Transcript Barrier Copula Functions

Copula Functions and Markov
Processes for Equity and Credit
Derivatives
Umberto Cherubini
Matemates – University of Bologna
Birbeck College, London 24/02/2010
Outline
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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.
Coupon determination
HH
HL
LH
LL
DCNky
1
1
0
0
DCNsd
1
0
1
0
v(t,T)
1
1
1
1
Coupon
1
0
0
0
Super-replication
• It is immediate to check that
Max[DCNky + DCNsd – v(t,T),0] ≤ Coupon
and
Coupon ≤ Min[DCNky,DCNsd ]
otherwise it will be possible to exploit
arbitrage profits.
• Fréchet bounds provide super-replication
prices and hedges, corresponding to
perfect dependence scenarios.
Copula pricing
• It may be easily proved that in order to rule out arbitrage
opportunities the price of the coupon must be
Coupon = v(t,T)C(DCNky/v(t,T),DCNsd /v(t,T))
where C(u,v) is a survival copula representing dependence between
the Nikkei and the Nasdaq markets.
• Intuition.Under the risk neutral probability framework, the risk neutral
probability of the joint event is written in terms of copula, thanks to
Sklar theorem,the arguments of the copula being marginal risk
neutral probabilities, corresponding to the forward value of univariate
digital options.
• Notice however that the result can be prooved directly by ruling out
arbitrage opportunities on the market. The bivariate price has to be
consistent with the specification of the univariate prices and the
dependence structure. Again by arbitrage we can easily price…
…a “bearish” coupon
HH HL LH LL
DCNky
1
1
0
0
DCNsd
1
0
1
0
v(t,T)
1
1
1
1
DC
1
0
0
0
v(t,T) – DCNky – DCNsd + DC
0
0
0
1
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.
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.
Conditional probability I
• The dualities above may be used to
recover the conditional probability of the
events.
PrH a  u, H b  v  C u, v 
PrH a  u H b  v  

PrH b  v 
v
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…
L v   PrFX  v FY  v  
PrFX  v, FY  v  C v, v 


PrFY  v 
v
…and in booms
• In the case of booms, we have instead
U v   PrFX  v FY  v  
PrFX  v, FY  v  1  2v  C v, v 


PrFY  v 
1 v
• 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.
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 II
• 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
Top down vs bottom up
• In credit risk applications, top down approaches denote
models that specify the joint distribution of the default
events and the marginal probability of default of each
“name” as an outcome. The main shortcoming of the
approach is to calibrate the model to single name
derivatives products.
• Bottom up approaches model the term structure of single
name default in the first place and joint default probability
after that. That can be done either using copula functions
or multivariate intensity models (Marshal Olkin, for
example). The main flaw of this approach is to ensure
temporal consistency (particularly if one uses copula
functions).
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
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 vs bottom up
• Top down approaches include: Wishart
processes, Jacobi processes for average
correlation, Radon transform to recover
the multivariate density form option prices
(multivariate Breeden and Litzenberger). In
this approach it may be difficult to calibrate
all univariate prices simultaneously.
• Bottom up approaches include copula
functions. For copula functions it may be
very difficult to ensure the martingale
requirement.
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.
SCOMDY dynamics
• The analysis is based on a very flexible
multivariate asset dynamics called SCOMDY
(Semi-Parametric Copula-based Multivariate
Dynamics) due to Chen and Fan (2006).
• The idea is a multivariate setting in which the
price increments are linked by copula functions.
• We build a model with this structure and we
build into it the features that enable to ensure
the martingale requirement. In a single world, we
design a market wiht SCOMDY dynamics and
independent increments.
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.
• Notice: the independent increment property guarantees
both the Markov property and no-Granger-causality
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, 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.
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).
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 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