Quantact workshop on risk management of …Quantact workshop on risk management Why replicate hedge funds? Replication techniques Strong replication Weak replication Modeling of returns

Quantactworkshop on risk

management

Why replicatehedge funds?

Replicationtechniques

Strong replication

Weak replication

Modeling ofreturns

Optimal Hedging

B-vines models

Empirical results

First experiment

Second experiment

Third experiment

Fourth experiment

Multidimensional example

Modeling

Target distributionfunction

Replication ofEDHEC indices

Conclusion

References

Replication methods for financial indexes

Bruno Remillard, HEC Montreal

Quantact workshop on risk management ofsegregated funds

March 9, 2018

Quantact workshop on risk management March 9 1 / 74


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Outline

1 Why replicate indexes

2 Replication techniques

3 Kat-Palaro approach and drawbacks

4 Normal mixtures

5 Optimal hedging

6 Some numerical results



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Outline




4 Normal mixtures

5 Optimal hedging




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Outline




4 Normal mixtures

5 Optimal hedging




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Outline




4 Normal mixtures

5 Optimal hedging




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Modeling ofreturns

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Outline




4 Normal mixtures

5 Optimal hedging




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Modeling ofreturns

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Outline




4 Normal mixtures

5 Optimal hedging




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References

Why replicate hedge funds?

Hedge fund industry is booming ⇒ Increasingpopularity

Number of funds increased from about 500 in early 90s,over 8000 in 2006, now 9803Assets under management presently 3.02 trillion dollarsalthough there were 1023 liquidations in 2016 (1471 in2008)

Lack of regulation and transparency

Use of leverage, derivatives, short sales and othernon-traditional investment strategies

Early sales pitch: offer superior returns



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Why replicate hedge funds?

Sales pitch has changed

Increased competition has led to lower risk-adjustedreturns (on average)

New sales pitch: Diversification benefits due to lowcorrelation with traditional assets

There are however a few drawbacks in hedge fundinvesting:

High management and performance feesLack of liquidity and significant lock-up periodsLack of transparency

Solution: Find a more efficient method to generate thesame returns

Clones are now used to create ETFs: IQ Hedge MultiStrategy ETF, Horizons HFF (targeting MorningstarBroad Hedge Fund Index SM).



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Replication techniques

Replication techniques

Strong replication: The target is the index

Naive or imitative approach

Factorial approach

Weak replication: The target is the distribution functionof the index



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Replication techniques Strong replication

Strong replication

Naive replication: try to imitate the HF managerinvestment strategy or index composition.Example: For a Merger Arbitrage Fund⇒ long sellers,short buyers.

”Alternative beta” approach: attempt to reproducehedge fund returns by investing in a portfolio of assetsthat provide similar end of month returns.

Portfolio weights are calculated using regression (e.g.,last 40 months moving window) or filters.

Alternative beta funds have been launched by severalinstitutions including Goldman Sachs, JPMorgan,Deutsche Bank, Innocap.

According to Wallerstein et al. (2010), the short versionVerso of Innocap perfomed best, from 2008 to 2009.



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Illustration of the factorial approach

R(Fund)t =

∑k=1

βt,kR(Factor ,k)t + εt

The unknown weights βt,k are evaluated using arolling-window regression, e.g., last 24 months, or usingfiltering.

For filtering, one must define a dynamic for the weights(Roncalli and Teıletche, 2007).

Example: The target is HFRI Fund Weighted CompositeIndex, and the factors: S&P500 Index TR, Russel 2000 IndexTR, Russell 1000 Index TR, Eurostoxx Index, Topix, US10-year Index, 1-month LIBOR (data from Innocap, fromApril 1997 to October 2008).



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May1997 May1998 May1999 May2000 May2001 May2002 May2003 May2004 May2005 May2006 May2007 May2008−2

−1.5

−1

−0.5

0

0.5

1

1.5

2Betas from regression with window size24 on HFRIFWI Index

SPTR IndexRU20INTR IndexRU10INTR IndexSX5E IndexTPX IndexTY1 ComdtyUS0001M Index

May1997 May1998 May1999 May2000 May2001 May2002 May2003 May2004 May2005 May2006 May2007 May2008−0.1

−0.05

0

0.05

0.1

0.15Returns

Figure: Evolution of βs with a 24-month rolling window.



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May1997May1998May1999May2000May2001May2002May2003May2004May2005May2006May2007May2008−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Betas for Kalman filter using100% of the sample on HFRIFWI Index

SPTR IndexRU20INTR IndexRU10INTR IndexSX5E IndexTPX IndexTY1 ComdtyUS0001M Index

May1997 May1998 May1999 May2000 May2001 May2002 May2003 May2004 May2005 May2006 May2007 May2008−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1Returns

HFRI Fund Weighted CompositeKalman

Figure: Evolution of βs using Kalman filter.



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Table: In-sample and out-of-sample statistics. Tracking error,mean and volatily are expressed in %/year .

