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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
  • Radial symmetry and non-exchangeability
  • Estimation and calibration
  • Copula pricing methods
  • Altiplanos, Everest co
  • Counterparty risk

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

5
Questions
  • In order to price and hedge this product both the
    investors and the issuer have to address three
    questions
  • Is the product long or short in the underlying
    assets?
  • Is the product long or short in the volatility of
    the underlying assets?
  • Is the product long or short in the correlation
    between the underlying asset?

6
Risks
  • A product like this is constructed to provide
    exposure to equity risk, with particular
    reference to correlation between Nasdaq and
    Nikkey (correlation products).
  • The product is also exposed to other, ancillary
    risks in particular, credit risk of the issuer
    and possible correlation with the underlying
    (counterparty risk)

7
First-to-default derivatives
  • Consider a credit derivative product that pays
    protection, to keep things simple in a fixed sum
    L, the first time that a company in a reference
    basket of credit risks gets into default. The
    reference credit risks included in the basket are
    called names in the structured finance jargon.
  • Again to keep things simple, let us assume that
    payment occurs at expiration date T of the
    derivative contract
  • If Q(?1 gt T, ?2 gt T…) denotes the joint survival
    probability of all the names in the basket, it is
    straightforward to check that the value of the
    derivative contract, named first-to-default
    turns out to be
  • First to default LP(t,T)(1 Q(?1 gt T, ?2 gt
    T…))

8
Risks
  • The first-to-default swap is constructed to
    provide exposure to default risk, with particular
    reference to correlation among the credit events
    (correlation products).
  • The product is also exposed to other, ancillary
    risks in particular, the risk of term structure
    movements and credit risk of the issuer and
    possible correlation with the underlying
    (counterparty risk)

9
Hybrids
  • Correlation among different risk factors is the
    main risk in hybrids, or risky swaps as are
    called in the market.
  • These are contracts between two parties, A and B,
    indexed to interest rates, currencies, or credit
    and contingent on survival of a third
    counterparty C.

10
Copula functions in finance
  • Given the prices of single-bet financial
    contracts, what is the price of a multiple-bet
    contract? There is no unique answer to that
    question… (Bernard Dumas)
  • The fortune of copula applications to finance is
    due to the massive increase in supply and demand
    of multiple-bet products.

11
Compatibility problems
  • In statistics compatibility refers to the
    relationship between joint distributions (say of
    dimension n) and marginal distributions (for all
    dimensions k lt n).
  • In finance compatibility means that the price of
    multi-variate derivatives, namely the value of
    contingent claims written on a set of events have
    to be consistent with the values of derivatives
    written on subsets of the events.

12
Single bets
  • Assume you get 1000 if 3 months from now the US
    stock market is at least 2 lower than today and
    zero otherwise. How much are you willing to pay
    for this bet? Say 200 . This price is linked to
    a 20 probability of success.
  • Assume you get 1000 if 3 months from now the
    Canadian stock market is at least 3 lower than
    today and zero otherwise. How much are you
    willing to pay for this bet? Say 200 . This
    price is linked to a 20 probability of success.

13
Multiple bets (Altiplano)
  • Assume you get 1000 iff
  • 3 months from now the US stock market is at least
    2 lower than today AND the Canadian market is at
    least 3 lower than today.
  • How much are you willing to pay for this bet? Say
    66.14 . This price is linked to a 6.614
    probability of success. Of course
  • 6.614 v(t,T)C(20,20)

14
Copula functions
15
Copula functions
  • Copula functions are based on the principle of
    integral probability transformation.
  • Given a random variable X with probability
    distribution FX(X). Then u FX(X) is uniformly
    distributed in 0,1. Likewise, we have v FY(Y)
    uniformly distributed.
  • The joint distribution of X and Y can be written
  • H(X,Y) H(FX 1(u), FY 1(v)) C(u,v)
  • Which properties must the function C(u,v) have in
    order to represent the joint function H(X,Y) .

16
Copula function Mathematics
  • A copula function z C(u,v) is defined as
  • 1. z, u and v in the unit interval
  • 2. C(0,v) C(u,0) 0, C(1,v) v and C(u,1) u
  • 3. For every u1 gt u2 and v1 gt v2 we have
  • VC(u,v) ? C(u1,v1) C (u1,v2) C (u2,v1)
    C(u2,v2) ? 0
  • VC(u,v) is called the volume of the copula C
  • Examples
  • C(u,v) uv
  • C(u,v) min(u,v)
  • C(u,v) max(u v 1,0)

17
Copula functions Statistics
  • Sklar theorem each joint distribution H(X,Y) can
    be written as a copula function C(FX,FY) taking
    the marginal distributions as arguments, and vice
    versa, every copula function taking univariate
    distributions as arguments yields a joint
    distribution.

