Content uploaded by Bouchra R. Nasri

Author content

All content in this area was uploaded by Bouchra R. Nasri on Nov 25, 2018

Content may be subject to copyright.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 1

1

Replication methods for ﬁnancial indexes

Bruno R´emillard∗, Bouchra Nasri and Malek Ben-Abdellatif

Department of Decision Sciences, HEC Montr´eal,

Montr´eal (Qu´ebec), Canada H3T 2A7

∗E-mail:bruno.remillard@hec.ca

E-mail: bouchra.nasri@hec.ca

E-mail: malek.ben-abdellatif@hec.ca

In this paper, we ﬁrst present a review of statistical tools that can be used in

asset management either to track ﬁnancial indexes or to create synthetic ones.

More precisely, we look at two important replication methods: the strong repli-

cation, where a portfolio of very liquid assets is created and the goal is to track

an actual index with the portfolio, and weak replication, where a portfolio of

very liquid assets is created and used to either replicate the statistical proper-

ties of an existing index, or to replicate the statistical properties of a custom

asset. In addition, for weak replication, the target is not an index but a payoﬀ,

and the replication amounts to hedge the portfolio so it is as close as possible

to the payoﬀ at the end of each month. For strong replication, the main tools

are predictive tools, so ﬁltering techniques and regression play an important

role. For weak replication, which is the main topic of this paper, in order to

determine the target payoﬀ, the investor has to ﬁnd or choose the distribution

function of the target index or custom index, as well as its dependence with

other assets, and use a hedging technique. Therefore, the main tools for weak

replication are modeling (estimation and goodness-of-ﬁt) and optimal hedging.

For example, an investor could wish to obtain Gaussian returns that are inde-

pendent of some ETFs replicating the Nasdaq and S&P 500 indexes. In order

to determine the dependence of the target and a given number of indexes, we

introduce a new class of easily constructed models of conditional distributions

called B-vines. We also propose to use a ﬂexible model to ﬁt the distribution of

the assets composing the portfolio and then hedge the portfolio in an optimal

way. Examples are given to illustrate all the important steps required for the

implementation of this new asset management methodology.

Keywords: ETF; Hedge Funds; Replication; Smart beta; Copulas; B-vines;

HMM; Hedging.

1. Introduction

Historically, hedge funds have been an important class of alternative in-

vestment assets for diversifying portfolios. The early sales pitch was that

hedge funds oﬀer superior returns, due to use of leverage, derivatives, short

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 2

2

sales and other non-traditional investment strategies. The new sales pitch

is that there are diversiﬁcation beneﬁts due to low correlation with tradi-

tional assets classes. However, investors are still often rebutted in investing

in hedge funds, mainly because of high management and performance fees,

lack of liquidity and signiﬁcant lock-up periods, and lack of transparency.

Mainly based on the work on [1], [2] and [3], major investors like ﬁnan-

cial institutions looked for more eﬃcient and aﬀordable methods to generate

the same kind of returns. This was mainly done by strong replication, i.e.,

by constructing portfolios of very liquid assets tracking a hedge fund index.

Nowadays, smart beta methods, a new brand name for replication tech-

niques, oﬀer even more ﬂexibility to small investors as well, through ETFs.

For example, Horizons HFF (hhf.to) is an ETF targeting the Morningstar

Broad Hedge Fund Index SM, while State Street SPDR ETF (spy) tracks

S&P 500 index.

In addition to strong replication, weak replication, based of the payoﬀ

distribution model of [4], was proposed by [5] and extended by [6]. This

innovative approach consists in constructing a dynamic strategy to track a

payoﬀ, in order to reproduce the statistical properties of hedge fund returns

together with their dependence with a selected investor portfolio. It can also

be used to construct synthetic indexes with tailor-made properties, which

is an advantage over strong replication since the latter can only replicate

an existing index.

In Section 2, we review the main statistical techniques to replicate in-

dexes, including a new “Smart Beta” approach that can be used to diversify

investors portfolios. In order to implement the proposed methodology, a

new family of conditional distributions called B-vines are introduced in Sec-

tion 3. The essential steps of modeling and hedging are discussed in Section

4. Examples of applications are then given in Section 5.

2. Replication methods

There are basically two replication approaches: strong replication, where

the target is the index (naive or imitative method, and factor-based

method), and weak replication, where the target is a payoﬀ determined

by the distribution of an existing index or a custom index, also called syn-

thetic index. In both cases, the idea is to construct a portfolio of liquid

assets with end of the month values as close as possible to the target.

Strong replication is divided in two sub-groups. On one hand, there is

the “naive replication”, where the investor try to imitate the hedge fund

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 3

3

manager investment strategy or the index composition. This is kind of

easy for indexes when their composition is known, but it is far from ob-

vious when the strategy or composition is unknown. For example, for a

Merger Arbitrage Fund index, the idea is to long (potential) sellers and

short (potential) buyers.

On the other hand, the factorial approach attempts to reproduce hedge

fund returns or indexes by investing in a portfolio of assets that provide

similar end of month returns. The implementation of the factorial approach

is described in Section 2.1, while the multi-asset extension of the weak

replication is discussed in Section 2.2.

Alternative beta funds based on the factorial approach have been

launched by several institutions including Goldman Sachs, JP Morgan,

Deutsche Bank, and Innocap, to name a few. According to [7], the short

version Verso of Innocap, based on ﬁltering methods, performed best in the

turbulent period 2008–2009. Note also that [8] showed that factor-based

replicators produce independent returns over time, which might be inter-

esting from an investor’s perspective. Furthermore, an investor can easily

track the performance of a given replicator. However, in a recent study, [9]

found very high correlations between factor-based replicators and indexes

like S&P 500. This undesirable dependence show that these replicators

cannot really be used for diversiﬁcation purposes, contrary to synthetic in-

dexes that can be built with weak replication techniques. An illustration

of this powerful technique is given in Section 5.4.

Before presenting the mathematical framework deﬁning strong and weak

replication, we summarize in Table 1 the main diﬀerences between the two

approaches.

Table 1. Main diﬀerences between strong replication and weak replication.

Method Target Tracking Synthetic index Controlled dependence

Strong Index Yes No No

Weak Payoﬀ Possible Possible Possible

Note: Tracking is possible for weak replication if the value of the payoﬀ is

posted at the end of the month. In this case, the analog of the tracking error

is the RMSE (root mean square error) of the hedging error. This important

value appears in our examples of implementation in Section 5. For synthetic

indexes, it is possible to control the dependence.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 4

4

2.1. Factorial approach for strong replication

To implement the factorial approach, one needs the returns1R?

tof the

target fund S?and one needs to select appropriate liquid assets (factors)

S=S(1), . . . , S (p)composing the replication portfolio. The returns of

Sare denoted by Rt=R(1)

t, . . . , R(p)

t, and the associated weights are

denoted by βt= (βt,1, . . . , βt,p). The model is written in the linear form

R?

t=β>

tRt+εt,(1)

where the εt’s are non-observable tracking error terms.