Portfolio In-sample statisticsTE Corr Mean Std Skew Kurt(Ex)

Target - 1.00 8.12 7.72 -0.59 2.45Reg. 10.58 0.93 8.79 8.32 -0.69 2.22Kalman 8.54 0.95 9.68 7.75 -0.59 2.53

Out-of-sample statistics

TE Corr Mean Std Skew Kurt(Ex)

Target - 1.00 8.12 7.72 -0.59 2.45Reg. 19.27 0.83 9.30 9.86 -0.11 3.34Kalman 14.71 0.86 9.97 8.20 -0.40 2.63



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Replication techniques Weak replication

Weak replication

Alternative replication method proposed by Amin and Kat(2003) and extended by Kat and Palaro (2005).

Based on the Payoff Distribution Model put forth by Dybvig(1988).

Aim: Replicate hedge fund returns not by identifying thereturn generating betas, but identifying a trading strategythat can be used to generate the distribution of the hedgefund returns.

Kat and Palaro (2005) show that for most hedge funds, theirstatistical properties can be replicated by investing in analternative dynamic strategy.

The implementation proposed by Kat and Palaro is howeversubject to several shortcomings and inconsistencies.



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Main ingredients of the Kat-Palaro approach

Assets

Payoff

“Option pricing”

Hedging strategy



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The assets

S (1) represents the reference portfolio of the investor(e.g., stock and bond portfolio).

S (2) represents the reserve asset used to generate thereplication strategy (e.g., equal weighted portfolio ofhighly liquid futures contracts).

S (3) represents the fund one seeks to replicate.



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Mathematical formulation

Given two assets S (1) (reference asset) and S (2) (reserveasset), it is possible to reproduce both the probabilitydistribution of the returns of a third risky asset S (3) (index)and its dependence with the returns of reference asset S (1).

� The aim is not to reproduce the monthly values of S (3)

but rather its statistical properties.

More precisely, there exists a (return) function g such thatthe joint distribution of the (monthly) returns

R(1)0,T = log

(S1T

S10

)of S

(1)T and g

(R

(1)0,T ,R

(2)0,T

)is the same as

the joint distribution of R(1)0,T and R

(3)0,T .



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Payoff

The goal is to determine the best method to generate the

return g(R

(1)0,T ,R

(2)0,T

)without investing in S (3).

Kat and Palaro (2005) propose to do so by hedging, i.e.,constructing a dynamic portfolio {Vt(v0, ϕ)}Tt=0 of the twoassets S (1) and S (2), traded daily, in order to generate thepayoff

G(S

(1)T ,S

(2)T

)= 100 exp

{g(R

(1)0,T ,R

(2)0,T

)}at the end of the month.



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Replication portfolio

ϕ(j)t : number of parts of asset S (j) invested during (t − 1, t].

ϕt may depend only on S0, . . . ,St−1, ϕt is assumed to bepredictable. Initially, ϕ0 = ϕ1, and the portfolio initial valueis v0.Setting βt = e−r(T−t)/T for the discounting factors,self-financing yields

βTVT = βTVT (v0, ϕ) = v0 +T∑t=1

ϕ>t (βtSt − βt−1St−1).



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K-P measure

If the goal is attained, i.e., VT = G(S

(1)T , S

(2)T

), then the

(log) return of the portfolio is

log(VT/v0) = log(100/v0) + g(R

(1)0,T ,R

(2)0,T

).

In Kat and Palaro (2005), the initial amount v0 to beinvested in the portfolio is viewed as a measure ofperformance of the HF, called the K-P measure.

One could set α = log(v0/100) to estimate manager’s alpha.



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Interpretation

If v0 = 100, i.e. α = 0, the strategy generates the samereturns as S (3) (in distribution).

If v0 < 100, i.e. α < 0, it is worth replicating,generating superior returns (in distribution), while ifv0 > 100, i.e., α > 0, it may be not worth replicating.

Note that centered moments like standard deviation,skewness, kurtosis, are not affected by the value of the K-Pmeasure v0.



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Expression for g

The return’s function g is easily shown to be calculableusing the marginal distribution functions F1, F2 and F3 of

S(1)T , S

(2)T , S

(3)T , and the copulas C1,2 and C1,3 associated

respectively with the joints returns(R

(1)0,T ,R

(2)0,T

)and(

R(1)0,T ,R

(3)0,T

)(Kat and Palaro, 2005).

From Sklar’s theorem (Sklar, 1959),

P(R

(1)0,T ≤ x ,R

(3)0,T ≤ y

)= H1,3(x , y) = C1,3 {F1(x),F3(y)} .

All the dependence between two assets is determined by theassociated copula.



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The exact expression for g is given by

g(x , y) = Q{x ,P

(R

(2)0,T ≤ y |R(1)

0,T = x)}

, (1)

where Q(x , α) is the order α quantile of the conditional law

of R(3)0,T given R

(1)0,T = x , i.e., for any α ∈ (0, 1), q(x , α)

satisfies

P{R

(3)0,T ≤ Q(x , α)|R(1)

0,T = x}

= α.



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Relationship with copulas

Using properties of copulas, e.g. Nelsen (1999), Joe (1997)or Cherubini et al. (2004), the conditional distributions canbe expressed in terms of the margins and the associatedcopulas.In particular,

P(R

(2)0,T ≤ y |R(1)

0,T = x)

=∂

∂uC1,2(u, v)

∣∣∣∣u=F1(x),v=F2(y)

.



