=Paper=
{{Paper
|id=Vol-2018/paper-10
|storemode=property
|title=Hedging and Risk Aversion on Russian Stock Market: Strategies Based on MGARCH and
MSV Models
|pdfUrl=https://ceur-ws.org/Vol-2018/paper-10.pdf
|volume=Vol-2018
|authors=Valeriya Lakshina
}}
==Hedging and Risk Aversion on Russian Stock Market: Strategies Based on MGARCH and
MSV Models==
https://ceur-ws.org/Vol-2018/paper-10.pdf
Hedging and Risk Aversion on Russian Stock
Market: Strategies Based on MGARCH and
MSV Models
Valeriya Lakshina[0000−0003−1447−9377]
National Research University Higher School of Economics,
Nizhny Novgorod, Russia
vlakshina@hse.ru
Abstract. The paper studies the problem of dynamic hedge ratio calcu-
lation for the portfolio consisted of two assets – futures and the underly-
ing stock. We apply the utility based approach to account for the degree
of risk aversion in the hedging strategy. Seventeen portfolios, consisted of
Russian blue-chip stocks and futures, are estimated in the paper. In order
to estimate the conditional covariances of hedged portfolio returns, such
multivariate volatility models as GO-GARCH, copula-GARCH, asym-
metric DCC and parsimonious stochastic volatility model are applied.
The hedging efficiency is estimated on the out-of-sample period using
the maximum attainable risk reduction, the financial result and the in-
vestor’s utility. It’s shown that for 60% of portfolios ADCC surpasses the
other models in hedging. Including the degree of risk aversion in the in-
vestor’s utility function together with above-mentioned volatility models
allows to reach hedging efficiency of 88%.
Keywords: dynamic hedge ratio, stock futures, multivariate volatility models,
risk aversion, hedging efficiency, copula, expected utility
1 Introduction
Hedging is one of the most common tasks in finance. It requires knowing both
hedging and hedged asset returns’ distribution. In other words, one should be
aware of the multivariate distribution of returns, or, at least, the first two mo-
ments of this distribution. Our paper is focused on modeling the second moment
because the variance-covariance matrix is needed to solve the aforementioned
financial task.
There are two main approaches to modeling volatility: generalized autoregres-
sive conditional heteroskedasticity or GARCH (a survey on multivariate GARCH
models see in [2]) and models of stochastic volatility or SV (a review of multivari-
ate SV models see in [1]). The latter take into account the volatility uncertainty
directly by including the random term in the volatility equation. This assump-
tion seems to be closer to the empirical evidence. But estimation of multivariate
SV poses a challenge both due to dimensionality problem and the lack of closed-
form likelihood function in general case.1 It’s worth mentioning that the first
problem is also acute for GARCH models and implies that the number of pa-
rameters grows quadratically (sometimes even faster as in VECH model, [8])
related to the number of assets in the portfolio.
The second issue, arising from the use of SV for building the hedging strategy,
is estimation. In contrast to GARCH, SV contains two sources of uncertainty
and in most cases it is not possible to derive the likelihood function analyti-
cally. However, there are a number of ways to estimate multivariate SV by the
maximum likelihood method. For example the likelihood function can be ap-
proximated by a Gaussian density (see, e.g., [19]), or simulated (see, e.g., [4,
13]).
In this paper we attempt to propose a multivariate SV model in which
both the aforementioned problems are remedied and apply it to stocks hedging.
The model suggests that the demeaned returns follow Student’s t-distribution,
whereas the volatility matrix is also random. This property follows from the fact
that Student’s t-distribution can be represented as a mixture of normal distribu-
tions. As a matter of fact, if the demeaned returns are distributed normally con-
ditionally on volatility matrix and the volatility matrix itself has inverse Wishart
distribution, then, according to [5], the volatility matrix can be marginalized out
from the returns’ distribution, which results in Student’s t-distribution for the
returns. Consequently, the demeaned returns distribution is known in contrast
to the majority of other multivariate SV models.
The paper contains the estimation of the described model via Markov chain
Monte Carlo algorithm implemented in Stan software [18]. The parameters are
obtained for seventeen stocks listed on Moscow Exchange futures market [14].