The unknown weights βtare then evaluated from a predictive method

using relation (1), e.g., by using a rolling-window regression over the last

24 months, or by using ﬁltering methods. Note that for ﬁltering, one must

also deﬁne the (Markovian) dynamics of the weights βt; see, e.g., [10].

To measure the performance of a replicating method, one uses the track-

ing error (TE), deﬁned in the in-sample case by

TEin =(1

n

n

X

t=1 R?

t−ˆ

β>

tRt2)1/2

,

while for the out-of-sample, it is deﬁned by

TEout =(1

n

n

X

t=1 R?

t−ˆ

β>

t−1Rt2)1/2

,

where ˆ

βtis the vector of predicted weights using returns (R?

t,Rt), (R?

t−1,

Rt−1), . . .. The out-of-sample tracking error is a more realistic measure of

performance, since the error R?

t−ˆ

β>

t−1Rtis the one monitored by investors.

As seen in the example below, ﬁltering usually yields better results than

regression in terms of tracking error.

Example 2.1. This example is taken from [11], Chapter 10. The target

is HFRI Fund Weighted Composite Index, and the factors are S&P500

Index TR, Russel 2000 Index TR, Russell 1000 Index TR, Eurostoxx Index,

Topix, US 10-year Index, 1-month LIBOR.2Here, two methods were used

to compute the dynamic weights β: a regression with a 24-month window,

and a Kalman ﬁlter, where the dynamics of the β’s is a random walk,

meaning that βt=βt−1+ηt, where the innovations ηtare assumed to be

independent and identically distributed.

1Typically monthly returns, especially in the case of hedge fund indexes.

2Data, from April 1997 to October 2008, were provided by Innocap.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 5

5

Table 2. In-sample and out-of-sample statistics.

Portfolio TE Corr Mean Std Skew Excess kurt

In-sample statistics

Target - 1.00 8.12 7.72 -0.59 2.45

Regression 10.58 0.93 8.79 8.32 -0.69 2.22

Kalman 8.54 0.95 9.68 7.75 -0.59 2.53

Out-of-sample statistics

Target - 1.00 8.12 7.72 -0.59 2.45

Regression 19.27 0.83 9.30 9.86 -0.11 3.34

Kalman 14.71 0.86 9.97 8.20 -0.40 2.63

Note: Values are expressed in annual percentage. The excess kurtosis

of the Gaussian distribution is 0.

This is a very basic and unrealistic model, but it can be improved, e.g., by

adding dependence in the increments or adding constraints on the portfolio

compositions. In this case, the Kalman ﬁlter assumptions are no longer met,

and one should use for example a particle ﬁlter; see, e.g., [11], Chapter 9.

However, even with a simple model and the Kalman ﬁlter, the results are

surprisingly good, better than the rolling-window regression. In-sample and

out-of-sample statistics for our example are displayed in Table 2.

In general, the βtare much less variable in the Kalman ﬁlter case,

leading to less expensive transactions, in addition to being a better tracking

method. See, e.g., [11], Chapter 10.

Before ending this section, it is worth noting that one could also use

machine learning methods for tracking purposes. It would be interesting

to compare the performance of machine learning vs ﬁltering. This will be

done in a forthcoming work.

2.2. Weak replication

Weak replication is an alternative replication method proposed by [5] and

later extended by [6] based on the payoﬀ distribution model of [4]. The aim

was to replicate hedge fund returns or hedge fund indexes not by identifying

the return generating betas as in the factor-based approach, but by building

a trading strategy that can be used to generate the (statistical) distribution

of the hedge fund returns or indexes. The implementation proposed in [6] is

subject to several shortcomings and inconsistencies. Improvements of the

Kat-Palaro method were proposed in [12] for a start.

In view of applications to asset management, and mainly for diversiﬁca-

tion purposes, it is desirable to generalize the Kat-Palaro approach (limited

to only one reference asset). To this end, it was suggested in [13] to con-

sider a multivariate asset Sof p=d+ 1 components, where S(1), . . . , S (d)

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 6

6

represent the value of reference portfolios of the investor, and the so-called

reserve asset S(d+1) 3. As before, S?is the index one seeks to replicate.

The aim is not to reproduce the monthly values of S?, which might not

even exists, but rather reproduce its statistical properties.

The steps required to implement the proposed weak replication method

are given next.

2.2.1. Implementation steps

(1) Determine the joint distribution of the (daily) returns Rkof Sk.

We suggest to use a Gaussian Hidden Markov Model (HMM). This

model is described in Section 4.1. However any dynamic model is per-

mitted, as long as it ﬁts the data.

(2) Find a compatible distribution for the monthly returns R0,T . In

particular, ﬁnd the marginal distributions F1, . . . , Fdof R(r ef)

0,T =

R(1)

0,T , . . . , R(d)

0,T , ﬁnd the copula of R(ref )

0,T , and ﬁnd the conditional

distribution F(·,x) of R(res)

0,T given R(ref)

0,T =R(1)

0,T , . . . , R(d)

0,T =x.

This can be done by simulation from daily returns, as suggested in

Section 4.2. Again, we suggest to use a Gaussian HMM. We strongly

advise against using real monthly returns to complete this step since in

general the sample size for estimation purposes is not long enough, and

in addition, there is a lack of compatibility between the distribution of

the daily and monthly returns, thus creating a bias.

(3) Find or choose the distribution function F?of the return R?

0,T of the

target index S?.

If the asset S?does not exists, i.e., we are creating a synthetic index,

then the investor must choose F?. Interesting choices of distributions

are the Gaussian, truncated Gaussian, and the Johnson SU distribu-

tion. Even if the index S?exists, one can try to ﬁt these three distri-

butions.

3E.g., equal weighted portfolio of highly liquid futures contracts

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 7

7

(4) Find or choose the conditional distribution function H(·,x) of R?

0,T

given R(ref)

0,T =x, which can be expressed as

H(y, x) = C{F?(y),F(x)},

where F(x) = (F1(x1), . . . , Fd(xd)), and C(·,v) is the conditional dis-

tribution of U=F?(R?

0,T ) given V=FR(ref)

0,T =v.

If the index S?does not exists, then the investor must choose C. In any

case, we recommend to choose or try to ﬁt a B-vine model, as deﬁned

in Section 3. The importance of the choice of Cis discussed in Section

2.2.3.

(5) Compute the return function ggiven by

g(x, y) = Q{F(y, x),x},(2)

where Q(·,x) is the conditional quantile function, deﬁned as the inverse

of H(·,x). For more details on copula-based conditional quantiles, see

the recent articles [14], [16]. The function gcan also be expressed as

g(x, y) = F−1

?C−1{F(y, x),F(x))}.(3)

The reason for deﬁning gthis way is that the joint distribu-

tion of R?