18
Copula functions and dependence structure in risks
  • Copula functions represent a tool to separate the
    specification of marginal distributions and the
    dependence structure.
  • Say two risks A and B have joint probability
    H(X,Y) and marginal probabilities FX and FY. We
    have that H(X,Y) C(FX , FY), and C is a copula
    function.
  • Examples
  • C(u,v) uv, independence
  • C(u,v) min(u,v), perfect positive dependence
  • C(u,v) max (u v - 1,0) perfect negative
    dependence
  • The perfect dependence cases are called Fréchet
    bounds.

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
Radial symmetry and exchangeability
32
Radial symmetry
  • Take a copula function C(u,v) and its survival
    version
  • C(1 u, 1 v) 1 v u C( u, v)
  • A copula is said to be endowed with the radial
    symmetry (reflection symmetry) property if
  • C(u,v) C(u, v)

33
Radial symmetry example
  • Take u v 20. Take the gaussian copula and
    compute N(u,v 0,3) 0,06614
  • Verify that
  • N(1 u, 1 v 0,3) 0,66614
  • 1 u v N(u,v 0,3)
  • Try now the Clayton copula and compute Clayton(u,
    v 0,2792) 0,06614 and verify that
  • Clayton(1 u, 1 v 0,2792) 0,6484 ? 0,66614

34
Radial symmetry economics
  • In economics and econometrics, radial symmetry
    has led to discover phenomena of correlation
    asymmetry.
  • Empirical evidence have been found that
    correlation is higher for downward moves of the
    stock market than for upward moves (Longin and
    Solnik, Ang and Chen among others).

35
Exceedance correlation
  • Longin and Solnik have first proposed the concept
    of exceedance correlation correlation measured
    on data sampled in the tails.
  • Step 1. Standardize data si (Si ?) /?
  • Step 2. Select sub-samples si gt ?, si lt ?
  • Step 3. Corr (si gt ?, sj gt ?), Corr (si lt ?, sj
    lt ?)
  • For (radial) symmetric distributions
  • Corr (si gt ?, sj gt ?) Corr (si lt ?, sj lt ?)

36
Conditioning bias
  • Correlation figures measured on conditional
    samples are different from the unconditional
    figure (conditioning bias). Ahn and Chen computed
    the bias for the gaussian distribution in closed
    form, and proposed a measure of average
    exceedance correlation.
  • The measure, H, is a weighted square difference
    of a set of empirical exceedance correlations
    ?(?) with respect to the theoretical figure
    ?(?).

37
Exceedance rank-correlation
  • Schmid and Schmidt propose a similar concept of
    conditional rank-correlation.

38
Exchangeable copulas
  • Most of the copula functions used in finance are
    symmetric or exchangeable, meaning
  • C(u,v) C(v,u)
  • In a recent paper, Nelsen proposes a measure of
    non-exchangeability
  • 0 ? 3 sup C(u,v) C(v,u) ? 1
  • Nelsen also identifies a class of maximum
    non-exchangeable copulas.

39
Non exchangeable copulas
  • A way to extend copula functions to account for
    non-exchangeability was suggested by Khoudraji
    (Phd dissertation, 1995).
  • Take copula functions C(.,.) and C(.,.), and 0 lt
    ?, ? lt 1 and define
  • C?, ?(u,v) C(u1 ?, v1 ?) C(u ?, v ?)
  • The copula function obtained is in general
    non-exchangeable. In particular, this was used by
    Genest, Ghoudi and Rivest (1998) taking C(.,.)
    the product copula and C(.,.) the Gumbel copula
  • C?, ?(u,v) u1 ?v1 ?C(u ?, v ?)

