On the term structure of multivariate equity derivatives

Umberto CherubiniMatemates – University of BolognaBloomberg, New York 24/03/2010

Outline

• 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 writtenH(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 interval2. C(0,v) = C(u,0) = 0, C(1,v) = v and C(u,1) = u3. 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

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

vuCyYxX21 ,

,Pr

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*B(u,v)

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

1

0

,, dttvtB

ttuA

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

uuutBt

tuuuA

0

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

Symmetric Markov processes

• Definition. A Markov process is symmetric if 1. Marginal distributions are symmetric2. 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

DependenceCross-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 equivalent1. 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 H-condition 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 meanings1. 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 no-Granger causality bring about Granger independence

Granger independence1

2

3

4

{1 2}

{3 4}

{1 2 3 4}

1+ 2 =2

1+ 2 =1

1+ 2 =1

1+ 2 =0

1+ 2 =2

1+ 2 =1

1+ 2 =0

1+ 2 =1

1= 1 =1

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 independenceP(1= 1 = 2 = 2 = 1) = P(2 = 2 = 1 1 = 1 = 1)P(1 = 1 = 1)…while under vector independenceP(1= 1 = 2 = 2 = 1) = P(2 = 2 = 1)P(1 = 1 = 1)

Pricing kernel recursion• A copula recursion can be used to establish no-

arbitrage 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

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Copula recursion: vector independence

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Correlation recursion

• If we are only interested in the recursion of correlation instead of the whole copula

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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 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. – 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