
at time t0 are Qi j - Assumption 2. Markov Property. Each asset is
generated by a first order Markov process.
Dependence of the price Si(tj -1) and asset
Si(tj) from time tj-1 to time tj is represented
by a copula function Tij 1,j(u,v) - Assumption 3. No Granger Causality. The future
price of every asset only depends on his current
value, and not on the current value of other
assets. This implies that the m x n copula
function admits the hierarchical decomposition - C(G1 (Q11, Q12, Q1n), Gm(Qm1, Qm2, Qmn))
113No-Granger Causality
- The no-Granger causality assumption, namely
- P(Si(tj)? S1(tj 1),, Sm(tj 1)) P(Si(tj)?
Si(tj 1)) - enables the extension of the martingale
restriction to the multivariate setting. - In fact, we assuming Si(t) are martingales with
respect to the filtration generated by their
natural filtrations, we have that - E(Si(tj)?S1(tj 1),, Sm(tj 1))
- E(Si(tj)?Si(tj 1)) S(t0)
- Notice that under Granger causality it is not
correct to calibrate every marginal distribution
separately.
114Running maxima and minima
- Due to the first order Markov assumption the
dynamics of each asset can be represented as a
function of the bivariate copulas Tij 1,j(u,v) - Running Maximum Gij (u1, u2, uj)
- Ti1,2(u1, u2) ? Ti2,3(u2,u3) ? Ti j 1,j(uj
1 , uj) - Running Minimum Gij (u1, u2, uj)
- Ti1,2(u1, u2) ? Ti2,3(u2,u3) ? Ti j 1,j(uj
1 , uj) - Notice that the operator implies recursions like
- Gij (u1, u2, uj)
- Gij 1(u1, u2, uj 1) ? Ti j 1,j(uj 1 ,
uj)
115Univariate barrier Altiplanos
- Risk neutral valuation implies compatibility
restrictions between barrier and European
options. In fact, set - DC(K, tk) forward price of the European digital
option paying one unit of cash iff S(tk) gt K. - NT(K, tk) forward price of the barrier digital
option paying one unit of cash iff S(tp) gt K, for
all p 1,2,,k. - Price compatibility requires then
- NT(K, tk) NT(K,tk1) ? Ti k 1,k(DC(K, tk1) ,
DC(K, tk)) - Notice that the Markov property assumption
implies this recursive structure (bootstrapping)
116Barrier BootstrappingBrownian motion and O-U
clock
117Cross-section compatibility
- Assume Q(Si(tm) gt H, Sj(tm) gt K)
- ?(Q(Si(tm) gt H),Q(Sj(tm) gt K)), then
- Q(mink?m Si(tk) gt H, mink?m Sj(tk) gt K)
- ?(Q(mink?m Si(tk) gt H),Q(mink?mSj(tk) gt K)))
- Sketch of proof for the symmetric case. Use the
reflection principle Pr(mink?m Si(tk) gt K) 2
Pr(Si(tm) gt K) 1 to prove - Pr(mink?m Si(tk) gt H, mink?m Si(tk) gt K)
- C(Pr(mink?m Si(tk) gt H), Pr(mink?m Sj(tk) gt K)
- C(Pr(Ui gt ui), Pr(Uj gt uj))
- C(Pr(Ui 1)/2gt (ui 1)/2, Pr(Uj 1)/2 gt (uj
1)/2 - Pr(Si(tm) gt H, Sj(tm) gt K) ?((ui 1)/2, (uj
1)/2) -
118European and Barrier Altiplanos
- DCi(K, tm) forward price of the European
digital option paying one unit of cash iff Si(tm)
gt K. - NTi(K, tm) forward price of the barrier digital
option paying one unit of cash iff Si(tp) gt K,
for all p 1,2,,m. - The price of a European Altiplano is
- EA ?(DC1(K, tm), DC2(K, tm),, DCn(K, tm))
- The price of a barrier Altiplano is
- BA ?(NT1(K, tm), NT2(K, tm),, NTn(K, tm))
119European and barrier AltiplanosTemporal
Dependence Shocks
120European and barrier AltiplanosCross-Section
Dependence Shocks
121Multivariate credit products
122Application to credit market
- Assume the following data are given
- The cross-section distribution of losses in every
time period ti 1,ti (Y(ti )). The
distribution is Fi. - A sequence of copula functions Ci(x,y)
representing dependence between the cumulated
losses at time ti 1 X(ti 1), and the losses
Y(ti ). - Then, the dynamics of cumulated losses is
recovered by iteratively computing the
