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On the term structure of
multivariate equity derivatives
Umberto Cherubini
Matemates – University of Bologna
Bloomberg, New York 24/03/2010
Outline
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Copula functions and Markov processes
Top-down/bottom up in credit and equity
A generale SCOMDY market model
Independent increments
Copulas and correlation recursions
Copula functions and Markov
processes
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 > u2 and v1 > 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
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 function and
dependence structure
• Copula functions are linked to non-parametric dependence
statistics, as in example Kendall’s  or Spearman’s 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.

 
  H x, y   F x F  y dxdy
X
  
1 1
 S  12  C u, v dudv  3
0 0
1 1
  4  C u, v dC u, v   1
0 0
Y
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 (a>0, b 0, Cmin
and Cind for the negative dependent (b>0, a 0.
Ellictical 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.
Archimedean 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.
Conditional probability
• The conditional probability of X given Y = y can
be expressed using the partial derivative of a
copula function.
C u, v 
Pr  X  x Y  y  
v u F1  x ,v  F2  y 
Stochastic increasing
• Take two variables X1 and X2.
• X1 is said to be stochastic increasing or
decreasing in X2 if
Pr(X1  x| X2 = y)
is increasing or decreasing in X2.
Reference: Joe (2007)
Copula product
• The product of a copula has been defined
(Darsow, Nguyen and Olsen, 1992) as
Au, t  Bt , v 
dt
A*B(u,v)  
t
t
0
1
and it may be proved that it is also a
copula.
Markov 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,un+1,…, un+k–1) 
Au1 , u2 ,..., un1 , t  Bt , un1 , um 2 ,..., um k 1 
dt
0
t
t
un
i
Symmetric Markov processes
• Definition. A Markov process is symmetric if
1. Marginal distributions are symmetric
2. 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.
Example: Brownian Copula
• Among other examples, Darsow, Nguyen
and Olsen give the brownian copula
 t  1 v   s  1 w 
dw
0  
ts

u
If the marginal distributions are standard
normal this yields a standard browian
motion. We can however use different
marginals preserving brownian dynamics.
Time Changed Brownian
Copulas
• Set h(t,) an increasing function of time t, given state .
The copula
 ht ,   1 v   hs,   1 w 
dw
0  
ht ,    hs,  


u
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)
CheMuRo 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
u



