Conference PaperPDF Available
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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
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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
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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.
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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?
tˆ
β>
tRt2)1/2
,
while for the out-of-sample, it is defined by
TEout =(1
n
n
X
t=1 R?
tˆ
β>
t1Rt2)1/2
,
where ˆ
βtis the vector of predicted weights using returns (R?
t,Rt), (R?
t1,
Rt1), . . .. The out-of-sample tracking error is a more realistic measure of
performance, since the error R?
tˆ
β>
t1Rtis 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=βt1+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.
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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)
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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
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(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) = F1
?C1{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.
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More precisely, letting βk=erkT /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(βkSkβk1Sk1),
where ϕ(j)
kis number of shares of asset S(j)invested during
((k1)T /n, kT/n], and ϕkmay depend only on S0,...,Sk1. 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?).
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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) = F1
?{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 .312α.
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 (µ3r+σ3xµ1
σ1 ρ13 ρ12s1ρ2
13
1ρ2
12 !
+σ3yµ2
σ2s1ρ2
13
1ρ2
12 ),
so using the Black-Scholes setting with associated risk neutral measure Q,
V0= 100er/12EQegR(1)
0,1/12,R(2)
0,1/12= 100eα,
with
α=µ3
12 r
12 1
12 (µ1
σ3
σ1
+µ2
σ3
σ2s1ρ2
13
1ρ2
12 σ2
3
2)
+1
12 (σ3
σ1rσ2
1
2 ρ13 ρ12s1ρ2
13
1ρ2
12 !
+σ3
σ2rσ2
2
2s1ρ2
13
1ρ2
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
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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 d2, 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) = F1
?h˜
C1{F(y, x),F(x))}i.
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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(VF(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 Rj1(v1, . . . , vj) be the conditional distribution
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of Vjgiven V1=v1, . . . , Vj1=vj1, with R0(v1) = v1, and for (u, v)
(0,1)d+1, set C0(u) = u, and
Cj(u, v1, . . . , vj) = Dj{Cj1(u, v1, . . . , vj1),Rj1(v1, . . . , vj)}.(7)
Note that E{Cj(u, v1, . . . , vj1, Vj)|V1=v1, . . . , Vj1=vj1}is given by
Z1
0Cj{u, v1,...,Rj1(v1, . . . , vj)}dRj1(v1, . . . , vj)
=Z1
0Dj{Cj1(u, v1, . . . , vj1), t}dt
=Dj{Cj1(u, v1, . . . , vj1),1}
=Cj1(u, v1, . . . , vj1).
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)=Γj1D1
j{u, Rj1(v1, . . . , vj)}, v1, . . . , vj1.(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,...,Rd1are known4, and the
conditional copulas D1,...,Ddhave to be chosen, in order to determine
4R0,...,Rd1are called the Rosenblatt’s transforms and are particularly important in
simulating copulas or for testing goodness-of-fit. See, e.g., [11].
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C1,...,Cd.
Level 1:
D1
C0
U|R0
V1=⇒ C1
Level 2:
D2
C1
U|V1|R1
V2|V1=⇒ C2
.
.
.·· · ...
Level j:
Dj
Cj1
U|V1, . . . , Vj1|Rj1
Vj|V1, . . . , Vj1=⇒ Cj
.
.
.·· · ...
Level d:
Dd
Cd1
U|V1, . . . , Vd1|Rd1
Vj|V1, . . . , Vd1, 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,...,Rd1
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
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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 τt1has 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,
t0. 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)
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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 (QhI)/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 aR, 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
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are aiming for applications with d2, 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 φk1
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(k1)T/n(Sk1,ˆτk1) + Gk1D1(Sk1)ρτk1)k1,(10)
˜
Vk=˜
Vk1+ϕ>
k(βkSkβk1Sk1),(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,
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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)
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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.
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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
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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 yαjβ>
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;
Φyµ?/12
σ?/12 Φaµ?/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 1h22hz. With a=
0.02, µ?= 0.08 and σ?= 0.05, one gets an annual mean of 0.0842, and an
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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, uθ+tθ11, 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
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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
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-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
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-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, 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.
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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.
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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(Tt){ΛD(`)}1. Next, define
(˜
Λt)ij =Λij γj(t)i(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) = er(Tt)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) = er(Tt)D1(s)EQ0(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)D1(s)ρ(i), and Gt=
ertCt(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)ertVtD1(Xt)ρ(Tt) (A.5)
=sCt(Xt,Tt) + ertGtD1(Xt)ρ(Tt).(A.6)
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