The sample covers the period from January 2006 to December 2016. All the
positions are hedged with futures and the dynamic hedging coefficients are cal-
culated using multivariate SV as well as several multivariate GARCH models.
The resulted hedging strategies are compared via different criteria of hedging
efficiency.
The rest of the article is organized as follows. Section 2 describes the method-
ology. Section 3 contains the estimation results and their discussion. Section 4
concludes.
2 Methodology
2.1 Hedge Ratio
The aim of hedging is the reduction of portfolio value fluctuations. This can be
achieved by opening the opposite position on the hedging instrument, usually a
futures. The main task of building the hedging strategy is finding the optimal
1
The likelihood function for multivariate SV could be obtained under certain condi-
tions, see [11].
amount of futures in the portfolio, i. e. calculation of optimal hedge ratio, which
shows the relation of the hedged asset value to the hedging asset value.
The return of the hedged position at time t is denoted by rt and equals to (1).
rt = rS,t − hrt∗ · rF,t , (1)
where rS,t – stock returns at time t; rF,t – futures returns at time t; hrt∗ – optimal
hedge ratio at time t.
We use the utility approach to implement the investor’s attitude to risk in
building the hedging strategy. We obtain the optimal hedge ratio from maxi-
mization of the investor’s expected utility E U (rt ), (2).
V (rt )
E U (rt ) = E(rt ) − τ , (2)
2
where τ – positive parameter for risk aversion (large values indicate that investor
dislikes risk), E(rt ) – expected portfolio returns, V(rt ) – portfolio variance. The
optimal hedge coefficient hrt∗ is defined in (3).
cov(rS,t , rF,t ) E(rF,t )
hrt∗ = − , (3)
V(rF,t ) 2τ V(rF,t )
where cov(·) – covariance. Evidently, if τ → inf, then hrt∗ coincides with tradi-
tional optimal hedge ratio, based on minimization of portfolio variance.
2.2 Multivariate Volatility
Historically the first methods of hedge ratio calculation assumed that the ratio is
constant. Further hedging strategies based on the dynamic hedge ratio appeared.
They allow to implement heteroskedasticity of returns in the model. In this study,
four volatility models have been taken to estimate the dynamic hedge ratio.
There are two main approaches to the covariance matrix modeling: GARCH
models and stochastic volatility models (MSV). Since the empirical evidence
shows that volatility is volatile itself, it seems more appropriate to use SV, while
modeling volatilty. SV has two sources of uncertainty – in mean and volatility
equations. This fact results in challenging estimation procedure of SV, because
it’s impossible to derive the likelihood function analytically in general case. Thus
SV models are usually estimated within the Bayesian framework.
Let xt , xt = (x1t , x2t , . . . , xnt )| be a portfolio consisted of n assets at time
moment t. xt is represented as a sum of its mathematical expectation E (xt |Ft ),
conditional on all available at t − 1 information, and innovations yt , (4).
xt = E (xt |Ft−1 ) + yt , t = 1, . . . , T, xt − (n × 1)-vector, (4)
Innovations yt conditional on volatility Σt are distributed normally with zero
mean, (5).
yt |Σt ∼ N (0, Σt ) (5)
At the same time, Σt is a random process itself, generated by inversed
Wishart distribution, (6).
Σt ∼ IW (ν, Ht ) (6)
Using properties of compound distributions, we obtain (7).
yt ∼ t (ν, 0, Ht ) , (7)
where tν (0, Ht ) – multivariate Student’s t-distribution with ν degrees of freedom
and variance Ht , [5].
It’s worth mentioning that, since the distribution of yt is known in the model,
the conjugate priors for the parameters could be derived. For univariate case con-
jugate priors for parameters with fixed ν are derived in [21], where the conjugate
prior for volatility is Fisher distribution with n − k (k – number of parameters)
and ν degrees of freedom. According to [6], multivariate beta distribution is the
generalization of Fisher distribution, which is a prior for Ht in (6).
We also use several multivariate GARCH models, namely general orthogo-
nalized GARCH (GO-GARCH), copula-GARCH and asymmetric dynamic con-
ditional correlations (ADCC).