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

66
Pricing strategies
  • The pricing of call and put options whose pay-off
    is dependent on more than one event may be
    obtained by
  • Integrating the the value of the pay-off with
    respect to the copula density times the marginal
    density
  • Integrating the conditional probability
    distribution times the marginal distribution of a
    risk
  • Integrating the joint probability distribution

67
From the pricing kernel to options
  • The idea relies on Breeden and Litzenberger
    (1978)
  • By integrating the pricing kernel (i.e. the
    cumulative or decumulative risk neutral
    distribution) we may recover put and call prices
  • From simple digital call and put options we can
    recover call and put prices simply set Pr(S(T)
    u) Q(u)

68
Joint probability distribution approach
  • Assume a product with pay-off
  • Maxf(S1(T), S2(T)) K, 0
  • The price can be computed as

69
Conditional probability distribution approach
  • Assume a product with pay-off
  • Maxf(S1(T), S2(T)) K, 0
  • The price can be computed as

70
AND/OR operators
  • Copula theory also features more tools, which are
    seldom mentioned in financial applications.
  • Example
  • Co-copula 1 C(u,v)
  • Dual of a Copula u v C(u,v)
  • Meaning while copula functions represent the AND
    operator, the functions above correspond to the
    OR operator.

71
Altiplanos, Everest co
72
Altiplanos
  • Assume a coupon is set and paid at time tj (reset
    date).
  • Assume a basket of n 1,2 assets, with prices
    Sn(tj).
  • Call Sn(t0) the reference prices, typically
    recorded at the beginning of the contract and
    used as strikes.
  • Call Ij the characteristic function taking value
    1 if Sn(tj)/Sn(t0) gt B and 0 otherwhise for both
    the assets.
  • Coupon is a bivariate option, for coupon rate c

73
Altiplanos with memory
  • Assume a coupon is set and paid at time tj (reset
    date).
  • Assume a basket of n 1,2 titoli, with prices
    Sn(tj).
  • Call Sn(t0) the reference prices, typically
    recorded at the beginning of the contract and
    used as strikes.
  • Call Ij the characteristic function taking value
    1 if Sn(tj)/Sn(t0) gt B and 0 otherwhise for both
    the assets.
  • Each time the characteristic function yields 1,
    coupons c are paid for that year and for all the
    previous years in which the trigger event had not
    taken place (memory feature).

74
Altiplano
75
Altiplano with memory B 70
76
Equity-linked bonds
  • Assume a coupon which is defined and paid at time
    T.
  • Assume a basket of n 1,2,…N assets, whose
    prices are Sn(T).
  • Denote Sn(t0) the reference prices, typically
    registered at the origin of the contract, and
    used as strike prices.
  • The coupon of a basket option is
  • maxAverage(Sn(T)/Sn(0),1k
  • (1 k) maxAverage(Sn(T)/Sn(0) (1k),0
  • with n 1,2,…,N and a minimum guaranteed return
    equal to k.

77
Everest
  • Assume a coupon which is defined and paid at time
    T.
  • Assume a basket of n 1,2,…N assets, whose
    prices are Sn(T).
  • Denote Sn(t0) the reference prices, typically
    registered at the origin of the contract, and
    used as strike prices.
  • The coupon of an Everest note is
  • maxmin(Sn(T)/Sn(0),1k
  • (1 k) maxmin(Sn(T)/Sn(0) (1k),0
  • with n 1,2,…,N and a minimum guaranteed return
    equal to k.
  • The replicating portfolio is
  • Everest note ZCB 2-colour rainbow (call on
    minimum)

78
Exercises
  • Verify that a product giving a call on the
    maximum of a basket is short correlation
  • Hint 1 ask whether the pay-off includes a AND or
    OR operator
  • Hint 2 verify the result writing the replicating
    portfolio in a bivariate setting
  • Verify that a long position in a first to
    default swap is short correlation

79
An Altiplano with barrier bet
  • Bet 1000 iff both the Canadian and the US
    markets do not lose more the 20 in a year from
    today.
  • The underlying assets of the contract
  • Running minimum of the Canadian market (must be
    greater than 80 of todays value)
  • Running minimum of the US market (must be greater
    than 80 of todays value)

80
The Altiplano barrier problem
  • Assume we observe a sequence of two viariables
  • X X(t1), X(t2),…, X(tm) Y Y(t1), Y(t2),…,
    Y(tm)
  • whose dependence is described by
  • Q(X(tm) gt K, Y(tm) gt H) Cm(QXm (K), QYm (H))
  • Which is the copula corresponding to the joint
    distribution
  • Q(minj?mX(tj) gt K, minj?m Y(tj) gt H) ?