0,T ,R(ref )

0,T is the same as the joint distribution of

g(R0,T ),R(ref )

0,T . This means that g(R0,T ) has distribution function

F?, and that the conditional distribution of g(R0,T ) given R(ref)

0,T =x

is H(·,x). In particular the statistical properties of R?

0,T are the same

as the statistical properties of g(R0,T ).

(6) Compute the payoﬀ function Gdeﬁned by

G(ST) = 100 exp {g(R0,T )}.

The interpretation of the payoﬀ function is the following: if one starts

by investing 100$, and one can replicate exactly this payoﬀ with a hedg-

ing portfolio, then one would get the return g(R0,T ). In particular the

distribution function of the portfolio return is F?, and it is obtained

without investing in S?.

(7) Construct a dynamic portfolio {Vk(V0,ϕ)}n

k=0 of the assets S, traded

daily, in order to generate the payoﬀ G(ST) at the end of the month.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 8

8

More precisely, letting βk=e−rkT /n be the discounting factors, the

discounted value of the portfolio at the end of the month is

βnVn=V0+

n

X

k=1

ϕ>

k(βkSk−βk−1Sk−1),

where ϕ(j)

kis number of shares of asset S(j)invested during

((k−1)T /n, kT/n], and ϕkmay depend only on S0,...,Sk−1. Ini-

tially, the portfolio initial value is V0.

This hedging problem is typical in ﬁnancial engineering, where V0can

be interpreted as the value of an option on Shaving payoﬀ Gat ma-

turity T, and one wants to replicate the payoﬀ. Usually, we are more

interested in the price of the option, while here the emphasis is on the

hedging portfolio, which is the object of the investment.

For hedging, we suggest to use the discrete time hedging method deﬁned

in Section 4.3. This strategy, adapted for a continuous time model, is

optimal with respect to minimizing the square hedging error.

2.2.2. K-P measure

If the goal is attained, i.e., Vn=G(ST), the return of the portfolio is

log(Vn/V0) = log(100/V0) + g(R0,T ),

which has the same distribution as α+S?, where α= log(V0/100) can

be used to estimate manager’s alpha or the feasibility of the replication.

In the context of replicating hedge funds, it is suggested in [6] that the

initial amount V0to be invested in the portfolio be viewed as a measure of

performance of the hedge fund manager. Here we prefer to use αwhich we

call the K-P measure. It can be interpreted as follows:

•If α= 0, i.e., V0= 100, the strategy generates the same returns as S?

(in distribution);

•If α < 0, i.e., V0<100, it is worth replicating, generating superior

returns (in distribution), while if α > 0, i.e., V0>100, it may be not

worth replicating.

Note that centered moments like standard deviation, skewness, kurtosis,

are not aﬀected by the value of the K-P measure α. However, the expected

value of the portfolio is α+E(S?).

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 9

9

Example 2.2. A simple example in risk management is an investor in-

terested in creating a portfolio S?with a speciﬁc distribution function F?,

which would be independent of several reference indexes, so that the return

of the hedging portfolio will not be aﬀected by extreme behavior of the

reference indexes. In this case, gis given by

g(x, y) = F−1

?{F(y, x)}.(4)

An example of implementation of this model is given in Section 5.4.

Remark 2.1. It makes sense that α > 0, especially if the target distribu-

tion of S?is not realistic. For example, one could wish to generate Gaussian

returns with annual mean of 30% and a volatility of 1% that is independent

of S(1), but the real distribution would be Gaussian with mean .3−12α.

In fact, if the joint distribution of the monthly returns is Gaussian, with

annual means µ1, µ2, µ3, annual volatilities σ1, σ2, σ3and correlations ρ12,

ρ13, then, according to Equation (2),

g(x, y) = 1

12 (µ3−r+σ3x−µ1

σ1 ρ13 −ρ12s1−ρ2

13

1−ρ2

12 !

+σ3y−µ2

σ2s1−ρ2

13

1−ρ2

12 ),

so using the Black-Scholes setting with associated risk neutral measure Q,

V0= 100e−r/12EQegR(1)

0,1/12,R(2)

0,1/12= 100eα,

with

α=µ3

12 −r

12 −1

12 (µ1

σ3

σ1

+µ2

σ3

σ2s1−ρ2

13

1−ρ2

12 −σ2

3

2)

+1

12 (σ3

σ1r−σ2

1

2 ρ13 −ρ12s1−ρ2

13

1−ρ2

12 !

+σ3

σ2r−σ2

2

2s1−ρ2

13

1−ρ2

12 ).

As a result, the genuine mean of the target is independent of µ3! For

example, if r= 1%, µ1= 8%, µ2= 6%, σ1= 10%, σ2= 8%, σ3= 1%,

ρ12 = 0.25 and ρ13 = 0, then α=µ3

12 −.02499

12 , and we would get a Gaussian

distribution with an annual mean of 2.499% and an annual volatility of

1% that is independent of S(1). It is interesting to look at the real annual

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 10

10

mean of the portfolio (assuming perfect hedging) as a function of ρ13 . This

is illustrated in Figure 1. Note that the maximum value 2.501% is attained

for ρ13 =−0.072.

-1 -0.5 0 0.5 1

ρ13

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

real annual mean (%)

Fig. 1. Real annual mean in percent of the Gaussian distribution of the monthly return

R?as a function of the correlation ρ13 with monthly return R(1).

2.2.3. Choice of C

First, note that Cis a function of the copula Cof (U, V) viz.

C(u, v) = ∂v1···∂vdC(u, v1, . . . , vd)

cV(1, v1, . . . , vd),(u, v)∈(0,1)1+d,(5)

where cVis the density of the copula CV(·) = C(1,·). When d= 1, we

can take C(u, v) = ∂vC(u, v) for any copula C. However, if d≥2, then the

copula of Vmatters. One cannot just take any d+ 1-dimensional copula C.

To solve this intricate problem, we propose to use a construction similar to

the one used for vine copulas. This new construction is described in Section

3, after we discuss why the choice of Cmatters.

To this end, let ˜

Cbe an arbitrary conditional distribution function of

Ugiven a d-dimensional random vector ˜

Vassociated with the copula ˜

Cof

(U, ˜

V), and deﬁne

˜g(x, y) = F−1

?h˜

C−1{F(y, x),F(x))}i.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 11

11

Setting Z=FR(res)

0,T ,R(ref )

0,T , one gets

Ph˜g(R0,T )≤y, R(ref )

0,T ≤xi=PhZ ≤ ˜

CnF?(y),FR(ref )

0,T o,R(r ef)

0,T ≤xi

=Eh˜

C {F?(y),V}I(V≤F(x))i

=Z(0,F(x)]

˜

C {F?(y),v}cV(v)dv,

since Zis uniformly distributed and is independent of R(ref )

0,T , according to

[15]. So, in general, ˜

F?(y) = Eh˜

C {F?(y),V}iis not the target distribution

function F?. However, ˜

F?=F?if ˜

C(1,v) = CV(v). This shows that one

then must be careful with the choice of Cin order to have compatibility.