Cˆ (u, v)   D1C w, F G 1 v   H 1 w dw
0
is a copula iff
 D Cw, F t  H wdw  H * F t   Gt 
1
1
1
0
C
Dependence
Cross-section/ Temporal
• Pricing strategies of multivariate derivatives, both
for credit and equity, should account for two types
of dependence
• Cross-section (spatial) dependence. Market or
asset co-movements at the same time.
– Example: CDS/CDX compatibility, univariate versus
multivariate digital options compatibility
• Temporal dependence. Dependence of market
movements at different times.
– Example: CDX term structure, univariate and
multivariate barrier derivatives.
Top/down vs bottom/up
• In credit the joint need to calibrated
multivariate derivatives (CDX) to univariate
ones (CDS) lead to the use of copula, as
the easiest among bottom up approaches
• The need determine the term structure of
CDX derivatives lead to the development
of top down approaches
• Bottom/up vs top/down approaches may
also be found in the equity literature.
A general SCOMDY model for
equity markets
Top-down vs bottom up
• When pricing multivariate equity derivatives one
is required to satisfy two conditions:
– Multivariate prices must be consistent with univariate
prices
– Prices must be temporally consistent and must be
martingale
• One approach, that we call top down, consists in
the specification of the multivariate distribution
and the determination of univariate distributions
• On another approach, that we call bottom up,
one first specifies the univariate distributions and
then the joint distribution in the second stage.
Top down approaches
• Johnson (1987) and Margrabe (1987)
multivariate Black-Scholes model
• Driessen, Maenhout and Vilkov (2005)
Jacobi process for average correlation.
• Da Fonseca, Grasselli and Tebaldi (2007):
Wishart processes (Bru, 1991)
• Carr and Lawrence (2009): Radon
transform to recover the multivariate
density form option prices (multivariate
Breeden and Litzenberger).
Bottom up approaches
• Rosemberg (2003): multivariate Plackett
distributions
• Cherubini and Luciano (2002): static arbitrage
free pricing using copula functions
• Van Der Goorbergh, Genest and Werker (2005):
time varying dependence copula
• Patton (2003): conditional copulas (FOREX)
• Fermanian and Wegkamp (2004): pseudo
copulas
• Cherubini and Romagnoli (2010): bootstrap of
univariate and multivariate barrier options.
The 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
– 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.
SCOMDY dynamics
• Analysis is carried out on a multivariate model of
dynamics called SCOMDY (Semi-Parametric
Copula-based Multivariate Dynamics, Chen and
Fan 2006).
• The idea is a multivariate setting in which the
log-price increments are linked by copula
functions. Namely, we model X(ti) = ln(S(ti))
Xj(ti) = Xj(ti-1) + i
Xk(ti) = Xk(ti-1) + i
Assumptions
• Assumption 1. Risk Neutral Marginal Distributions The
logarithm of each price follows a process with
independent increments. For each asset, there exists a
probability measure under which the price is a
martingale under its own natural filtration.
• Assumption 2. No Granger Causality. Each asset is not
Granger caused by others. The future price of every
asset only depends on his current value, and not on the
current value of other assets.
• Assumption 3. Risk Neutral Joint Distribution There
exists a probability measure under which the price of
each asset is a martingale under the enlarged filtration of
all assets,
No-Granger Causality
• The following are equivalent
1. Xj is not Granger-caused by X1,…, Xj –1, …, Xj +1
,…, Xm.
2. If Xj is a Markov process with respect to its natural
filtration, it is a Markov process with respect to the
enlarged filtration generated by X1, X2,…, Xm
• The no-Granger causality assumption, enables
the extension of the martingale restriction to the
multivariate setting…
H-condition
• H-condition denotes the case in which a process which is a
martingale with respect to a filtration remains a martingale
with respect to an enlarged filtration
• H-condition and no-Granger-causality are very close
concepts. No Granger causality enables to say that if a
process is Markov with respect to an enlarged filtration it
remains Markov with respect to rhe natural filtration. Based
on this, a result due to Bremaud and Yor states that the Hcondition holds.
• Notice that the H-condition allows to obtain martingales by
linking martingale processes with copulas. It justifies
mixing cross-section analysis (to calibrate martingale
prices) and time series analysis (to estimate dependence).
A SCOMDY model with
independent increments
Independent increments
•
In a multivariate setting the term independent
increments may have several meanings
1. Component-wise independent increments: the
increments of each variable Xj are independent
on its own past history.
2. Vector independent increments. The vector of
increments is independent of the past history
of the whole vector.
3. Granger-independence. the increments of
each variable Xj are independent on the past
history of all variables.
Granger vs vector dependence
• Vector independence implies both Granger
and component-wise independence
• Granger independence implies component
-wise independence.
• Component-wise independence and noGranger causality bring about Granger
independence
Granger independence
1
1+ 2 =2
1+ 2 =2
2
1+ 2 =1
1+ 2 =1
3
1+ 2 =1
1+ 2 =0
4
1+ 2 =0
1+ 2 =1
1= 1 =1
{1 2}
{1 2 3 4}
{3 4}
1= 1 =0
Granger vs vector independence
• Consider the price of a claim paying 1 unit
of cash if 1= 1 = 2 = 2 = 1.
• Under Granger independence
P(1= 1 = 2 = 2 = 1) =
P(2 = 2 = 1 1 = 1 = 1)P(1 = 1 = 1)
…while under vector independence
P(1= 1 = 2 = 2 = 1) =
P(2 = 2 = 1)P(1 = 1 = 1)
Pricing kernel recursion
• A copula recursion can be used to establish noarbitrage relationships between multivariate
pricing kernels of successive maturities
• Notice that in the case of vector independence
(but only in that case) this recursion reduces to
the product of multivariate cross-section
probabilities.
• The price under Granger independence will be
higher (lower) than that under vector
independence is the price of a longer product is
stochastic increasing (decreasing) in the
previous maturity product.
Multivariate equity derivatives
• Pricing algorithm:
– Estimate the dependence structure of log-increments
from time series
– Simulate the copula function linking levels at different
maturities.
– Draw the pricing surface of strikes and maturities
• Examples:
– Multivariate digital notes (Altiplanos), with European
or barrier features
– Rainbow options, paying call on min (Everest
– Spread options
Copula recursion: Granger
independence
CX j X k 
i
i
1 1

0 0
Ci X j
k
i 1 X i 1
 2Ci u1 , v1 , u2 , v2 
u1u2

 2Ci u1 , v1 ,1,1
u1u2
1




FX j t    FX j t  Fi 1  
i
0
i 1
1
FX k t    FX j t  Fi 1  
i

  ,  , F  F 1 u   F 1  , F F 1 v   F 1  dC
 i X k
 X ij 1 X ik1  ,  
k 
i  X j
X ik1
Xj
i 1 X i 1 
i
i 1
 i


   Ci X j
0
i 1
Copula recursion: vector
independence
CX j X k 
i
i

1 1
   Ci  F i  FX1j u   FX1j  , F i FX1k v   FX1k 
i
i 1
i 1

  i
0 0
Ci  C i ,i
1




FX j t    FX j t  Fi 1  
i
0
i 1
1
FX k t    FX j t  Fi 1  
i
0
i 1
dC
X ij 1 X ik1
 ,  
Correlation recursion
• If we are only interested in the recursion of
correlation instead of the whole copula
X
j
i
, X ik
Hi 
 H i  Ki  X j
k
i 1 , X i 1
X X
j
i 1
X X
j
i
i 
k
i 1
k
i
cov i ,i 
  
i
i
Ki 
i   
i
X X
j
i
k
i
i
Reference 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. – 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
Reference bibliography II
• Cherubini U. – S. Mulinacci – S. Romagnoli (2009): “A Copula
Based Model of Speculative Price Dynamics”, working paper.
• Cherubini U. – Mulinacci S. – S. Romagnoli (2008): “A CopulaBased 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 (2010): “The Dependence Structure of
Running Maxima and Minima: Results and Option Pricing
Applications”, Mathematical Finance,
• Cherubini U. – S. Romagnoli (2009): “Computing Copula Volume in
n Dimensions”, Applied Mathematical Finance, 16(4).307-314
• Cherubini U. – F. Gobbi – S. Mulinacci – S. Romagnoli (2010): “On
the Term Structure of Multivariate Equity Derivatives”, working paper
• Cherubini U. – F. Gobbi – S. Mulinacci (2010): “Semiparametric
Estimation and Simulation of Actively Managed Funds”, working
paper