The initial setup is analogous to MSV model, (see (4), (5)), except that the
conditional distribution of innovations could be different.
For the GO-GARCH model (8) holds.
ΣtGO−GARCH = XVt X | , Vt = diag(vt ), (8a)
vt = C + A(yt yt ) + Bvt−1 , (8b)
where X is the matrix whose parametrization is based on the singular decomposi-
tion of the unconditional variance of returns (for details, see [20]), Vt – a diagonal
matrix, which nonzero elements are portfolio assets volatilities, given by any one-
dimensional GARCH model. For example, in (8b) A, B are diagonal matrices of
parameters, C is a n × 1 parameter vector, is an element-wise multiplication.
As a result, each row of vt represents a standard univariate GARCH.
ADCC volatility is modeled as in (9).
ΣtADCC = Dt Rt Dt , (9a)
Dt = diag(dt ), dt dt = vt (9b)
−1/2 −1/2 −1/2 −1/2
Rt = diag q11,t . . . qnn,t Qt diag q11,t . . . qnn,t , (9c)
| |
Qt = (1 − α − β)Q̄ + αyt−1 yt−1 + βQt−1 + γe
yt−1 yet−1 , (9d)
where Rt is conditional correlation matrix of returns, α, β, γ – parameters and
γ is responsible for asymmetry effects in volatility, yet−1 are the zero-threshold
innovations which are equal to yt when less than zero and are equal to zero
otherwise. More details are in [3].
Copula-GARCH model is similar to (9), but differs by the fact that joint
distribution of returns are modeled via Student’s t copula function, see [12, 16].
To sum up, our hedging strategies are based on four multivariate volatility
models, three GARCH and one SV. We obtain the dynamic optimal hedge ratio
from conditional covariance matrices, estimated by these models. Utility ap-
proach allows to account for investor’s risk aversion, while building the hedging
strategy.
3 Empirical Results
3.1 Data Description
For our empirical study we take 17 companies, listed on Moscow Exchange [14],
which stocks are also traded on the futures market. The companies included in
the sample are presented in Table 1.
Table 1. Companies under consideration
Ticker Company name Ticker Company name
chmf Severstal rosn Rosneft Oil Company
fees Federal Grid Company rtkm Rostelecom
gazp Gazprom sber Sberbank of Russia
gmkn Norilsk Nickel sngs Surgutneftegas
hydr RusHydro tatn Tatneft
lkoh Lukoil trnf Transneft
mgnt Magnit urka Uralkali
nlmk Novolipetsk Steel vtbr Bank VTB
nvtk Novatek
It’s worth mentioning that there are about 100 participants on the stock
section of MOEX futures market, but reasonable prices’ history is available only
for stocks in Table 1. Each bivariate “stock-futures” price series has its own
length and we do not take stocks with historical prices, which amount less than
200 observations. The longest series, belonging to lkoh, has 3586 observations
and covers the period from the 9th of August 2002 till the 30th of December
2016. The rest of the series are within this period. The source of the data is
Finam investment company website [9]. Short descriptive statistics of the data
under consideration is presented in Table 2.
3.2 Estimation Results
To calculate the dynamic hedge ratio, the following volatility models are eval-
uated in this paper: ADCC, GO-GARCH, cop-GARCH and MSV. Conditional
mean of returns is modeled using ARMA. For each asset the whole sample is
divided into two parts: in-sample period includes the first 80% of the series, the
Table 2. Descriptive statistics. N – number of observations, Mean – mean of daily
logarithmic returns, St.dev. – standard deviation, Skew. – skewness coefficient, Kurt. –
kurtosis coefficient
Stocks Futures
Ticker N Mean St.dev. Skew. Kurt. Mean St.dev. Skew. Kurt.