81
The idea
82
Dependence structure
  • If event A(i) occurs (the running minimum of Y at
    time m is greater than H) then B(i) should take
    place as well (the level of Y at time m is
    greater than H) but not vice versa
  • A(i) ? B(i)
  • This implies
  • C(Q(B(i) )?Q(A(i) )) min(Q(B(i) )?Q(A(i) ))
    Q(A(i) )

83
Cross-section compatibility
  • Assume Q(X(tm) gt K, Y(tm) gt H)
  • C(Q(X(tm) gt K),Q(Y(tm) gt H))), then
  • Q(minj?m X(tj) gt K, minj?m Y(ti) gt H)
  • C(Q(minj?m X(tj) gt K),Q(minj?m Y(tj) gt H)))
  • Proof
  • Q(X(tm) gt K, Y(tm) gt H) C (Q(B(1)),Q(B(2)))
  • Q(A(1)? B(1)?A(2) ?B(2)
  • C(min(Q(A(1)), Q(B(1)), min(Q(A(2)), Q(B(2)))
  • Q(A(1)?A(2))
  • Q(minj?m X(ti) gt K, minj?m Y(tj) gt H)

84
Altiplano
  • Assume an Altiplano product that pays one unit of
    cash if all the assets Si are above a given
    barrier Bi at a future date tm.
  • Denote Qmi(Bi) Pr((Si (tm) gt Bi), the
    probability that asset Si be above the barrier at
    time tm under the risk neutral measure.
  • Denote C(Qm1(B1), Qm2(B2),… Qmk(Bk)) the
    dependence structure among the assets
  • The price of the Altiplano is
  • Altiplano v(t,tm)C(Qm1(B1), Qm2(B2),… Qmk(Bk))

85
Altiplano with barrier
  • Assume an Altiplano product that pays one unit of
    cash if all the assets Si are above a given
    barrier Bi by a future date tm (on a set of
    monitoring dates t1, t2,…, tm).
  • The price of the Altiplano is
  • Altiplano
  • v(t,tm)C(Qm1(A1), Qm2(A2),… Qmk(Ak))

86
Counterparty risk
87
Counterpart risk in derivatives
  • Most of the derivative contracts, particularly
    options, forward and swaps, are traded on the OTC
    market, and so they are affected by credit risk
  • Credit risk may have a relevant impact on the
    evaluation of these contracts, namely,
  • The price and hedge policy may change
  • Linear contracts can become non linear
  • Dependence between the price of the underlying
    and counterparty default should be accounted for

88
The replicating portfolio approach
  • The idea is to design a replicating portfolio to
    hedge and price counterparty derivatives
  • Goes back to Sorensen and Bollier (1994)
    approach counterparty risk in swaps represented
    as a sequence of swaptions
  • Copula functions may be used to extend the idea
    to dependence between counterparty risk and the
    underlying

89
Credit risk in a nutshell
  • Expected loss EL
  • It is the discount required to account for losses
    due to default of the issuer or the counterparty
  • Default probability DP
  • It is the probability of default of the issuer or
    the counterparty
  • Loss given default Lgd
  • It is the percentage of value to be lost in case
    of default of the issuer or the counterparty
    (alternatively, the recovery rate RR is the
    amount recovered)
  • EL DP X Lgd

90
A baseline model
  • Assume a forward contract stipulated at time 0
    with for delivery time T, and assume default may
    occur only at T. It is easy to check that the
    value of the contract in this case is
  • CF(t) EQv(t,T)Lgdi1imax?(S(T) F(0)),0
  • with ? 1, - 1 for long and short positions,
    v(t,T) the discount factor, Lgdi and 1i the
    loss-given-default and default indicator function
    for counterparty i. CF(t) is instead the
    default-free value of the contract.
  • Counterparty risk is a short position in options,
    of the call type for long positions and of the
    put type for short ones.

91
Sources of risk
  • Counterparty risk is evaluated as
  • EQv(t,T)Lgdi1imax?(S(T) F(0)),0
  • and is made up by four sources of risk
  • Discount factor risk (interest rate risk)
  • Underlying asset risk
  • Counterparty default risk
  • Recovery risk
  • All of these sources of risk may be correlated.