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9496

98100

102104

106

96

98

100

102

10440

60

80

100

120

140

Reserve assetReference asset

Rep

licat

ed p

ayof

f

Figure: Payoff function G corresponding to marginals Normal,Johnson, and Student, with Gaussian and Gumbel copulas.



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Implementation of the Kat-Palaro approach

Model the marginal distributions F1,F2 and F3 of R(1)0,T ,

R(2)0,T , R

(3)0,T .

They suggest to use Normal, Student or Johnsondistributions.

Model the copula C1,2 between R(1)0,T and R

(2)0,T , and the

copula C1,3 between R(1)0,T and R

(3)0,T .

They suggests to use Gaussian and Student copulas andcertain Archimedean copula families.

Solve for g .

Solve for the cheapest dynamic replicating strategy that

produces G(S

(1)T , S

(2)T

)(using daily returns).

They suggest using the Black-Scholes framework, i.e.,they use delta-hedging.



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Hedging strategy

Even if the returns are not normal, Kat-Palaro suggests totreat them within the Black-Scholes framework.Denoting by C

(BS)t the value of an option with payoff G in a

bivariate Black-Scholes model, they obtain

v(KP)0 = C

(BS)0 ,

and

ϕ(KP,i)t+1 =

∂C(BS)t

∂S(i)t

, i = 1, 2.

or all t ∈ {0, . . . ,T − 1}.



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Evaluation of v(KP)0 and ϕ(KP)

The values C(BS)t and ∂C

(BS)t

∂S(i)t

, t ∈ {0, . . . ,T − 1}, can be

calculated from multinomial trees or Monte Carlosimulations.� Because rebalancing of the portfolio is not done

continuously, the value of the portfolio V(KP)t is in

general different from C(BS)t when t > 0.

For an analysis of the (discrete) hedging error in aBlack-Scholes setting, see, e.g., Wilmott (2006).



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Major drawbacks of K-P methodology I

1 Although returns of the hedge funds and traded assetsare clearly non-normal, the replicating strategy isdeduced from the BS methodology. This can lead tolarge hedging errors and biased in replication.

After finding the optimal hedging strategy thatminimize the root mean square hedging error (RMSHE),we propose a methodology adapted from Americanoption pricing to evaluate the strategy.



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Major drawbacks of K-P methodology II

2 Kat-Palaro approach can lead to inconsistencies becausethe law of the daily returns, which is essential in findingoptimal hedging strategies, might not be compatiblewith estimated law of the monthly returns.

We use Gaussian HMM to model the daily returns andthen solve for the appropriate law of the monthlyreturns. We also propose a new goodness-of-fit test forselecting the number of regimes, marginals and copulas.Recently, Webanck (2011) used Gaussian HMM toaccount for serial dependence. Goodness-of-fit testswere proposed in Remillard et al. (2014).



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Our modeling approach

Modeling part:

Estimation of the parameters of the (daily) bivariateGaussian mixtures, using historical daily returns(R

(1)t ,R

(2)t

).

If replicating an existing hedge fund, use historical

monthly returns of(R(1),R(3)

)to estimate F3 and the

copula C1,3.If creating a synthetic fund, choose F3 and the copulaC1,3.

Replication part:

Choose a replication method (e.g., Monte Carlointerpolation).Find v0 and the optimal hedging strategy ϕ using dailyreturns.



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Modeling of returns

Johnson distributions

Kat and Palaro (2005) propose to use a Johnson SU familydefined by X = a + b sinh

(Z−cd

), where Z ∼ N(0, 1).

Its interest lies in the property that the first 4 moments ofany distribution can be fitted. However, Johnson’s law is notinfinitely divisible.

Therefore, it should not serve as a model for the distributionof R

(1)0,T or R

(2)0,T if the daily returns are assumed to be

independent.

Best suited for modeling the returns of the (synthetic) fund.



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Modeling of returns

Gaussian mixtures

We overcome the inconsistency issue by using Gaussianmixtures to model the daily returns and then solve for thelaw of the monthly returns, also given by a normal mixture.

The interpretation of a mixture is easy: With probability πk ,choose a regime k and observe a bivariate Gaussian vectorwith mean µk and covariance Ak .

One can add serial dependence by choosing the regimesaccording to a Markov chain. This a treated in aforthcoming paper. In that case, the strategy depends on theunknown regimes (predicted by filtering methods).



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Modeling of returns

Definition

A random vector X is a Gaussian mixture with m regimesand parameters (πk)mk=1, (µk)mk=1 and (Ak)mk=1, if its densityis given by

f (x) =m∑

k=1

πkφ(x ;µk ,Ak)

where φ(x ;µ,A) = e−12 (x−µ)>A−1(x−µ)

(2π)d/2|A|1/2 is the density of a

Gaussian vector with mean vector µ and covariance matrix A.

Interpretation: Choose regime k at random with probabilityπk and then generate X ∼ N(µk ,Ak).



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Modeling of returns

Estimation and goodness-of-fit

Estimation for mixtures is based on the EM algorithm(Dempster et al., 1977).

For the selection of the number m of regimes, a newgoodness-of-fit test, based on Rosenblatt’s transforms(Genest et al., 2009), is used. P-values are calculated usingparametric bootstrap.



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Optimal Hedging

Optimal hedging

Expected square hedging error is chosen as a measure ofquality of replication (Remillard and Rubenthaler, 2013,2016).

