Estimation of VIX futures through Gaussian factor models.
| Autor | Fernandes, Felipe do Nascimento |
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Introduction
Since their emergence in 2004, VIX future contracts have become an important instrument for investment decisions, hedging strategies and risk management. As typically occurs in commodities markets, VIX future contracts are negotiated in futures markets. Moreover, there are VIX traded options, and more recently, instruments like exchanged traded products (ETPs), such as exchange traded notes (ETNs) and exchanged traded funds (ETFs). Hence, modeling the dynamics of these instruments is of crucial importance for traders to develop successful strategies involving that index. The need for an accurate description of the index has aroused the interest of the academics. Several researchers have demonstrated their interest in the subject for better understanding the behavior of the VIX and its derivatives.
Gregory and Timcenko (2016) developed an initial study of VIX futures contracts. Hill et al. (2015) and Alexander and Korovilas (2013) described strategies and historical performance of ETPs. For details on the operational dynamics of ETPs, see Boroujerdi and Fogertey (2015).
Zhang and Zhu (2006) used the stochastic volatility to model spot VIX prices following the Heston (1993) approach. They developed an expression for the VIX future prices. The discrepancy with the observed data was between 2 to 12% when the estimation was carried out based on the last year of the sample. Using the whole sample the authors reported a difference of 12 to 44%.
Wang et al. (2017) proposed a closed-form pricing formula for VIX futures based on discrete-time Heston-Nandi GARCH model. The estimation is based on data from different sources: S&P 500 returns, VIX spot and VIX future prices. They found that the joint information based on VIX and VIX futures provided the best results.
Avellaneda and Papanicolaou (2019) developed a two-factor Gaussian model to describe VIX futures contracts dynamics. First, the authors constructed the constant maturity futures (CMFs) based on VIX futures contracts. After conducting unit root tests on VIX futures, they found evidence of a stationary process. The main hypothesis in the study was that the CMFs are stationary for each maturity contract. Hence, they modeled the spot VIX by decomposing it as the sum of two latent factors, both following a meanreverting process. To establish the number of factors, the authors performed principal components analysis (PCA). They found that two factors accounted for the 90% of the variance of the term-structure shape. To estimate the model they used Kalman filtering and analyzed the behavior of rolling strategies involving ETNs and ETFs.
In this study, we use the original two-factor model of Avellaneda and Papanicolaou (2019) to model VIX prices. However, our analysis is based on two approaches. The first one considers the model with the market price of risk (MPR) held constant. In the second approach, we include a time-varying market price of risk (TVMPR). In both cases, we perform the estimation considering the following: (i) VIX futures contracts (future prices) and also the spot VIX (spot price); and (ii) only VIX futures contracts (future prices). The intuition behind this procedure is the same as in commodity models where the spot price is an unobservable factor and it is estimated as being an output of the model.
Unlike many studies in the literature of VIX prices that use stochastic volatility models, this paper is in line with that of Avellaneda and Papanicolaou (2019) focusing on Gaussian factor models. It contributes on VIX estimation procedures investigating the behavior of a more parsimonious model considering two different approaches when compared to the Avellaneda and Papanicolaou (2019) study: (i) excluding the VIX spot price from the sample, that is, considering only VIX future prices series; (ii) considering a constant market price of risk in the model. We compare these approaches to their counterparts. We found evidence that removing the spot VIX results in a better adjustment to empirical data, mainly for medium and long term maturities. Moreover, the best adjustment when one removes the spot VIX from the sample, occurs with the time-varying market price of risk model. The usefulness of these results is a simpler procedure when dealing with VIX estimation.
The remainder of this paper is organized as follows. Section 2 specifies the model; Section 3 presents the data; Section 4 presents the results; Section 5 illustrates an application to an ETN; and Section 6 concludes.
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The models
Consider ([OMEGA], F, P) the probability space. Furthermore, consider that the market is complete and arbitrage free, so that the uniqueness of an equivalent martingale measure (EMM) Q is well-established by the fundamental theorems of Finance.
Avellaneda and Papanicolaou (2019) run the PCA analysis and found that the first component accounts for 72% of the variance of the term-structure shape. Including the second component increases the explanation to 90%. The third and fourth components explain 96% and 97% of the variance of the term-structure, respectively. They considered a parsimonious model with two components to describe the behavior of VIX future contracts. We followed their study and decomposed the logarithm of the spot VIX as the sum of two latent components, [X.sub.1,t] and [X.sub.2,t]. These two factors are described through an Ornstein-Uhlenbeck process, following the assumption that the time series of CMFs are stationary. This is coherent with the well-known fact, based on empirical analysis, that the variance of financial time series is a meanreverting process.
In the P-measure, the logarithm of the spot VIX is given by:
ln [VIX.sub.t] = [X.sub.1,t] + [X.sub.2,t], (1a)
[dX.sub.1,t] = [K.sub.1]([[micro].sub.1]--[X.sub.1,t]) dt + [[sigma].sub.1] [dB.sub.1,t], (1b)
[dX.sub.2,t] = [K.sub.2]([[micro].sub.2]--[X.sub.2,t]) dt + [[sigma].sub.2] [dB.sub.2,t], (1c)
where the factors are driven by standard Brownian motions [B.sub.1] and [B.sub.2] respectively, which are correlated with pdt = [dB.sub.1,t], [dB.sub.2,t]. In addition, [k.sub.1], [k.sub.2] > 0 are the speeds of mean reversion and [[sigma].sub.1], [[sigma].sub.2] > 0 are the volatilities of each factor.
Define [[lambda].sub.1,t] and [[lambda].sub.2,t] as the time-varying risk premiums of both latent factors, respectively
[[lambda].sub.1,t] = [p.sub.1] + [q.sub.1] [X.sub.1,t], (2a)
[[lambda].sub.2,t] = [p.sub.2] + [q.sub.2] [X.sub.2,t], (2b)
where [p.sub.1], [p.sub.2], [q.sub.1] and [q.sub.2] [member of] R.
Changing to the EMM Q, we obtain:
[Please download the PDF to view the mathematical expression], (3a)
[Please download the PDF to view the mathematical expression]. (3b)
The change of measure preserves the mean-reverting dynamics of both factors in the Q-measure, where the new long-run means are [[mu].sup.*.sub.1] = [k.sub.1][[mu].sub.1]-- [[sigma].sub.1][p.sub.1]/[k.sub.1][[mu].sub.1] and [[mu].sup.*.sub.2] = [k.sub.2][[mu].sub.2]--[[sigma].sub.2][p.sub.2]/[k.sub.2][[mu].sub.2]. In the same manner, the new speeds of mean reversion are given by [K.sup.*.sub.1] = [K.sub.1] + [[sigma].sub.1] [q.sub.1] and [k.sup.*.sub.2] = [k.sub.2] + [[sigma].sub.2] [q.sub.2].
Let [Please download the PDF to view the mathematical expression] denote the jth VIX future price observed at time t maturing at time [T.sub.j]. We represent by [Please download the PDF to view the mathematical expression] the CMF contract constructed for the jth contract based on VIX future prices of adjacent maturities [Please download the PDF to view the mathematical expression]. Using the fact that [Please download the PDF to view the mathematical expression], one can prove that the logarithm of [Please download the PDF to view the mathematical expression] is given by:
[Please download the PDF to view the mathematical expression], (4)
where
[Please download the PDF to view the mathematical expression]
which is a function of the hyperparameters and the remaining time to maturity for each contract.
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The data
The analyzed sample consists of seven VIX futures contracts and the spot VIX, from 21st April, 2008 to 28th January, 2019, forming a complete panel. Each series contains 510 weekly observations...
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