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 as1. 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.

1,,4

3,12

,

1

0

1

0

1

0

1

0

vudCvuC

dudvvuC

dxdyyFxFyxH

S

YX

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 copulasPr(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) =Cmin +(1 – – )Cind + Cmax , , [0,1] Cmin= max (x + y – 1,0), Cind= xy, Cmax= min(x,y)

• The parameters ,are linked to non-parametric dependence measures by particularly simple analytical formulas. For example

S = • Mixture copulas (Li, 2000) are a particular case in

which copula is a linear combination of Cmax and Cind for positive dependent risks (>0, Cmin and Cind for the negative dependent (>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.

yFvxFuv

vuCyYxX

21 ,

,Pr

Copula product

• The product of a copula has been defined (Darsow, Nguyen and Olsen, 1992) as

A*B(u,v)

and it may be proved that it is also a copula.

1

0

,,dt

t

vtB

t

tuA

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)

i

nu

kmmnn dtt

uuutB

t

tuuuA

0

121121 ,...,,,,,...,,

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

If the marginal distributions are standard normal this yields a standard browian motion. We can however use different marginals preserving brownian dynamics.

u

dwst

wsvt

0

11

Time 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)

u

dwshth

wshvth

0

11

,,

,,

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

is a copula iff

u

dwwHvGFwCDvuC0

111 ,),(ˆ

tGtFHdwwHtFwCDC

*,1

0

11

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

0

11 , dwwHzFwCD

A 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.1. Start with the probability distribution of increments in the first

period F1 and set F1 = H1.2. 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…

zHdwwHzFwCD 2

1

0

11

21 ,

Distribution of losses: 10 y

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0,45

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Cross section rho = 0.4

Temporal Dep. Kendall tau = 0.2

Temporal Dep. Kendall tau = -0.2

Temporal dependence

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

0,4

0-3% 3-7% 7-10% 10-15% 15-30%

Product

Temporal Frank rho = 0.2

Temporal Frank rho = 0.4

Temporal Frank rho = -0.2

Temporal Frank rho = - 0.4

Equity tranche: term structure

0,15

0,2

0,25

0,3

0,35

0,4

3 4 5 6 7 8 9 10

Cross Section rho = 0.4

Cross Section rho = 0.2

Temporal Dep. Kendall tau = -0.2

Senior tranche: term structure

-0,001

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

3 4 5 6 7 8 9 10

Cross Section rho = 0.4

Cross Section rho = 0.2

Temp. Dep Kendall tau = -0.2

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 Ti

j – 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 (Q1

1, Q12,… Q1

n)…, Gm(Qm1, Qm

2,… 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 isY = + 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

0

11 dwwFzFzF ZXY

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