The replication strategy problem for a given payoff C is tofind an initial investment amount V0 and a predictableinvestment strategy −→ϕ = (ϕt)

Tt=1 that minimize the

expected quadratic hedging error E[{G (V0,

−→ϕ )}2], where

G = G (V0,−→ϕ ) = βTC − VT ,

Vt = V0 +t∑

j=1

ϕ>j ∆j , t = 0, . . . ,T ,

and∆t = βtSt − βt−1St−1, t = 1, . . . ,T .



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Optimal Hedging

Solution

Set PT+1 = 1, and for t = n, . . . , 1, define

At = E(

∆t∆>t Pt+1|Ft−1

),

bt = A−1t E (∆tPt+1|Ft−1) ,

αt = A−1t E (βTC∆tPt+1|Ft−1) ,

Pt =T∏j=t

(1− b>j ∆j

).

Theorem 1

Suppose that E (Pt |Ft−1) 6= 0 P-a.s., for t = 1, . . . ,T . Thenthe solution (V0,

−→ϕ ) of the minimization problem isV0 = C0, and

ϕt = αt − Vt−1bt , t = 1, . . . ,T ,

where βtCt = E(βTCPt+1|Ft)E(Pt+1|Ft)

, t=0,. . . ,T.Quantact workshop on risk management March 9 34 / 74


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Optimal Hedging

Markovian case

If the price process is Markov and C = f (ST ), then Ct is adeterministic function of St , and αt and bt are deterministicfunctions of St−1.

Having deterministic functions make implementation a littlebit easier.

Once St is observed, one gets Vt , so ϕt+1 can be calculated.



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Optimal Hedging

Implementation in the Markovian case

Algorithm based on Monte Carlo simulations.

Extension of the methodology developed by Del Moralet al. (2012) to calculate Ct−1, αt and bt (as a functionof Ct , αt+1 and bt+1), for all points of a grid.

Interpolate linearly on the grid to approximate functionsCt−1, αt and bt for possible value of the assets.



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Optimal Hedging

Generalized payoff approach

Joint work with B. Nasri and M. Ben-Abdellatif (Remillardet al., 2017b)

Given investor’s portfolios P1, . . . ,Pp and a reserve assetportfolio Pp+1, one can find a (return) function gdetermined by the joint distribution of the portfolios, amarginal distribution function F and a copula C so that

Y = g(R

(1)T , . . . ,R

(p+1)T

)has a given distribution F ;

The conditional distribution of Y given

X =(R

(1)T , . . . ,R

(p)T

)is

P (Y ≤ y |X = x) =∂u1 · · · ∂upC (u, v)

∂u1 · · · ∂upC (u, 1),

evaluated at u = (F1(x1), . . . ,Fp(xp)), v = F (y).



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Optimal Hedging B-vines models

B-vines I

Aim: find a flexible way to construct a conditionaldistribution of a random variable Y given a d-dimensionalrandom vector X.

Using the representation of conditional distributions in termsof copulas, this problem amounts to constructing theconditional distribution C of a uniform random variable Ugiven a random vector V (with uniform margins) that iscoherent with the distribution function CV of V.

Unfortunately, the usual vines models for multivariatecopulas cannot be used here, because of this compatibilityconstraint.

For more details on unconstrained vine models applied toconditional distributions, see, e.g., Kraus and Czado (2017).



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B-vines II

As noted before, when d = 1, the compatibility condition isnot a constraint at all since CV (v) = v , v ∈ [0, 1], and thesolution is simply to take C(u, v) = ∂vC (u, v), for a copulaC that is smooth enough.

Next, in the case d = 2, if D1 and D2 are bivariate copulas,with conditional distributions Dj(u, t) = ∂tDj(u, t),j ∈ {1, 2}, and CV is the copula of V = (V1,V2), then

C(u, v) = D2 {D1(u, v1), ∂v1CV(v1, v2)} , (2)

v = (v1, v2) ∈ (0, 1)2, defines a conditional distribution for Ugiven V = v , compatible with the law of V.

This construction is a particular case of a D-vine copula, asdefined in Joe (1996), Aas et al. (2009).



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B-vines III

Guided by formula (2), let Dj , j ∈ {1, . . . , d} be bivariatecopulas and let Dj(u, t) = ∂tDj(u, t) be the associatedconditional distributions.

For j ∈ {1, . . . , d}, further let Rj−1(v1, . . . , vj) be theconditional distribution of Vj givenV1 = v1, . . . ,Vj−1 = vj−1, with R0(v1) = v1, and for(u, v) ∈ (0, 1)d+1, set C0(u) = u, and

Cj(u, v1, . . . , vj) = Dj {Cj−1(u, v1, . . . , vj−1),Rj−1(v1, . . . , vj)} .(3)



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B-vines IV

Note thatE{Cj(u, v1, . . . , vj−1,Vj)|V1 = v1, . . . ,Vj−1 = vj−1} is givenby∫ 1

0Cj{u, v1, . . . ,Rj−1(v1, . . . , vj)}dRj−1(v1, . . . , vj)

=

∫ 1

0Dj{Cj−1(u, v1, . . . , vj−1), t}dt

= Dj{Cj−1(u, v1, . . . , vj−1), 1}= Cj−1(u, v1, . . . , vj−1).