3. B-vines models

The aim of this section is to ﬁnd a ﬂexible way to construct a conditional

distribution of a random variable Ygiven a d-dimensional random vector

X. Using the representation of conditional distributions in terms of cop-

ulas, this problem amounts to constructing the conditional distribution C

of a uniform random variable Ugiven a random vector V(with uniform

margins) that is coherent with the distribution function CVof V. Unfortu-

nately, the usual vines models for multivariate copulas cannot be used here,

because of this compatibility constraint. For more details on unconstrained

vine models applied to conditional distributions, see, e.g., [16].

As noted before, when d= 1, the compatibility condition is not a con-

straint at all since CV(v) = v,v∈[0,1], and the solution is simply to take

C(u, v) = ∂vC(u, v), for a copula Cthat is smooth enough.

Next, in the case d= 2, if D1and D2are bivariate copulas, with

conditional distributions Dj(u, t) = ∂tDj(u, t), j∈ {1,2}, and CVis the

copula of V= (V1, V2), then

C(u, v) = D2{D1(u, v1), ∂v1CV(v1, v2)},v= (v1, v2)∈(0,1)2,(6)

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, as

deﬁned in [17], [18].

Guided by formula (6), let Dj,j∈ {1, . . . , d}be bivariate copulas and

let Dj(u, t) = ∂tDj(u, t) be the associated conditional distributions. For

j∈ {1, . . . , d}, further let Rj−1(v1, . . . , vj) be the conditional distribution

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 12

12

of Vjgiven V1=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)}.(7)

Note that E{Cj(u, v1, . . . , vj−1, Vj)|V1=v1, . . . , Vj−1=vj−1}is given by

Z1

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

=Z1

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 Cjis the conditional distribution of Ugiven V1, . . . , Vj. The

conditional quantile of Ugiven V1, . . . , Vjis also easy to compute, satisfying

a recurrence relation similar to (7). In fact, if the conditional quantile of

Cjis denoted by Γj, then for any j∈ {1, . . . , d}, and for any u, v1,...vd∈

(0,1),

Γj(u, v1, . . . , vj)=Γj−1D−1

j{u, Rj−1(v1, . . . , vj)}, v1, . . . , vj−1.(8)

In general, this construction does not lead to a proper vine copula since

all copulas involved are not bivariate copulas, the copula of Vbeing given.

In fact, it is more general than the pair-copula construction method used

in vines models. Nevertheless, this type of model will be called B-vines

and its construction is illustrated below, where the underlined variables

(in red) mean that their distributions R0,...,Rd−1are known4, and the

conditional copulas D1,...,Ddhave to be chosen, in order to determine

4R0,...,Rd−1are called the Rosenblatt’s transforms and are particularly important in

simulating copulas or for testing goodness-of-ﬁt. See, e.g., [11].

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 13

13

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

Note that B-vines can be particularly useful in conditional mean regres-

sion (OLS, GAM, GLM, etc,) and conditional quantile settings, where the

distribution of the covariates is often given; see, e.g., [14]. It can also be

used in our replication context when the target S?exists; in this case, we

could look at B-vines constructed from popular bivariate families like Clay-

ton, Gumbel, Frank, Gaussian and Student, and ﬁnd the ones that ﬁt best

the data, in the same spirit as the choice of vines for copula models in

the R packages CDVine or VineCopula. In a future work we will propose

goodness-of-ﬁt tests for these models.

4. Modeling and hedging

In what follows, building on [12], we propose a model to ﬁt the data and

deal with numerical problems arising from using a larger number of assets

for hedging.

To implement successfully the proposed replication approach, one needs

to model the distribution of the returns Rtand R0,T for steps (1) and (2) de-

scribed in Section 2.2.1. Once this is done, we will have as a by-product the

conditional distribution Fand the Rosenblatt’s transforms R0,...,Rd−1

used for computing the conditional distribution C, as in Section 3. For

replicating an existing asset S?, one further needs the joint distribution of

R?

0,T ,R(ref )

0,T . To do this, we propose to use Gaussian Hidden Markov

Models (HMM) as deﬁned in [19]. This model is described next in Section

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 14

14

4.1. Next, one needs to ﬁnd a distribution of the monthly returns compat-

ible with the distribution of the daily returns. A solution to this problem

is proposed in Section 4.2. Finally, a replication method is suggested in

Section 4.3.

4.1. Gaussian HMM

Regime-switching models are quite intuitive. First, the regimes {1, . . . , l}

are not observable and are modeled by a ﬁnite Markov chain with transition

matrix Q. At period t, given that the previous regime τt−1has value i,

the regime τt=jis chosen with probability Qij, and given τt=j, the

log-returns Rthave a Gaussian distribution with mean µjand covariance

matrix Bj.

The law of most ﬁnancial time series can be modeled adequately by

a Gaussian HMM, provided the number of regimes is large enough. In-

deed, the serial dependence in regimes propagates to returns and captures

the observed autocorrelation in ﬁnancial time series. Also, the conditional

distribution is time-varying, leading to conditional volatility, as well as con-

ditional asymmetry and kurtosis. Finally, the Black-Scholes framework is

a particular case of this model when the number of regimes is 1. Parame-

ters are quite easy to estimate and there is also an easy way to choose the

number of regimes, depending on the results of goodness-of-ﬁt tests; see,

e.g., [19] for more details.

In the next section, we introduce the continuous time limit of a Gaussian

HMM, the main reason being that for this limiting process, one can show

that there exists an equivalent martingale measure that is optimal in the

sense of [20] and that can be used for pricing and hedging; see, e.g., [21].

4.1.1. Continuous time limiting process

Under weak conditions, the continuous time limit of a Gaussian HMM

is a regime-switching geometric Brownian motion (RSGBM). Using the

same notations as in [21], let Tbe a continuous time Markov chain on

{1, . . . , l}, with inﬁnitesimal generator Λ. In particular, P(Tt=j|T0=

i) = Pij (t), where the transition matrix Pcan be written as P(t) = etΛ,

t≥0. Then, the (continuous) price process Xmodeled as a RSGBM

satisﬁes the stochastic diﬀerential equation

dXt=D(Xt)υ(Tt)dt +D(Xt)σ(Tt−)dWt,(9)

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 15

15

where D(s) is the diagonal matrix with diagonal elements (sj)d

j=1 and W

is a d-dimensional Brownian motion, independent of T. Note that the time

scale is in years, and we assume that a(j) = σ(j)σ(j)>is invertible for any

j∈ {1, . . . , l}.

