Quantact workshop on risk management Why replicate hedge funds? Replication techniques Strong replication Weak replication Modeling of returns Optimal Hedging B-vines models Empirical results First experiment Second experiment Third experiment Fourth experiment Multidimensional example Modeling Target distribution function Replication of EDHEC indices Conclusion References Replication methods for financial indexes Bruno R´ emillard, HEC Montr´ eal Quantact workshop on risk management of segregated funds March 9, 2018 Quantact workshop on risk management March 9 1 / 74
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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
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
Outline
1 Why replicate indexes
2 Replication techniques
3 Kat-Palaro approach and drawbacks
4 Normal mixtures
5 Optimal hedging
6 Some numerical results
Quantact workshop on risk management March 9 2 / 74
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
Outline
1 Why replicate indexes
2 Replication techniques
3 Kat-Palaro approach and drawbacks
4 Normal mixtures
5 Optimal hedging
6 Some numerical results
Quantact workshop on risk management March 9 2 / 74
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
Outline
1 Why replicate indexes
2 Replication techniques
3 Kat-Palaro approach and drawbacks
4 Normal mixtures
5 Optimal hedging
6 Some numerical results
Quantact workshop on risk management March 9 2 / 74
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
Outline
1 Why replicate indexes
2 Replication techniques
3 Kat-Palaro approach and drawbacks
4 Normal mixtures
5 Optimal hedging
6 Some numerical results
Quantact workshop on risk management March 9 2 / 74
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
Outline
1 Why replicate indexes
2 Replication techniques
3 Kat-Palaro approach and drawbacks
4 Normal mixtures
5 Optimal hedging
6 Some numerical results
Quantact workshop on risk management March 9 2 / 74
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
Outline
1 Why replicate indexes
2 Replication techniques
3 Kat-Palaro approach and drawbacks
4 Normal mixtures
5 Optimal hedging
6 Some numerical results
Quantact workshop on risk management March 9 2 / 74
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
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
Quantact workshop on risk management March 9 3 / 74
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
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).
Quantact workshop on risk management March 9 4 / 74
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 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
Quantact workshop on risk management March 9 5 / 74
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 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.
Quantact workshop on risk management March 9 6 / 74
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 techniques Strong replication
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).
Quantact workshop on risk management March 9 7 / 74
Quantact workshop on risk management March 9 10 / 74
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 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.
Quantact workshop on risk management March 9 11 / 74
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 techniques Weak replication
Main ingredients of the Kat-Palaro approach
Assets
Payoff
“Option pricing”
Hedging strategy
Quantact workshop on risk management March 9 12 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 13 / 74
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 techniques Weak replication
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 .
Quantact workshop on risk management March 9 14 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 15 / 74
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 techniques Weak replication
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).
Quantact workshop on risk management March 9 16 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 17 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 18 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 19 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 20 / 74
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 techniques Weak replication
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)
.
Quantact workshop on risk management March 9 21 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 22 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 23 / 74
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 techniques Weak replication
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}.
Quantact workshop on risk management March 9 24 / 74
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 techniques Weak replication
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).
Quantact workshop on risk management March 9 25 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 26 / 74
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 techniques Weak replication
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).
Quantact workshop on risk management March 9 27 / 74
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 techniques Weak replication
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.
Quantact workshop on risk management March 9 28 / 74
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
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.
Quantact workshop on risk management March 9 29 / 74
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
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).
Quantact workshop on risk management March 9 30 / 74
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
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).
Quantact workshop on risk management March 9 31 / 74
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
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.
Quantact workshop on risk management March 9 32 / 74
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
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 .
Quantact workshop on risk management March 9 33 / 74
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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
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
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
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.
Quantact workshop on risk management March 9 35 / 74
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
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.
Quantact workshop on risk management March 9 36 / 74
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
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).
Quantact workshop on risk management March 9 37 / 74
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
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).
Quantact workshop on risk management March 9 38 / 74
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
Optimal Hedging B-vines models
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).
Quantact workshop on risk management March 9 39 / 74
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
Optimal Hedging B-vines models
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
It follows that Cj is the conditional distribution of U givenV1, . . . ,Vj .
Quantact workshop on risk management March 9 41 / 74
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
Optimal Hedging B-vines models
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.
Quantact workshop on risk management March 9 42 / 74
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
Optimal Hedging B-vines models
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
Quantact workshop on risk management March 9 43 / 74
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
Optimal Hedging B-vines models
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
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
Optimal Hedging B-vines models
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.
Quantact workshop on risk management March 9 45 / 74
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
Optimal Hedging B-vines models
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).
Quantact workshop on risk management March 9 46 / 74
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
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.
Quantact workshop on risk management March 9 47 / 74
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
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.
Quantact workshop on risk management March 9 48 / 74
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
Empirical results First experiment
Table: Replication results based on 10 000 trajectories for the firstscenario for g (R0,T ).
Table: Replication results based on 10 000 trajectories for thesecond scenario for the payoff G and log(VT/100) under optimalhedging and delta hedging.
Table: Replication results based on 10 000 trajectories for thesecond scenario for the payoff G and log(VT/100) under optimalhedging and delta hedging.
Quantact workshop on risk management March 9 52 / 74
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
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.
Quantact workshop on risk management March 9 53 / 74
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
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.
Quantact workshop on risk management March 9 54 / 74
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
Empirical results Fourth experiment
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.
Quantact workshop on risk management March 9 55 / 74
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
Empirical results Fourth experiment
−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.
Quantact workshop on risk management March 9 56 / 74
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
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)
Quantact workshop on risk management March 9 57 / 74
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
Empirical results Multidimensional example
Multidimensional example II
Table: Summary statistics for the 3 portfolios. Values are reportedon an annual basis.
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.
Quantact workshop on risk management March 9 58 / 74
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
Empirical results Multidimensional example
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)
Quantact workshop on risk management March 9 59 / 74
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
Empirical results Multidimensional example
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.
Quantact workshop on risk management March 9 60 / 74
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
Empirical results Multidimensional example
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.
τ (2) -0.061 -0.060 0Quantact workshop on risk management March 9 61 / 74
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
Empirical results Multidimensional example
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.
Quantact workshop on risk management March 9 62 / 74
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
Empirical results Multidimensional example
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.
Quantact workshop on risk management March 9 63 / 74
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 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
Quantact workshop on risk management March 9 66 / 74
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
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.
Quantact workshop on risk management March 9 67 / 74
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
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.
Quantact workshop on risk management March 9 67 / 74
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
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 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.
Quantact workshop on risk management March 9 67 / 74
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
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.
Quantact workshop on risk management March 9 67 / 74
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
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.
Quantact workshop on risk management March 9 68 / 74
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
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.
Quantact workshop on risk management March 9 69 / 74
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
Conclusion
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.
Quantact workshop on risk management March 9 70 / 74
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
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.
Quantact workshop on risk management March 9 71 / 74
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
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.
Quantact workshop on risk management March 9 72 / 74
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
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.
Quantact workshop on risk management March 9 73 / 74
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
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.
Quantact workshop on risk management March 9 74 / 74