It follows that Cj is the conditional distribution of U givenV1, . . . ,Vj .



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B-vines V

The conditional quantile of U given V1, . . . ,Vj is also easyto compute, satisfying a recurrence relation similar to (3).

In fact, if the conditional quantile of Cj is denoted by Γj ,then for any j ∈ {1, . . . , d}, and for any u, v1, . . . vd ∈ (0, 1),

Γj(u, v1, . . . , vj) = Γj−1

[D−1

j {u,Rj−1(v1, . . . , vj)} , v1, . . . , vj−1

].

(4)This construction does not lead to a proper vine copula sinceall copulas involved are not bivariate copulas, the copula ofV being given.

In fact, it is more general than the pair-copula constructionmethod used in vines models.



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B-vines VI

Nevertheless, this type of model will be called B-vines andits construction is illustrated below, where the underlinedvariables (in red) mean that their distributions R0, . . . ,Rd−1

are known1, and the conditional copulas D1, . . . ,Dd have to



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B-vines VII

be chosen, in order to determine C1, . . . , Cd .

Level 1:

D1

C0

U |R0

V1 =⇒ C1

Level 2:

D2

C1

U|V1 |R1

V2|V1 =⇒ C2

... · · · . . .

Level j :

Dj

Cj−1

U|V1, . . . ,Vj−1 |Rj−1

Vj |V1, . . . ,Vj−1 =⇒ Cj... · · · . . .

Level d :

Dd

Cd−1

U|V1, . . . ,Vd−1 |Rd−1

Vj |V1, . . . ,Vd−1,Vd =⇒ Cd

1R0, . . . ,Rd−1 are called the Rosenblatt’s transforms.Quantact workshop on risk management March 9 44 / 74


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B-vines can be particularly useful in conditional meanregression (OLS, GAM, GLM, etc,) and conditional quantilesettings, where the distribution of the covariates is oftengiven; see, e.g., Remillard et al. (2017).

It can also be used in our replication context when the targetS (3) exists; in this case, we could look at B-vines constructedfrom popular bivariate families like Clayton, Gumbel, Frank,Gaussian and Student, and find the ones that fit best thedata, in the same spirit as the choice of vines for copulamodels in the R packages CDVine or VineCopula.

In a future work we will propose goodness-of-fit tests forthese models.



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Typical computational settings

When d = 1,

Vertices discretization : $0.05

Simulations per grid point : 20000

Hedging periods : 22 days

When d > 1, no discretization is needed since one uses thecontinuous time limiting process and the optimal martingalemeasure (Remillard et al., 2017b).



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Empirical results

In order to illustrate the advantages of the optimal hedgingand modeling approach put forth in the paper, we firstpresent three cases:

Case 1: the returns of assets S (1) and S (2) follow amixture of 4 regimes.The monthly returns areapproximated by a Gaussian distribution. The monthly

return R(3)0,T also follow a Gaussian distribution (0,12).

The dependence between S (3) and S (1) is modeled witha Gaussian copula.

Case 2: Daily returns of Rt are the same as in case 1.However, the distribution of the monthly returns iscompatible with the daily returns.



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Empirical results

Case 3: The daily returns Rt are iid as an infinitemixture (the difference between two independentGamma variables). The monthly returns Rt are thusdifferences between two independent exponential

random variables. The monthly returns R(3)0,T also follow

a double exponential distribution.



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Empirical results First experiment

Table: Replication results based on 10 000 trajectories for the firstscenario for g (R0,T ).

Parameter True value g

Mean 0 0.000155102Std. dev. 0.0346 0.035653449Skewness 0 -0.059435597Kurtosis 0 0.040573771ρ 0.3 0.199636044

� There is no need to do replication here because g is not

calculated correctly!



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Empirical results Second experiment

Table: Replication results based on 10 000 trajectories for thesecond scenario for g (R0,T ) and log(VT/100), under optimalhedging and delta hedging.

Parameter True value g Opt hedging BS hedging

Mean 0 3.96E-07 3.57E-07 -4.23E-04Std. dev. 0.0346 0.0350 0.0350 0.0350Skewness 0 -0.0589 -0.0641 -0.0640Kurtosis 0 0.0300 0.0325 0.0324ρ 0.3 0.3028 0.3028 0.3029

Table: Replication results based on 10 000 trajectories for thesecond scenario for the payoff G and log(VT/100) under optimalhedging and delta hedging.

Parameter Opt hedging BS hedging |OH/BS |Mean hedging error 4.0E-5 -4.2E-2 10491.67Square root MSE 1.8E-2 4.6E-2 2.56



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Empirical results Second experiment

−0.2 −0.15 −0.1 −0.05 0 0.050

5

10

15

20

25

30Kernel density estimation of Hedging Error

Optimal Hedging ErrorBS Hedging Error

Figure: Kernel density estimation of hedging errors for optimalhedging and delta hedging for the second experiment.



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Table: Replication results based on 10 000 trajectories for thesecond scenario for g (R0,T ) and log(VT/100), under optimalhedging and delta hedging.