4.1.2. Relationship between discrete time and continuous time

parameters

The relationship between the continuous-time parameters (υ,a,Λ) of the

limiting RSGBM and the parameters of the Gaussian HMM is the following:

if the parameters µh,Bh,Qhof the discrete time model are obtained from

data sampled 1/h times a year, then υ(j)≈µh(j) + 1

2diag{Bh(j)}/h,

where diag(B) is the vector of the diagonal elements of a matrix B,a(j)≈

Bh(j)/h, and Λ ≈(Qh−I)/h. For example, for daily data, one usually

takes h= 1/252.

Note that if we deﬁne Xh,t =Sbt/hcand Th,t =τbt/hc, where bacstands

for the integer part of a∈R, then the processes (Xh,t,Th,t ) converge in law

to (X,T). Note also that the optimal hedging strategy converges as well;

see, e.g., [22].

4.2. Monthly returns compatibility

Compatibility means that the distribution of the monthly returns R0,T is

the same as the distribution of the sum of typically n= 21 consecutive daily

returns. Since the hedging will be done under a continuous time RSGBM,

there is no compatibility problem. However, since we need the distribution

of log(XT) to construct the payoﬀ, and the latter is not known explicitly,

we propose to simulate a large number of monthly returns log(XT), say

10000, which is impossible to get in practice, and then ﬁt a Gaussian HMM

to these simulated data. The joint distribution of the monthly returns is

then approximated by a mixture of (multivariate) Gaussian distributions,

and the conditional distribution function Fis also a (univariate) Gaussian

mixture. See, e.g., [19] for more details.

4.3. Discrete time hedging

Since we ﬁtted a Gaussian HMM to the daily returns, an obvious solution of

the hedging problem would be to use the results of [23] for optimal hedging

in discrete time; see also [11]. However, implementing this methodology

requires interpolating functions on a (d+ 1)-dimensional grid. Since we

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 16

16

are aiming for applications with d≥2, this approach leads to too much

imprecision. For example, a (too) small grid of 100 points for each asset

would require computing and storing 102(d+1) points, while a relatively

precise grid of 1000 points for each asset requires 103(d+1) points. Even

with d= 2, this means storing 109points, which is way too much.

This is why we consider a continuous-time approximation, which does

not require any interpolation or grid construction and works in any dimen-

sion. It is easy to show, see, e.g., [22] that many interesting discrete time

models can be approximated by continuous time models. In particular, this

is true for the Gaussian HMM whose continuous time limit is the RSGBM.

Option pricing and optimal quadratic hedging have been studied recently

for this process in [21], and it turns out that the optimal hedging strategy

and option price can be deduced from an equivalent martingale measure.

Under this equivalent martingale measure, assets still follow a RSGBM,

with the additional feature that the distribution of the regimes is now an

inhomogeneous continuous time Markov time. Nevertheless, this model is

quite easy to simulate and does not require any calibration to option prices.

4.3.1. Continuous time approximation

Because we have possibly more than 2 risky assets, and based on the results

in [21], [22], we approximate ϕkby φk−1

nT, where φis the optimal hedging

strategy of the RSGBM obtained from [21], Lemma 4.1.

To get nearly optimal hedging strategies in discrete time, we ﬁrst use

Monte Carlo methods by simulating the process Xunder the optimal mar-

tingale measure, as given by Equation (A.2), to obtain the values CkT/n(s, i)

and ∇sCkT /n(s, i) given by formulas (A.3) and (A.4). Then we simply dis-

cretize the continuous time optimal hedging values (A.5)–(A.6) to get, for

k∈ {1, . . . , n},

ϕk=∇sC(k−1)T/n(Sk−1,ˆτk−1) + Gk−1D−1(Sk−1)ρ(ˆτk−1)/βk−1,(10)

˜

Vk=˜

Vk−1+ϕ>

k(βkSk−βk−1Sk−1),(11)

Gk=βkCkT /n(Sk,ˆτk)−˜

Vk,(12)

where ˜

V0=V0=C0(S0,ˆτ0), G0= 0, and ˜

Vk=βkVkare the discounted

portfolio values. In particular, ϕ1=∇sC0(S0,ˆτ0).

Remark 4.1. One could replace CkT /n(Sk,ˆτk) by the weighted average

l

X

j=1

CkT /n(Sk, j )ηk(j), where ηk(j) is the predicted probability of τk=j,

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 17

17

given the past observations.

We now have the necessary tools to tackle the implementation problem.

Two examples of application are presented next.

5. Examples of application

In this section, we provide some empirical evidence regarding the ability

of the model to replicate a synthetic index. In the implementation of the

replication model, we consider a 3-dimensional problem.

5.1. Assets

The ﬁrst step is to select two reference portfolios P(1) and P(2) and the

reserve asset P(3). These 3 portfolios are dynamically traded on a daily

basis, so we choose very liquid instruments with low transaction costs.

We therefore restrict the components of the portfolios to be Futures con-

tracts. The cash rate is the BBA Libor 1-month rate. Log-returns on

futures are calculated from the reinvestment of a rolling strategy in the

front contract. The front contract is the nearest to maturity, on the

March/June/September/December schedule and is rolled on the ﬁrst busi-

ness day of the maturity month at previous close prices. Each future con-

tract is fully collateralized, so that, the total return is the sum of the rolling

strategy returns and the cash rate.

The ﬁrst investor portfolio is related to equities while the second is

related to bonds. The reserve asset is a diversiﬁed portfolio. The compo-

sition of these portfolios is detailed in Table 3. As in [13], we use daily

returns from 01/10/1999 to 30/04/2009 (115 months). Table 4 presents

some descriptive statistics of the daily returns R(1), R(2), R(3).

Table 3. Portfolios’ composition.

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

40% S&P500 EMINI future

P(2) 100% CAN 10YR BOND future

P(3) 10% E-mini NASDAQ-100 futures

20% Russell 2000 TR

20% MSCI Emerging Markets TR

10% GOLD 100 OZ future

10% WTI CRUDE future

30% US 2YR NOTE (CBT)

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 18

18

Table 4. Summary statistics for the 3 portfolios.

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

Daily returns

Mean 0.0198 0.0209 0.0363

Volatility 0.1327 0.0592 0.1238

Skewness -0.6447 -0.3261 -0.4418

Excess kurtosis 8.5478 2.0583 5.1415

Note: Values are reported on an annual basis.