92
Default at maturity
  • Under the further assumption of orthogonality
    between counterparty risk and the underlying
    asset we have
  • v(t,T)EQLgdi1i EQmax?(S(T) F(0)),0
  • where Q(T) is time T forward martingale measure.
    Notice that in case the pay-off of the product is
    a constant the formula yields the price of a
    defaultable zero-coupon-bond, namely
  • Di(t,T) v(t,T) v(t,T)EQ(T)Lgdi1i,
  • or else
  • Di(t,T) v(t,T) v(t,T)ELi,
  • with ELi EQLgdi1i the expected loss.
  • Under the independence assumption above we have
    then
  • v(t,T) Di(t,T) EQ(T)max?(S(T) F(0)),0

93
Default before maturity
  • Let us now relax the hypothesis of default at
    maturity. To keep things simple, partition time
    in a grid of dates t1,t2,…tn
  • Denote by Gj(ti) the survival probability of
    counterparty j A, B beyond ti.
  • Compute
  • GB(ti -1) GB(ti) Call(S(ti), ti v(ti
    ,T)F(0), ti )
  • GA(ti -1) GA(ti) Put(S(ti), ti v(ti
    ,T)F(0), ti )
  • respectively for long and short positions.
  • Sum these values across all the exercise dates.

94
Dependence structure
  • A more general approach is to account for
    dependence between the two main events under
    consideration
  • Exercise of the option
  • Default of the counterparty
  • Copula functions can be used to describe the
    dependence structure between the two events
    above.

95
Vulnerable digital call option
  • Consider a vulnerable digital call (VDC) option
    paying 1 euro if S(T) gt K (event A). In this
    case, if the counterparty defaults (event B), the
    option pays the recovery rate RR.
  • The payoff of this option is
  • VDC v(t,T)H(A,Bc)RR H(A,B)
  • v(t,T) Ha H(A,B)RR H(A,B)
  • v(t,T)Ha (1 RR)H(A,B)
  • DC v(t,T) Lgd C(Ha, Ha)

96
Vulnerable digital put option
  • Consider a vulnerable digital put (VDP) option
    paying 1 euro if S(T) K (event Ac). In this
    case, if the counterparty defaults (event B), the
    option pays the recovery rate RR.
  • The payoff of this option is
  • VDP DP v(t,T)(1 RR)H(Ac,B)
  • P(t,T)Ha v(t,T)(1 RR)H(Ac,B)
  • P(t,T)Ha v(t,T)(1 RR)Hb C(Ha, Hb)
  • v(t,T)(1 Ha) v(t,T) Lgd Hb
    C(Ha, Hb)
  • v(t,T) VDC v(t,T) Lgd Hb

97
Vulnerable digital put call parity
  • Define the expected loss EL Lgd Hb.
  • If D(t,T) is a defaultable ZCB issued by the
    counterparty we have
  • D(t,T) v(t,T)(1 EL)
  • Notice that copula duality implies a clear
    no-arbitrage relationship
  • VDC VDP v(t,T) v(t,T) EL D(t,T)
  • Buying a vulnerable digital call and put option
    from the same counterparty is the same as buying
    a defaultable zero-coupon bond

98
Vulnerable call and put options
99
Vulnerable put-call parity
100
Swap credit risk
  • In a swap contract, both the parties are exposed
    to counterparty risk. Credit risk should account
    for the joint event of the counterparty
    defaulting over the life on the contract, and the
    contract being in the money for the other
  • If counterparty A pays fixed the risk for her is…
  • while for B…

The approach due to Sorensen and Bollier, 1994
suggests to represent swap credit risk as a
portfolio of payer or receiver swaptions. Their
approach rests on the hypothesis of independence
of the swap rate term structure and credit risk
of the counterparties. Strike rate k of the
swaptions equals the swap rate at inception or
depends on the collateral where applicable
101
Pricing swap credit risk with copulas
  • Denoting GB(tj) the survival probability of
    counterparty B beyond time tj. Then her default
    probability between time tj-1 and tj is GB(tj-1)
    GB(tj). Furthermore, assume that, under the
    appropriate swap measure Q(u) Pr(sr(tj) u)
  • Swap credit risk for the fixed payer can be
    evaluated as

102
Credit risk for the fixed payer
103
Credit risk for the fixed receiver
104
Counterparty risk spread ratings
105
Counterparty risk spread ratings
106
The effects of dependence
107
Swap credit risk Aaa
108
Swap credit risk Aa
109
Swap credit risk A
110
Swap credit risk Baa
111
Swap credit risk Ba
112
Swap credit risk B
113
Swap credit risk Caa
114
Reference Bibliography
  • Nelsen R. (2006) Introduction to copulas, 2nd
    Edition, Springer Verlag
  • Joe H. (1997) Multivariate Models and Dependence
    Concepts, Chapman Hall
  • Cherubini U., E. Luciano and W. Vecchiato (2004)
    Copula Methods in Finance, John Wiley Finance
    Series.
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