Parameter True value g Opt hedging BS hedging

Mean 0 3.93E-4 3.92E-4 -5.68E-3Std. dev. 0.0433 0.0429 0.0429 0.0438Skewness 0 0.0839 0.0717 0.0552Kurtosis 3 2.9930 2.9932 2.9600ρ 0.3 0.3039 0.3087 0.3375

Table: Replication results based on 10 000 trajectories for thesecond scenario for the payoff G and log(VT/100) under optimalhedging and delta hedging.

Parameter Opt hedging BS hedging |OH/BS |Mean hedging error -2.91E-4 -9.20E-2 315.8Square root MSE 0.3534 0.4247 1.202



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Empirical results Third experiment

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5Kernel density estimation of Hedging Error

Optimal Hedging ErrorBS Hedging Error

Figure: Kernel density estimation of hedging errors for optimalhedging and delta hedging for the third experiment.



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Empirical results Fourth experiment

The goal of this fourth experiment (Hocquard et al., 2007).is to compare again hedging error, this time in function ofthe dependence of the target asset S (3).

To illustrate the advantage of the optimal hedging strategyproposed in Papageorgiou et al. (2008), we compare themean hedging error and the RMSHE for the optimal hedgingand for the Kat-Palaro approach.



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Assets

Asset S (1) is a proxy for the typical institutionalCanadian pension fund.

Asset S (2) is a diversified portfolio of typical marketexposures, specifically global equity indices, creditindices and commodity indices

Asset S (3) that is being replicated is chosen to begaussian distribution with an annual volatility of 12%.

Bivariate daily and monthly distributions of assets S (1) andS (2) (2000–2007) are modeled using normal mixtures.

We make hedging comparisons under various dependencelevels and copulas (Gaussian, Clayton and Frank), using 10000 scenarios of 22 daily returns.



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−1 −0.5 0 0.5 1−0.1

0

0.1

0.2

0.3

Kendall’s Tau for IP Copula − GaussianH

edg

ing

Err

or

OHKP

−1 −0.5 0 0.5 1

0.8

1

1.2

1.4

Kendall’s Tau for IP Copula − Gaussian

sqM

SH

E

OHKP

−1 −0.5 0 0.5 1−0.2

−0.1

0

0.1

0.2

0.3

Kendall’s Tau for IP Copula − Clayton

Hed

gin

g E

rro

r

OHKP

−1 −0.5 0 0.5 10.8

1

1.2

1.4

1.6

Kendall’s Tau for IP Copula − Clayton

sqM

SH

E

OHKP

−1 −0.5 0 0.5 1−0.2

0

0.2

0.4

0.6

Kendall’s Tau for IP Copula − Frank

Hed

gin

g E

rro

r

OHKP

−1 −0.5 0 0.5 1

0.8

1

1.2

1.4

1.6

Kendall’s Tau for IP Copula − Frank

sqM

SH

E

OHKP

Figure: Hedging Error Measures.



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Empirical results Multidimensional example

Multidimensional example I

We now provide some empirical evidence regarding theability of the model to replicate a synthetic index. In theimplementation of the replication model, we consider a3-dimensional problem.

Table: Portfolios’ composition.

P(1) 60% S&P/TSE 60 IX future40% S&P500 EMINI future

P(2) 100% CAN 10YR BOND future

P(3) 10% E-mini NASDAQ-100 futures20% Russell 2000 TR20% MSCI Emerging Markets TR10% GOLD 100 OZ future10% WTI CRUDE future30% US 2YR NOTE (CBT)



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Multidimensional example II

Table: Summary statistics for the 3 portfolios. Values are reportedon an annual basis.

Statistics R(1) R(2) R(3)

Daily returnsMean 0.0198 0.0209 0.0363Volatility 0.1327 0.0592 0.1238Skewness -0.6447 -0.3261 -0.4418Excess kurtosis 8.5478 2.0583 5.1415

We use a Gaussian HMM to model the joint distribution ofthe returns of the 3 portfolios. The choice of the number ofregimes is done as suggested in Remillard et al. (2017a).This leads to a selection of 6 regimes for the daily returns.Usually, for non-turbulent periods, 4 regimes are sufficientfor fitting daily returns.



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Multidimensional example III

In order to have monthly returns compatible with dailyreturns, we simulated 10 000 values of monthly returnsunder the estimated RSGBM and we fitted a Gaussian HMM(3 regimes were necessary).

It then follows that the conditional distribution F(·, x) is amixture of 3 Gaussian distributions.For this example, the target distribution F? is a truncatedGaussian distribution at −a, with (annual) parameters µ?and σ?, meaning that

F?(y) =

0, y ≤ −a;

Φ(

y−µ?/12

σ?/√

12

)−Φ

(−a−µ?/12

σ?/√

12

)Φ(

a+µ?/12

σ?/√

12

) , y ≥ −a.(5)



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Multidimensional example IV

The density is displayed in Figure 7.

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.060

5

10

15

20

25

30

Figure: Target density for the monthly returns.



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Multidimensional example V

We try to replicate the monthly returns of a synthetic hedgefund having distribution F? given by (5).

We will rebalance the portfolio once a day, so n = 21. Forsimplicity, we take S0 = (1, 1, 1) and r = 0.01. We considerthe independence model, meaning that C(u, v1, v2) = u.

Finally, for each model, we simulated 1 000 replicationportfolios.