convolution-like relationship
123A temporal aggregation algorithm
- Denote X(ti 1) level of a variable at time ti
1 and Hi 1 the corresponding distribution. - Denote Y(ti ) the increment of the variable in
the period ti 1,ti. The corresponding
distribution is Fi. - Start with the probability distribution of
increments in the first period F1 and set F1
H1. - Numerically compute
- where z is now a grid of values of the variable
- 3. Go back to step 2, using F3 and H2 compute
H3
124Distribution of losses 10 y
125Temporal dependence
126Equity tranche term structure
127Senior tranche term structure
128Houston, we have a problem
- The application of the algorithm to credit leads
to a problem. As the support of the amount of
default is bounded, the algorithm must be
modified accordingly, including constraints. - Continuous distribution of losses
- D1C (w,FY(K FX1(w))) 1, ? w ? 0,1
- Discrete distribution of losses
- C(FX(j),FY(K j)) C(FX(j 1),FY(K j)) P(X
j) - j 0,1,,K
- These constraints define a recursive system that
given the initial distribution of losses and the
temporal dependence structure yields the
distribution of losses in future periods. -
129Risk management applications
130Value-at-Risk Aggregation
- Assume you want to compute the Value at Risk of
an investment on a hedge funds over different
time horizons. - Applying the square root law would imply
independent increments, which is inconsistent
with findings for hedge funds. - One could estimate the dependence structure
between increments and levels and apply the
aggregation algorithm described above.
131Value-at-Risk for Hedge Funds
132Counterpart 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
133The 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
134Dependence 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.
135Vulnerable 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)
136Vulnerable 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
137Vulnerable 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 -
138Vulnerable call and put options
139Vulnerable put-call parity
140Example swap credit risk Counterparty BBB
141Reference Bibliography I
- Nelsen R. (2006) Introduction to copulas, 2nd
Edition, Springer Verlag - Joe H. (1997) Multivariate Models and Dependence
Concepts, Chapman Hall - Cherubini U. E. Luciano W. Vecchiato (2004)
Copula Methods in Finance, John Wiley Finance
Series. - Cherubini, U. (2004) Pricing Swap Credit Risk
with Copulas, working paper - Cherubini U. E. Luciano (2003) Pricing and
Hedging Credit Derivatives with Copulas,
Economic Notes, 32, 219-242. - Cherubini U. E. Luciano (2002) Bivariate
Option Pricing with Copulas, Applied
Mathematical Finance, 9, 69-85 - Cherubini U. E. Luciano (2002) Copula
Vulnerability, RISK, October, 83-86 - Cherubini U. E. Luciano (2001) Value-at-Risk
Trade-Off and Capital Allocation with Copulas,
Economic Notes, 30, 2, 235-256
142Reference bibliography II
- Cherubini U. S. Mulinacci S. Romagnoli
(2008) Copula Based Martingale Processes and
Financial Prices Dynamics, working paper. - Cherubini U. Mulinacci S. S. Romagnoli
(2008) A Copula-Based Model of the Term
Structure of CDO Tranches, in Hardle W.K., N.
Hautsch and L. Overbeck (a cura di) Applied
Quantitative Finance,,Springer Verlag, 69-81 - Cherubini U. S. Romagnoli (2008) The
Dependence Structure of Running Maxima and
Minima Results and Option Pricing Applications,
Mathematical Finance, forthcoming - Cherubini U. S. Romagnoli (2008) Computing
Copula Volume in n Dimensions, Applied
Mathematical Finance, forthcoming