chmf 1395 0.040 2.215 -0.366 6.237 0.040 2.402 -0.592 10.494
fees 1318 -0.050 2.941 -0.324 9.748 -0.048 3.184 0.189 8.993
gazp 2730 -0.013 2.458 -0.084 19.300 -0.013 2.591 0.213 24.388
gmkn 3041 0.058 2.747 -1.001 20.524 0.058 2.907 -1.112 26.078
hydr 1404 -0.032 2.172 0.199 6.239 -0.031 2.273 -0.092 6.818
lkoh 3586 0.055 2.317 -0.055 16.122 0.055 2.393 -0.335 25.889
mgnt 587 0.036 2.027 -0.089 5.140 0.040 2.202 -0.042 5.573
nlmk 231 0.244 2.030 0.229 3.906 0.242 2.879 0.936 12.178
nvtk 1973 0.084 2.925 -1.362 31.895 0.084 3.712 -0.530 14.097
rosn 2547 0.025 2.610 0.929 36.043 0.025 2.770 0.536 47.209
rtkm 3020 0.030 2.284 0.293 12.500 0.030 3.031 -0.676 26.798
sber 2753 0.068 2.995 0.129 17.146 0.068 3.122 0.210 17.946
sngs 3583 0.031 2.692 0.963 24.704 0.031 2.799 2.272 54.004
tatn 1402 0.065 2.170 -0.020 4.212 0.067 2.145 -0.407 7.613
trnf 2367 0.065 3.082 0.023 18.438 0.066 3.150 0.039 9.705
urka 1383 -0.028 2.136 -1.621 23.653 -0.024 2.537 -0.649 13.852
vtbr 2357 -0.026 2.919 0.576 45.755 -0.027 3.286 2.680 80.832
rest 20% are for out-of-sample period. The number of lags is chosen by mini-
mizing Schwartz information criterion, according to which ARMA(1,0) for mean
and GARCH(1,1) for all volatility models give the best fit.
The parameters of MSV model are obtained by one of the Markov chain
Monte Carlo methods – Hamiltonian Monte Carlo algorithm, also known as Hy-
brid Monte Carlo [7, 15]. The convergence of Markov chains is checked by Geweke
Z-test [10]. The test is based on the idea that the means, calculated on the first
and the last parts of a Markov chain (usually 10% and 50% correspondingly),
are equal. In that case the parameter samples are drawn from the stationary
distribution of the Markov chains and Geweke’s statistics has an asymptotically
standard normal distribution. The Markov chain converges under the null. For
some important parameters (namely, covariance of stock and futures returns,
futures variance and conditional return at a specific time point, see (3)) p-values
are presented in Table 3. Evidently that Geweke Z-test reveals the convergence
for the parameters under consideration.
Mean hedge ratios for τ = 4 are presented in Table 4. The average hedge
ratios range from 19% for lkoh to 94% for sngs.
In order to compare hedging strategies, obtained from different volatility
models, we calculate three measures of hedging efficiency – maximum risk re-
duction, financial result (or profit) and investor’s utility. The first measure is
Table 3. Geweke Convergence Z-testtext
Ticker cov(rS,t , rF,t ) V(rF,t ) E(rF,t )
chmf 0.612 0.554 0.505
fees 0.832 0.829 0.937
gazp 0.844 0.929 0.557
gmkn 0.896 0.897 0.562
hydr 0.630 0.545 0.790
lkoh 0.914 0.944 0.759
mgnt 0.834 0.545 0.984
nlmk 0.709 0.548 0.822
nvtk 0.534 0.644 0.803
rosn 0.931 0.887 0.666
rtkm 0.664 0.598 0.611
sber 0.876 0.849 0.551
sngs 0.933 0.933 0.674
tatn 0.610 0.569 0.898
trnf 0.885 0.913 0.580
urka 0.591 0.690 0.969
vtbr 0.872 0.821 0.637
Table 4. Mean Hedge Ratios
Ticker ADCC GO-GARCH cop-GARCH MSV
chmf 0.812 0.778 0.799 0.768
fees 0.836 0.837 0.843 0.820
gazp 0.862 0.810 0.851 0.889
gmkn 0.893 0.817 0.897 0.905
hydr 0.867 0.874 0.861 0.844
lkoh 0.934 0.821 0.192 0.935
mgnt 0.914 0.825 0.911 0.805
nlmk 0.612 0.611 0.608 0.644
nvtk 0.673 0.732 0.678 0.567
rosn 0.883 0.815 0.863 0.864
rtkm 0.562 0.618 0.609 0.785
sber 0.891 0.824 0.899 0.883
sngs 0.910 0.816 0.910 0.939
tatn 0.935 0.903 0.246 0.896
trnf 0.885 0.820 0.879 0.797
urka 0.555 0.559 0.514 0.679
vtbr 0.871 0.753 0.864 0.835
defined as in (10).
var(r)
E =1− (10)
var(rS )
Financial result is calculated as the sum of the logarithmic returns of the portfolio
for the forecast period [17]. The formula for investor’s utility is described in (2).