Table: Descriptive statistics for the independence model.

Statistics HE G(S21) V21 g(R21) log(V21/V0) TargetAverage -0.012 100.770 100.782 0.0076 -0.0001 0.0078Median -0.012 100.741 100.760 0.0074 -0.0003 0.0073Volatility 0.035 1.299 1.290 0.013 0.013 0.013Skewness 0.431 0.201 0.192 0.172 0.162 0.267Kurtosis 7.939 2.581 2.614 2.559 2.593 2.760Minimum -0.145 98.083 98.013 -0.019 -0.028 -0.02Maximum 0.241 104.987 104.926 0.049 0.040RMSE 0.037

τ (1) 0.023 0.024 0

τ (2) -0.061 -0.060 0Quantact workshop on risk management March 9 61 / 74


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Multidimensional example VI

As can be seen from the statistics displayed in Table 9, thetracking error given by the RMSE is very good.

The mean of the hedging error is significantly smaller that 0,meaning that the portfolio is doing better on average thanthe target payoff, even if α = 0.0078 > 0.

The target distribution is also quite well replicated and thedistribution of the hedging errors is also quite good, as canbe seen from Figure 8.

Finally, letting τ (j) be the estimated Kendall’s tau betweenthe variable and the returns of portfolio P(j), one can seethat the returns of the hedged portfolio are independent ofthe returns of the reference portfolios, meaning that thesynthetic asset has the desired properties.



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Multidimensional example VII

-0.15 -0.1 -0.05 0 0.05 0.1 0.150

2

4

6

8

10

12

14

Figure: Estimated density of the hedging error G (S21)− V21 forthe independence model based on 1000 replications. HereV0 = 100.7864 and α = logV0/100 = 0.007833.



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Replication of EDHEC indices

Role of the reserve asset

As a final illustration, we want to address the problem of thereserve asset. Using the methodology of Papageorgiou et al.(2008), one replicated EDHEC indices, over the period2002-2007, for two reserve assets.

Table: Regression of the HFRI indices returns with the replicationreturns for reserve asset 1, for several parameters

Target Intercept Slope R2

Volatility 0.000548103 1.007158135 99.50Skewness -1.007452845 1.122119166 65.99

Robust Skew 0.00904498 0.592606993 46.09Kurtosis 2.463617274 1.159951552 15.32

Robust Kurt 1.311517175 0.724346845 47.37Kendall’s Tau 0.043331675 1.014053643 99.05Pearson’s Rho 0.036896001 1.055609145 96.75



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Table: Regression of the HFRI indices returns with the replicationreturns for reserve asset 2, for several parameters

Target Intercept Slope R2

Volatility 0.000446726 1.005457776 99.61Skewness -0.642431289 0.891289249 69.66

Robust Skew 0.039253185 0.790688855 69.94Kurtosis 1.502116477 1.023389918 72.97

Robust Kurt 1.459565597 0.587177989 40.79Kendall’s Tau 0.035762895 1.030032998 99.43Pearson’s Rho 0.036595462 1.055630986 96.39



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Table: Initial investment v0 in the replication of EDHEC indices forboth reserve assets.

Fund V0

Reserve 1 Reserve 2

EDHEC-Convertible Arbitrage 99.54307232 99.91754606EDHEC-CTA Global 99.00591261 99.85286557EDHEC-Distressed Securities 100.2752537 100.645535EDHEC-Emerging Markets 100.0757102 100.4515871EDHEC-Equity Market Neutral 99.85680498 99.99383759EDHEC-Event Driven 99.87363087 100.3162115EDHEC-Fixed Income Arbitrage 99.88868645 100.0907229EDHEC-Global Macro 99.84474995 100.2384539EDHEC-Long/Short Equity 99.60337666 100.1087833EDHEC-Merger Arbitrage 99.70200103 99.99705359EDHEC-Relative Value 99.81967336 100.109444EDHEC-Short Selling 98.05685558 99.04396197EDHEC-Funds of Funds 99.74332198 100.0559835

Percentage of v0 under 100$ 84.62% 38.46%



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Conclusion

Conclusion

We propose an improved methodology that can be usedto replicate the returns of existing financial indexes orto generating synthetic funds with certain desiredproperties.

Two main improvements on the Kat-Palaro approach:

Solve for dynamic trading strategy that minimizessquared hedging error. ⇒ We provide some evidencethat minimizing the squared hedging error is moreefficient than the delta hedging approachUse Gaussian HMM to overcome inconsistenciesbetween the law of daily and monthly returns, and wepropose a new test to evaluate the goodness-of-fit inorder to evaluate the correct number of regimes.



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Solve for dynamic trading strategy that minimizessquared hedging error. ⇒ We provide some evidencethat minimizing the squared hedging error is moreefficient than the delta hedging approach

Use Gaussian HMM to overcome inconsistenciesbetween the law of daily and monthly returns, and wepropose a new test to evaluate the goodness-of-fit inorder to evaluate the correct number of regimes.



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Conclusion

References I

Aas, K., Czado, C., Frigessi, A., and Bakken, H. (2009).Pair-copula constructions of multiple dependence.Insurance Math. Econom., 44(2):182–198.