5.2. Modeling

As discussed in Section 4.1, we use a Gaussian HMM to model the joint

distribution of the returns of the 3 portfolios. The choice of the number of

regimes is done as suggested in [19]: we choose the lowest number of regimes

mso that the goodness-of-ﬁt test for mregimes has a P-value larger than

5%. This leads to a selection of 6 regimes for the daily returns. The large

number of regimes for the daily returns is due to the fact that the sample

period contains the last ﬁnancial crisis. Usually, for non-turbulent periods,

4 regimes are suﬃcient for ﬁtting daily returns. The estimated parameters

are given in Table 5. The associated transition matrix for daily returns of

the Gaussian HMM is

Qdaily =

0.9608 0.0000 0.0181 0.0000 0.0000 0.0211

0.0160 0.1494 0.3384 0.0000 0.4962 0.0000

0.0000 0.0579 0.6746 0.0108 0.2567 0.0000

0.0000 0.0000 0.0000 0.9823 0.0177 0.0000

0.0176 0.0993 0.2753 0.0175 0.5882 0.0021

0.0599 0.0000 0.0000 0.0000 0.0071 0.9330

,

and the inﬁnitesimal generator associated with the limiting RSGBM is

Λdaily =

−9.8765 0.0000 4.5658 0.0000 0.0000 5.3107

4.0402 −214.3435 85.2680 0.0000 125.0353 0.0000

0.0000 14.5863 −81.9990 2.7205 64.6922 0.0000

0.0000 0.0052 0.0004 −4.4624 4.4569 0.0000

4.4414 25.0201 69.3636 4.4157 −103.7633 0.5226

15.0866 0.0000 0.0000 0.0000 1.7858 −16.8724

.

Finally, for the last observation, corresponding to the beginning of the

hedging, the estimated probability of occurrence of each regime is

ηdaily = (0.9433,0.0003,0.0246,0.0000,0.0006,0.0312).

Therefore, we will take for granted that at time t= 0, we are in regime 1.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 19

19

Table 5. Estimated parameters for the Gaussian

HMM ﬁtted on daily returns.

Daily returns

Regime µjBj

-0.0182 0.0250 -0.0026 0.0157

1 0.0409 -0.0026 0.0028 -0.0021

-0.1706 0.0157 -0.0021 0.0200

0.1709 0.0131 -0.0000 0.0114

2 -1.6439 -0.0000 0.0040 0.0009

0.1790 0.0114 0.0009 0.0170

0.6694 0.0050 0.0002 0.0018

3 0.0619 0.0002 0.0036 -0.0006

0.9667 0.0018 -0.0006 0.0040

0.1486 0.0042 -0.0002 0.0028

4 0.0286 -0.0002 0.0018 0.0000

0.2178 0.0028 0.0000 0.0047

-0.6934 0.0084 -0.0013 0.0049

5 0.2548 -0.0013 0.0023 -0.0009

-0.9222 0.0049 -0.0009 0.0067

-0.4565 0.1169 -0.0115 0.0788

6 0.0749 -0.0115 0.0099 -0.0110

-0.4082 0.0788 -0.0110 0.0889

Note: Values are expressed on an annual basis.

5.2.1. Monthly returns

As suggested in Section 4.2, we simulated 10 000 values of monthly returns

under the estimated RSGBM. We ﬁtted a Gaussian HMM and found that 3

regimes were necessary, which is larger than usual, but we have to remember

that we are ﬁtting 10 000 values. The estimated parameters are given in

Table 6, and the associated transition matrix is

Qmonthly =

0.1209 0.6788 0.2003

0.1719 0.6184 0.2097

0.1926 0.5846 0.2229

.

Finally, for the last observation, corresponding to the beginning of the hedg-

ing, the estimated probability of occurrence of each regime is ηmonthly =

(0.1796,0.7635,0.0569). In particular, it means that the probability πnext

of being in each regime next month is

πnext =ηmonthlyQmonthly = (0.1639,0.6273,0.2088).(13)

It then follows that the conditional distribution F(·,x) is mixture of 3

Gaussian distributions, with mean αj+β>

jxand standard deviation σj,

j∈ {1,2,3}, and weights given by (13), where the values of the parameters

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 20

20

Table 6. Estimated parameters for the Gaussian

HMM ﬁtted on 10 000 simulated monthly returns un-

der RSGBM.

Regime µjBj

0.0728 0.0085 -0.0006 0.0067

1 0.0320 -0.0006 0.0027 -0.0004

0.1081 0.0067 -0.0004 0.0096

-0.4201 0.0726 -0.0117 0.0396

2 0.0050 -0.0117 0.0067 -0.0067

-0.2813 0.0396 -0.0067 0.0421

-0.4201 0.0726 -0.0117 0.0396

3 0.0050 -0.0117 0.0067 -0.0067

-0.2813 0.0396 -0.0067 0.0421

Note: The values are expressed on an annual basis.

are given in Table 7. More precisely,

F(y, x) =

3

X

j=1

πnext(k)Φ y−αj−β>

jx

σj!,(y, x)∈R3,(14)

where Φ is the distribution function of the standard Gaussian.

Table 7. Parameters of the conditional distribution of R(3)

0,T

given R(1)

0,T , R(2)

0,T .

Regime αjβjσjπj

1 0.0037 ( 0.6343 , -0.3353 ) 0.0463 0.1639

2 -0.0014 ( 0.6090 , -0.1828 ) 0.0296 0.6273

3 -0.0063 ( 0.6876 , 0.1661 ) 0.0231 0.2088

5.3. Target distribution function

For this example, the target distribution F?is a truncated Gaussian distri-

bution at −a, with (annual) parameters µ?and σ?, meaning that

F?(y) =

0, y ≤ −a;

Φy−µ?/12

σ?/√12 −Φ−a−µ?/12

σ?/√12

Φa+µ?/12

σ?/√12 , y ≥ −a.

(15)

Setting z=a+µ?/12

σ?/√12 and κ= Φ0(z).Φ (z), the mean of this distribution is

µ

12 +σ

√12 h, while the standard deviation is σ?

√12 √1−h2−2hz. With a=

0.02, µ?= 0.08 and σ?= 0.05, one gets an annual mean of 0.0842, and an

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 21

21

annual volatility of 0.0477. Note that F?(0) = 1 −Φµ?

√12σ?/Φ(z)=0.3.

The density is displayed in Figure 2.

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

0

5

10

15

20

25

30

Fig. 2. Target density for the monthly returns.

In the remaining of the section, we try to replicate the monthly returns

of a synthetic hedge fund having distribution F?given by (15). We will

rebalance the portfolio once a day, so n= 21. For simplicity, we take

S0= (1,1,1) and r= 0.01. We will consider two models: the ﬁrst one is

the independence model, meaning that C(u, v1, v2) = u, so that the return

function gis given by (4). This model is studied in Section 5.4. We

consider another model, called the Clayton model, deﬁne using the B-vines

representation by D1(u, t) = max 0, u−θ+t−θ−1−1/θ, which is the so-

called Clayton copula of parameter θ∈(−1,1), with Kendall’s τ=θ

θ+2 , and

D2(u, t) = ut, the independence copula. For this case, we take θ=−2/3,

leading to a Kendall’s tau of −0.5. This means that we require a negative

dependence with asset P(1).

Finally, for each model, we simulated 1 000 replication portfolios.

