Copula Functions and Markov Processes for Equity and Credit

Derivatives

Umberto CherubiniMatemates – University of Bologna

Birbeck College, London 24/02/2010

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 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

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.

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 thatMax[DCNky + DCNsd – v(t,T),0] ≤ Coupon

andCoupon ≤ 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 asDP = 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.

vvuC

vHvHuHvHuH

b

baba

,Pr

,PrPr

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…

vvvC

vFvFvF

vFvFv

Y

YX

YXL

,Pr

,Pr

Pr

…and in booms

• In the case of booms, we have instead

• 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.

v

vvCvvFvFvF

vFvFv

Y

YX

YXU

1,21

Pr,Pr

Pr

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 II

• The conditional probability of X given Y = y can be expressed using the partial derivative of a copula function.

yFvxFuv

vuCyYxX21 ,

,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

,, 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 ,...,,,,,...,,

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 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

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

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 variable3. 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 = 0.4Temporal Dep. Kendall tau = 0.2Temporal 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%

ProductTemporal Frank rho = 0.2Temporal Frank rho = 0.4Temporal Frank rho = -0.2Temporal 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 = 0.4Cross Section rho = 0.2Temporal 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 = 0.4Cross Section rho = 0.2Temp. Dep Kendall tau = -0.2

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 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.

• 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 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).

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 = + 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 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