The paper studies the problem of dynamic hedge ratio calculation for the portfolio consisted of two assets { futures and the underlying 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, asymmetric DCC and parsimonious stochastic volatility model are applied. The hedging e ciency is estimated on the out-of-sample period using the maximum attainable risk reduction, the nancial result and the investor'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 investor's utility function together with above-mentioned volatility models allows to reach hedging e ciency of 88%.
Hedging is one of the most common tasks in nance. 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 rst two moments of this distribution. Our paper is focused on modeling the second moment because the variance-covariance matrix is needed to solve the aforementioned nancial task.
There are two main approaches to modeling volatility: generalized autoregressive conditional heteroskedasticity or GARCH (a survey on multivariate GARCH models see in [2]) and models of stochastic volatility or SV (a review of multivariate SV models see in [1]). The latter take into account the volatility uncertainty directly by including the random term in the volatility equation. This assumption 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 closedform likelihood function in general case.1 It's worth mentioning that the rst problem is also acute for GARCH models and implies that the number of parameters 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 analytically. However, there are a number of ways to estimate multivariate SV by the maximum likelihood method. For example the likelihood function can be approximated 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 distributions. As a matter of fact, if the demeaned returns are distributed normally conditionally 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 coe cients are calculated using multivariate SV as well as several multivariate GARCH models. The resulted hedging strategies are compared via di erent criteria of hedging e ciency.
The rest of the article is organized as follows. Section 2 describes the methodology. Section 3 contains the estimation results and their discussion. Section 4 concludes. 2 2.1
Hedge Ratio The aim of hedging is the reduction of portfolio value uctuations. 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 nding the optimal 1 The likelihood function for multivariate SV could be obtained under certain conditions, 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 (
E U (rt) = E(rt)
V (rt) ;
2 rt = rS;t hrt rF;t; 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
maximization of the investor's expected utility E U (rt), (
hrt = cov(rS;t; rF;t)
V(rF;t)
E(rF;t) 2 V(rF;t) ; where cov( ) { covariance. Evidently, if ! inf, then hrt coincides with traditional optimal hedge ratio, based on minimization of portfolio variance. 2.2
Multivariate Volatility ytj t
N (0; t)
(
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 (xtjFt),
conditional on all available at t 1 information, and innovations yt, (
mean, (
At the same time,
Wishart distribution, (
t is a random process itself, generated by inversed
Using properties of compound distributions, we obtain (
t
IW ( ; Ht) yt
t ( ; 0; Ht) ; where t (0; Ht) { multivariate Student's t-distribution with 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
conjugate priors for parameters with xed are derived in [21], where the conjugate
prior for volatility is Fisher distribution with n k (k { number of parameters)
and degrees of freedom. According to [6], multivariate beta distribution is the
generalization of Fisher distribution, which is a prior for Ht in (
We also use several multivariate GARCH models, namely general orthogonalized GARCH (GO-GARCH), copula-GARCH and asymmetric dynamic conditional correlations (ADCC).
The initial setup is analogous to MSV model, (see (
For the GO-GARCH model (
tGO GARCH = XVtX|; Vt = diag(vt);
vt = C + A(yt
yt) + Bvt 1;
(
ADCC volatility is modeled as in (
tADCC = DtRtDt; Dt = diag(dt); dt
dt = vt Rt = diag q111;t=2 : : : qnn1;=t2
Qt diag q111;t=2 : : : qnn1;=t2 ;
Qt = (
Qt 1 + yet 1yet| 1; where Rt is conditional correlation matrix of returns, ; ; { parameters and is responsible for asymmetry e ects in volatility, yt 1 are the zero-threshold e 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 (
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 approach allows to account for investor's risk aversion, while building the hedging strategy. 3 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.
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.
To calculate the dynamic hedge ratio, the following volatility models are
evaluated 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 rst 80% of the series, the
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rest 20% are for out-of-sample period. The number of lags is chosen by
minimizing Schwartz information criterion, according to which ARMA(
The parameters of MSV model are obtained by one of the Markov chain
Monte Carlo methods { Hamiltonian Monte Carlo algorithm, also known as
Hybrid 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 rst
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 speci c time point, see (
Mean hedge ratios for = 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 di erent volatility
models, we calculate three measures of hedging e ciency { maximum risk
reduction, nancial result (or pro t) and investor's utility. The rst measure is
Ticker ADCC GO-GARCH cop-GARCH
de ned as in (
E = 1 var(r) var(rS)
(
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 . They vary from very small values, almost equal to zero, till ten and are presented in the rst column of Table 5, denoted by \ra". Maximum 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
mrr pro t util mrr pro t util mrr pro t util mrr pro t util The article considers the development of a hedging strategy based on maximizing investor's expected utility, taking into account the level of risk aversion. The optimal hedge ratio is time-dependent and is calculated using four multivariate 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 companies. The e ciency of hedging strategies is assessed by maximum risk reduction, nancial result of hedged position and investor's utility with risk aversion parameter 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 pro t of the hedged position for small values of risk aversion in 70% cases. To summarize, ADCC and MSV models are recommended to use for constructing hedging strategies on Russian stock market according to maximum risk reduction and utility for the former and pro t 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 timevarying degree of risk aversion, introducing the heterogeneity of investors by their attitude to risk and using other hedging instruments. 13. Kleppe, T.S., Yu, J., Skaug, H.J.: Simulated maximum likelihood estimation of continuous time stochastic volatility models. Advances in Econometrics 26, 137{ 161 (2010) 14. MOEX Moscow exchange. http://moex.com (2017) 15. Neal, R.M.: MCMC using hamiltonian dynamics. In: Brooks, S., Gelman, A., Jones, G.L., Meng, X.L. (eds.) Handbook of Markov Chain Monte Carlo, pp. 113{162.
Chapman & Hall/CRC. (2011)
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