Amin, G. and Kat, H. M. (2003). Hedge fund performance1990-2000: Do the “money machines” really add value.Journal of Financial and Quantitative Analysis,38(2):251–275.

Cherubini, U., Luciano, E., and Vecchiato, W. (2004).Copula Methods in Finance. Wiley Finance. Wiley, NewYork.

Del Moral, P., Remillard, B., and Rubenthaler, S. (2012).Monte Carlo Approximations of American Options thatPreserve Monotonicity and Convexity. In NumericalMethods in Finance, pages 117–145. Springer.



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Conclusion

References II

Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977).Maximum likelihood from incomplete data via the EMalgorithm. J. Roy. Statist. Soc. Ser. B, 39:1–38.

Dybvig, P. H. (1988). Distributional analysis of portfoliochoice. The Journal of Business, 61(3):369–393.

Genest, C., Remillard, B., and Beaudoin, D. (2009).Omnibus goodness-of-fit tests for copulas: A review and apower study. Insurance Math. Econom., 44:199–213.

Hocquard, A., Papageorgiou, N., and Remillard, B. (2007).Optimal hedging strategies with an application to hedgefund replication. Wilmott Magazine, (Jan-Feb):62–66.



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References III

Joe, H. (1996). Families of m-variate distributions withgiven margins and m(m − 1)/2 bivariate dependenceparameters. In Ruschendorf, L., Schweizer, B., and Taylor,M. D., editors, Distributions with fixed marginals andrelated topics, volume 28 of Lecture Notes–MonographSeries, pages 120–141. Institute of MathematicalStatistics, Hayward, CA.

Joe, H. (1997). Multivariate Models and DependenceConcepts, volume 73 of Monographs on Statistics andApplied Probability. Chapman & Hall, London.

Kallianpur, G. and Leadbetter, M. R. (1995). Proceedings ofan International Workshop on Stochastic PartialDifferential Equations. Center fo Stochastic Processes. 2VOLUMEs.



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Conclusion

References IV

Kat, H. M. and Palaro, H. P. (2005). Who needs hedgefunds? A copula-based approach to hedge fund returnreplication. Technical report, Cass Business School, CityUniversity.

Kraus, D. and Czado, C. (2017). D-vine copula basedquantile regression. Computational Statistics & DataAnalysis, 110:1–18.

Nelsen, R. B. (1999). An Introduction to Copulas, volume139 of Lecture Notes in Statistics. Springer-Verlag, NewYork.

Papageorgiou, N., Remillard, B., and Hocquard, A. (2008).Replicating the properties of hedge fund returns. Journalof Alternative Invesments, 11:8–38.



management



Strong replication

Weak replication

Modeling ofreturns

Optimal Hedging

B-vines models

Empirical results

First experiment

Second experiment

Third experiment

Fourth experiment


Modeling



Conclusion

References

Conclusion

References V

Remillard, B., Hocquard, A., Lamarre, H., andPapageorgiou, N. A. (2014). Option pricing and hedgingfor discrete time regime-switching model. Technicalreport, SSRN Working Paper Series No. 1591146.

Remillard, B., Hocquard, A., Lamarre, H., andPapageorgiou, N. A. (2017a). Option pricing and hedgingfor discrete time regime-switching model. ModernEconomy, 8:1005–1032.

Remillard, B., Nasri, B., and Ben-Abdellatif, M. (2017b).Replication methods for financial indexes. In Min, A.,Scherer, M., and Zagst, R., editors, Innovations inInsurance, Risk- and Asset Management, SpringerProceeding in Mathematics & Statistics. Springer. inpress.



management



Strong replication

Weak replication

Modeling ofreturns

Optimal Hedging

B-vines models

Empirical results

First experiment

Second experiment

Third experiment

Fourth experiment


Modeling



Conclusion

References

Conclusion

References VI

Remillard, B., Nasri, B., and Bouezmarni, T. (2017). Oncopula-based conditional quantile estimators. Statistics &Probability Letters, 128:14–20. September 2017.

Remillard, B. and Rubenthaler, S. (2013). Optimal hedgingin discrete time. Quantitative Finance, 13(6):819–825.

Remillard, B. and Rubenthaler, S. (2016). Option pricingand hedging for regime-switching geometric Brownianmotion models. Working paper series, SSRN WorkingPaper Series No. 2599064.

Roncalli, T. and Teıletche, J. (2007). An alternativeapproach to alternative beta. Technical report, SocieteGenerale Asset Management.

Sklar, M. (1959). Fonctions de repartition a n dimensions etleurs marges. Publ. Inst. Statist. Univ. Paris, 8:229–231.



management



Strong replication

Weak replication

Modeling ofreturns

Optimal Hedging

B-vines models

Empirical results

First experiment

Second experiment

Third experiment

Fourth experiment


Modeling



Conclusion

References

Conclusion

References VII

Wallerstein, E., Tuchschmid, N. S., and Zaker, S. (2010).How do hedge fund clones manage the real world? TheJournal of Alternative Investments, 12(3):37–50.

Webanck, G. (2011). Le modele de chaınes de markovcachees dans une strategie de replication de fonds decouvertures. Master’s thesis, HEC Montreal.

Wilmott, P. (2006). Paul Wilmott on Quantitative Finance,volume 3. John Wiley & Sons, second edition.


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