Transcript Barrier Copula Functions

Copula Models and Speculative
Price Dynamics
Umberto Cherubini
University of Bologna
RMI Workshop
National University of Singapore, 5/2/2010
Outline
•
•
•
•
•
Copula functions: main concepts
Copula functions and Markov processes
Application to credit (CDX)
Application to equity
Application to managed funds
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
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 Ha=1 – Ha and
Hb=1 – 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.
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 
Copula product
• The product of a copula has been defined
(Darsow, Nguyen and Olsen, 1992) as
Au , t  B t , v 
dt
0 t
t
1
A*B(u,v) 
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
Properties 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, 2010)
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



1
1
ˆ


C (u, v)   D1C w, F G v  H 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
Cross-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.
Temporal 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.
Credit market applications
Application 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
1



1
w dw
D
C
w
,
F
z

H
 1
0
A temporal aggregation
algorithm
•
•
1.
2.
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
1



1
D
C
w
,
F
z

H
1 w dw  H 2  z 
2
 1
0
3.
where z is now a grid of values of the variable
Go back to step 2, using F3 and H2 compute H3…
Distribution of losses: 10 y
0,45
0,4
0,35
0,3
0,25
Cross section rho = 0.4
Temporal Dep. Kendall tau = 0.2
Temporal Dep. Kendall tau = -0.2
0,2
0,15
0,1
0,05
0
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
Temporal dependence
0,4
0,35
0,3
0,25
Product
Temporal Frank rho = 0.2
Temporal Frank rho = 0.4
Temporal Frank rho = -0.2
Temporal Frank rho = - 0.4
0,2
0,15
0,1
0,05
0
0-3%
3-7%
7-10%
10-15%
15-30%
Equity tranche: term structure
0,4
0,35
0,3
Cross Section rho = 0.4
Cross Section rho = 0.2
Temporal Dep. Kendall tau = -0.2
0,25
0,2
0,15
3
4
5
6
7
8
9
10
Senior tranche: term structure
0,008
0,007
0,006
0,005
0,004
Cross Section rho = 0.4
Cross Section rho = 0.2
Temp. Dep Kendall tau = -0.2
0,003
0,002
0,001
0
3
-0,001
4
5
6
7
8
9
10
A general dynamic model for
equity markets
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 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.
Assumptions
• 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))
No-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.
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
Performance measurement of
managed funds
Performance measurement
• Denote X the return on the market, Y the
return due to active fund management and
Z = X + Y the return on the managed fund
• In performance measurement we may be
asked to determine
– The distribution of Z given the distribution od
X and that of Y
– The distribution of Y given the distribution of Z
and that of X measures from historical data.
Asset management style
• The asset management style is entirely
determined by the distribution Y and its
dependence with X.
– Stock picking: the distribution of Y (alpha)
– Market timing: the dependence of X and Y
• The analysis of the return Z can be performed as
a basket option on X and Y.
• Passive management: X and Y are independent
and Y has zero mean
• Pure stock picking: X and Y are independent
Henriksson Merton copula
• In the Heniksson Merton approach, it is
Y = a +  max(0, – X) + 
and the market timing activity results in a
“protective put strategy”
• Notice that market timing does not imply positive
dependence between the return on the strategy
Y and the benchmark X
• HM copula is particularly cumbersome to write
down (see paper), but it is only a special case of
market timing. In general market timing means
association (positive or negative) of X and Y
Hedge funds
• Market neutral investment is part of the
picture, considering that market neutral
investment means
H(Z, X) = FZ FX
• For this reason the distribution of the
investment return FY is computed by
1


FY z   1   FX z  FZ1 w dw
0
Multicurrency equity fund
Corporate bond fund
Reference Bibliography
•
•
•
•
•
•
•
•
Cherubini U. – E. Luciano – W. Vecchiato (2004): Copula Methods in Finance, John Wiley
Finance Series.
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 (2010): “The Dependence Structure of Running Maxima and
Minima: Results and Option Pricing Applications”, Mathematical Finance,
Cherubini U. – F. Gobbi – S. Mulinacci – S. Romagnoli (2010): “A Copula-Based Model For
Spatial and Temporal Dependence of Equity Makets”, in Durante et. Al.
Cherubini U. – F. Gobbi – S. Mulinacci – S. Romagnoli (2010): “A Copula-Based Model For
Spatial and Temporal Dependence of Equity Makets”, in Durante et. Workshop on Copula
Theory and Its Applications, Proceedings, Springer, forthcoming
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): “Semi-parametric Estimation and Simulation
of Actively Managed Portfolios” working paper