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 efficiency is estimated on the out-of-sample period using the maximum attainable risk reduction, the financial 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 efficiency of 88%.
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 moments 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 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 first 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 coefficients are calculated 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 methodology. Section 3 contains the estimation results and their discussion. Section 4 concludes.
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 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 r t and equals to (1).
where r S,t -stock returns at time t; r F,t -futures returns at time t; hr * t -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 (r t ), (2).
where τ -positive parameter for risk aversion (large values indicate that investor dislikes risk), E(r t ) -expected portfolio returns, V(r t ) -portfolio variance. The optimal hedge coefficient hr * t is defined in (3).
where cov(•) -covariance. Evidently, if τ → inf, then hr * t coincides with traditional optimal hedge ratio, based on minimization of portfolio variance.
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 x t , x t = (x 1t , x 2t , . . . , x nt ) be a portfolio consisted of n assets at time moment t. x t is represented as a sum of its mathematical expectation E (x t |F t ), conditional on all available at t − 1 information, and innovations y t , (4).
Innovations y t conditional on volatility Σ t are distributed normally with zero mean, (5).
At the same time, Σ t is a random process itself, generated by inversed Wishart distribution, (6).
Using properties of compound distributions, we obtain (7).
where t ν (0, H t ) -multivariate Student's t-distribution with ν degrees of freedom and variance H t , [5].
It's worth mentioning that, since the distribution of y t is known in the model, the conjugate priors for the parameters could be derived. For univariate case conjugate 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 H t in (6).
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 ( 4), ( 5)), except that the conditional distribution of innovations could be different.
For the GO-GARCH model ( 8) holds.
where X is the matrix whose parametrization is based on the singular decomposition of the unconditional variance of returns (for details, see [20]), V t -a diagonal matrix, which nonzero elements are portfolio assets volatilities, given by any onedimensional 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 v t represents a standard univariate GARCH. ADCC volatility is modeled as in (9).
where R t is conditional correlation matrix of returns, α, β, γ -parameters and γ is responsible for asymmetry effects in volatility, y t−1 are the zero-threshold innovations which are equal to y t 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 approach allows to account for investor's risk aversion, while building the hedging strategy.
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 first 80% of the series, the rest 20% are for out-of-sample period. The number of lags is chosen by minimizing 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 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 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 reduction, financial result (or profit) and investor's utility. The first measure is defined as in (10).
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". 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 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.
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 efficiency of hedging strategies is assessed by maximum risk reduction, financial 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 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 constructing 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 timevarying degree of risk aversion, introducing the heterogeneity of investors by their attitude to risk and using other hedging instruments.