5.4. Synthetic index independent of the reference portfolios

The results of this ﬁrst experiment are quite interesting, as can be seen from

the statistics displayed in Table 8, especially the tracking error given by the

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 22

22

RMSE. Note also that the mean of the hedging error is signiﬁcantly smaller

that 0, meaning that the portfolio is doing better on average than the target

payoﬀ, even if the K-P measure α= 0.0078 is positive. The target distri-

bution is also quite well replicated. The distribution of the hedging errors

is also quite good, as can be seen from the estimated density displayed in

Figure 3. Finally, letting τ(1) and τ(2) represent the estimated Kendall’s

tau between the variable and the returns of portfolio P(1) and P(2) respec-

tively, one can see that the returns of the hedged portfolio are independent

of the returns of the reference portfolios, as measured by Kendall’s tau,

meaning that the synthetic asset has the desired properties.

Table 8. Descriptive statistics for the independence model.

Statistics HE G(S21)V21 g(R21) log(V21 /V0) Target

Average -0.012 100.770 100.782 0.0076 -0.0001 0.0078

Median -0.012 100.741 100.760 0.0074 -0.0003 0.0073

Volatility 0.035 1.299 1.290 0.013 0.013 0.013

Skewness 0.431 0.201 0.192 0.172 0.162 0.267

Kurtosis 7.939 2.581 2.614 2.559 2.593 2.760

Minimum -0.145 98.083 98.013 -0.019 -0.028 -0.02

Maximum 0.241 104.987 104.926 0.049 0.040

RMSE 0.037

τ(1) 0.023 0.024 0

τ(2) -0.061 -0.060 0

Note: The hedging error HE is deﬁned by HE = G(S21 )− V21, and τ(j),j∈

{1,2}, is the estimated Kendall’s tau between the variable and the returns of

portfolio P(j). The results are based on 1 000 repetitions. Here V0= 100.645

and α= log V0/100 = 0.0064. The statistics for the target distribution are also

displayed for sake of comparison. Note also that ϕ1= (−26.464,5.630,42.050),

showing that we are short of the ﬁrst asset at the beginning.

5.5. Synthetic index with Clayton level-1 dependence

The results of this second experiment are also quite interesting, but for

diﬀerent reasons. As can be seen from the results displayed in Table 9, our

goal of replicating the distribution is not achieved. The tracking error given

by the RMSE is too large, the average gain of the portfolio is negative and

its volatility is too large to be interesting for an investor, even if the K-P

measure α= 0.0064 is smaller than in the independence model. This might

be due to the fact that initially, the weight of the assets in the portfolio

are quite large, since ϕ1= (−724.845,84.394,648.811). Furthermore, the

distribution of the hedging errors is not good at all, as can be seen from

the estimated density displayed in Figure 4. The conclusion is that the

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 23

23

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

0

2

4

6

8

10

12

14

Fig. 3. Estimated density of the hedging error G(S21)−V21 for the independence model

based on 1000 replications. Here V0= 100.7864 and α= log V0/100 = 0.007833.

target distribution is not quite well replicated, and one should not invest in

this strategy. The only positive point is that the dependence between the

returns of the payoﬀ and portfolio seems to match the theoretical one, as

measured by Kendall’s tau.

Table 9. Descriptive statistics for the Clayton model.

Statistics HE G(S21)V21 g(R21) log(V21/V0) Target

Average -1.152 100.689 101.842 0.00680 -0.0271 0.0078

Median 3.399 100.608 97.071 0.0061 -0.0362 0.0073

Volatility 27.739 1.235 28.917 0.0122 0.2784 0.0133

Skewness -0.772 0.268 0.753 0.240 0.077 0.267

Kurtosis 3.697 2.610 3.616 2.586 2.525 2.760

Minimum -140.008 98.126 48.336 -0.019 -0.733 -0.02

Maximum 50.675 104.836 244.844 0.047 0.889

RMSE 27.763

τ(1) -0.443 -0.461 -0.5

τ(2) 0.093 0.111

Note: HE = G(S21)− V21 , and τ(j),j∈ {1,2}, is the estimated Kendall’s tau

between the variable and the returns of portfolio P(j). The results are based on 1

000 repetitions. Here V0= 100.645 and α= log V0/100 = 0.0064. The statistics for

the target distribution are also displayed for sake of comparison.

To conclude this section, we computed the K-P measure for Clayton

models as a function of Kendall’s τ. This is illustrated in Figure 5 and it is

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 24

24

-150 -100 -50 0 50 100

0

0.005

0.01

0.015

Fig. 4. Estimated density of the hedging error G(S21)−V21 for the Clayton model with

τ=−0.5 based on 1000 replications.

coherent with the fact that the conditional distribution D1,τ , with τ=θ

θ+2 ,

are ordered according to Lehmann’s order. It then follows from (3) that

the payoﬀ are ordered as well, so the value of the option increases with τ.

-1 -0.5 0 0.5 1

τ

0.04610

0.04611

0.04612

0.04613

0.04614

0.04615

α

Fig. 5. Graph of the K-P measure α= log(V0/100) as a function of Kendall’s τfor the

Clayton model.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 25

25

5.6. Discussion

Before deciding to replicate an asset S?, we should always perform a Monte

Carlo experiment as we did in Sections 5.4–5.5. Using simulations, we can

decide in advance if an asset S?is worth replicating. For example, for

our data, it is worth using the independence model, but it is not worth

using the Clayton model. Simulations can also be useful in tracking a more

realistic P&L since transactions costs can be included in the Monte Carlo

experiment.

We notice that in all cases, the initial investment is more than 100,

meaning that the K-P measure is positive. This can be attributed to the

choice of the reserve asset. Indeed, [12] showed that the choice of the

reserve asset can aﬀect the replication results especially the mean return,

which depends linearly on the K-P measure. Nevertheless, at least in the

case of the independence, we were able to achieve our goal.

It is also worth mentioning that due to (3), if two dependence models

C1and C2are ordered according to Lehmann’s order, i.e., for any v∈

(0,1)d,C1(u, v)≤ C1(u, v), for all u∈[0,1], then the K-P measures are

also ordered.

6. Conclusion

We looked at two important methods of replication of indexes: strong and

weak replication. For strong replication, the aim is to construct a portfolio

of liquid assets that is as close as possible to an existing index, so statistical

methods related to prediction like regression and ﬁltering play an important

role. For weak replication, the aim is to construct a portfolio of liquid

assets that is as close as possible to a payoﬀ constructed in such a way that

the portfolio returns have predetermined distributional properties, such as

the marginal distribution and the conditional distribution relative to some

reference assets entering in the construction of the portfolio.

We also introduced a new family of conditional distribution models

called B-vines that can be useful in many ﬁelds, not just weak replication

of indexes.