The comparison is conducted for the out-of-sample period and summarized
in Table 5. We compute the measures of hedging performance for various levels
of risk aversion τ . They vary from very small values, almost equal to zero, till
ten and are presented in the first column of Table 5, denoted by “ra”. Maxi-
mum risk reduction is abbreviated by “mrr”. The rest of the column labels are
self-explanatory. Table 5 reveals the numbers of assets, for which the model in
the corresponding column maximizes the corresponding criterion. According to
Table 5. Summary of hedging efficiency
ADCC GO-GARCH cop-GARCH MSV
ra mrr profit util mrr profit util mrr profit util mrr profit util
0.00 4 0 1 4 2 2 7 6 8 2 9 6
1.11 10 0 13 7 2 3 0 3 1 0 12 0
2.22 10 2 10 7 1 6 0 5 1 0 9 0
3.33 10 2 10 7 1 6 0 6 1 0 8 0
4.44 10 2 10 6 5 6 1 5 1 0 5 0
5.56 10 2 10 6 5 6 1 5 1 0 5 0
6.67 10 2 10 6 5 6 1 5 1 0 5 0
7.78 10 2 10 6 6 6 1 5 1 0 4 0
8.89 10 2 10 6 6 6 1 5 1 0 4 0
10.00 10 2 10 6 6 6 1 5 1 0 4 0
Table 5, ADCC model clearly outperforms the other models by the maximum
risk reduction and investor’s utility. MRR in average amounts to 74% for this
model and ranges from 47% to 88%. GO-GARCH, copula-GARCH and MSV
reach their maximum MRR at the levels of 84%, 85% and 83% respectively.
The performance level of ADCC model seems to be stable and remains the
same for τ larger than 2. The dynamics of GO-GARCH hedge efficiency criteria
values also stabilizes for higher risk aversion levels. GO-GARCH performance
is the same among different efficiency measures and is relatively lower than
in ADCC case. Copula-GARCH demonstrates even lower performance by all
criteria except the profit of the hedged position. Stochastic volatility clearly
provides the highest financial result, if the investor prefers risk. At the same
time, with the growth of τ performance level of MSV declines.
It’s also worth mentioning that on small values of τ ADCC and MSV reach
their maximum performance by utility and profit correspondingly.
4 Conclusion
The article considers the development of a hedging strategy based on maximiz-
ing investor’s expected utility, taking into account the level of risk aversion. The
optimal hedge ratio is time-dependent and is calculated using four multivari-
ate volatility models ADCC, GO-GARCH, copula-GARCH with the Student’s
copula and multivariate stochastic volatility. The calculation is conducted for
seventeen portfolios consisted of stocks and futures of seventeen Russian compa-
nies. The efficiency of hedging strategies is assessed by maximum risk reduction,
financial result of hedged position and investor’s utility with risk aversion param-
eter varying from zero to ten. The most stable performance of hedging strategies
according to the chosen criteria demonstrates ADCC model, which provides the
highest maximum risk reduction and utility for about 60% of portfolios. MSV
maximizes profit of the hedged position for small values of risk aversion in 70%
cases. To summarize, ADCC and MSV models are recommended to use for con-
structing hedging strategies on Russian stock market according to maximum risk
reduction and utility for the former and profit for the latter. MSV gives better
results for risk-lovers and ADCC outperforms the other models if risk aversion
parameter is larger than 2.
The possible directions of the future research include implementing time-
varying degree of risk aversion, introducing the heterogeneity of investors by
their attitude to risk and using other hedging instruments.
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