We showed how to implement weak replication in general framework,

and we showed that it is possible to construct eﬃciently a synthetic asset

that is independent of prescribed asset classes, with a predetermined distri-

bution. Using simulations, we can decide in advance if an asset S?is worth

replicating. For example, for our data, it is worth using the independence

model, but it is not worth using the Clayton model.

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 26

26

For future work, we plan to investigate the performance of machine

learning methods compared to ﬁltering methods for strong replication pur-

poses. We will also propose goodness-of-ﬁt tests for the B-vines models

introduced in Section 3.

Acknowledgments

The authors thank the referee and the editors for their useful comments

and suggestions. Partial funding in support of this work was provided by

the Natural Sciences and Engineering Research Council of Canada, by the

Fonds Qu´eb´ecois de Recherche sur la Nature et les Technologies, and by

the Groupe d’´etudes et de recherche en analyse des d´ecisions (GERAD).

Appendix A. Optimal hedging in continuous time

For j∈ {1, . . . , l}, let m(j)=(υ(j)−r1), where 1is the vector with all

components equalled to 1, ρ(j) = a(j)−1m(j), and set `j=ρ(j)>m(j) =

ρ(j)>a(j)ρ(j)≥0. Further set γ(t) = e(T−t){Λ−D(`)}1. Next, deﬁne

(˜

Λt)ij =Λij γj(t)/γi(t), i 6=j, (˜

Λt)ii =−X

j6=i

(˜

Λt)ij .(A.1)

Then ˜

Λt,t∈[0, T ], is the inﬁnitesimal generator of a time inhomogeneous

Markov chain.

In [21], it is shown that the optimal hedging problem is related to an

equivalent martingale measure Q, in the sense that under the risk neutral

measure Q, if the price process Xsatisﬁes

dXt=rD(Xt)dt +D(Xt)σ(Tt−)dWt,(A.2)

and Tis a time inhomogeneous Markov chain with generator ˜

Λt, then the

value of an option with payoﬀ Φ at maturity Tis given by

Ct(s, i) = e−r(T−t)EQ[Φ(XT)|Xt=s,Tt=i].(A.3)

If the payoﬀ is smooth enough so that it is diﬀerentiable almost everywhere,

then

∇sCt(s, i) = e−r(T−t)D−1(s)EQ[Φ0(XT)XT|Xt=s,Tt=i], i ∈ {1, . . . , l}.

(A.4)

Since Ctand ∇sCtare related to expectations, one can use Monte Carlo

methods to obtain unbiased estimates of these values.

Next, setting αt(s, i) = ∇sCt(s, i) + Ct(s, i)D−1(s)ρ(i), and Gt=

e−rtCt(Xt,Tt)− Vt, with G0= 0, where Vtis the discounted value of the

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 27

27

(continuous time) hedging portfolio at time t, then the optimal hedging

strategy is

φt=αt(Xt,Tt−)−ertVt−D−1(Xt)ρ(Tt−) (A.5)

=∇sCt(Xt,Tt−) + ertGt−D−1(Xt)ρ(Tt−).(A.6)

References

[1] W. Fung and D. Hsieh, The risk in hedge fund strategies: Theory

and evidence from trend followers, Review of Financial Studies 14, 313

(2001).

[2] W. Fung and D. Hsieh, Hedge fund benchmarks: A risk-based approach,

Financial Analysts Journal 60, 65 (2004).

[3] J. Hasanhodzic and A. W. Lo, Can hedge-fund returns be replicated?:

The linear case, Journal of Investment Management 5, 5 (2007).

[4] P. H. Dybvig, Distributional analysis of portfolio choice, The Journal

of Business 61, 369 (1988).

[5] G. Amin and H. M. Kat, Hedge fund performance 1990-2000: Do the

“money machines” really add value, Journal of Financial and Quantita-

tive Analysis 38, 251 (2003).

[6] H. M. Kat and H. P. Palaro, Who needs hedge funds? A copula-based ap-

proach to hedge fund return replication, tech. rep., Cass Business School,

City University (2005).

[7] E. Wallerstein, N. S. Tuchschmid and S. Zaker, How do hedge fund

clones manage the real world?, The Journal of Alternative Investments

12, 37 (2010).

[8] P. Laroche and B. R´emillard, On the Serial Dependence of Hedge Fund

Indices Returns, tech. rep., Innocap (2008).

[9] M. Towsey, Hedge Fund Replication, tech. rep., Aon Hewitt (2013).

[10] T. Roncalli and J. Te¨ıletche, An alternative approach to alternative

beta, tech. rep., Soci´et´e G´en´erale Asset Management (2007).

[11] B. R´emillard, Statistical Methods for Financial EngineeringChapman

and Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC

Financial Mathematics Series (Taylor & Francis, 2013).

[12] N. Papageorgiou, B. R´emillard and A. Hocquard, Replicating the prop-

erties of hedge fund returns, Journal of Alternative Invesments 11, 8

(2008).

[13] M. Ben-Abdellatif, R´eplication des fonds de couverture, Master’s the-

sis, HEC Montr´eal (2010).

August 7, 2018 15:47 WSPC Proceedings - 9in x 6in replication20171128rev2 page 28

28

[14] B. R´emillard, B. Nasri and T. Bouezmarni, On copula-based condi-

tional quantile estimators, Statistics & Probability Letters 128, 14 (2017),

September 2017.

[15] M. Rosenblatt, Remarks on a multivariate transformation, Ann. Math.

Stat. 23, 470 (1952).

[16] D. Kraus and C. Czado, D-vine copula based quantile regression, Com-

putational Statistics & Data Analysis 110, 1 (2017).

[17] H. Joe, Families of m-variate distributions with given margins and

m(m−1)/2 bivariate dependence parameters, in Distributions with ﬁxed

marginals and related topics, eds. L. R¨uschendorf, B. Schweizer and

M. D. Taylor, Lecture Notes–Monograph Series, Vol. 28 (Institute of

Mathematical Statistics, Hayward, CA, 1996) pp. 120–141.

[18] K. Aas, C. Czado, A. Frigessi and H. Bakken, Pair-copula construc-

tions of multiple dependence, Insurance Math. Econom. 44, 182 (2009).

[19] B. R´emillard, A. Hocquard, H. Lamarre and N. A. Papageorgiou, Op-

tion pricing and hedging for discrete time regime-switching model, Mod-

ern Economy 8, 1005 (September 2017).

[20] M. Schweizer, Mean-variance hedging for general claims, Ann. Appl.

Probab. 2, 171 (1992).

[21] B. R´emillard and S. Rubenthaler, Option pricing and hedging for

regime-switching geometric Brownian motion models, working paper se-

ries, SSRN Working Paper Series No. 2599064 (2016).

[22] B. R´emillard and S. Rubenthaler, Optimal hedging in discrete and

continuous time, Tech. Rep. G-2009-77, Gerad (2009).

[23] B. R´emillard and S. Rubenthaler, Optimal hedging in discrete time,

Quantitative Finance 13, 819 (2013).