Modelling the implied volatility surface based on Shanghai 50ETF options

Economic Modelling 64 (2017) 295–301

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Economic Modelling journal homepage: www.elsevier.com/locate/econmod

Modelling the implied volatility surface based on Shanghai 50ETF options a

a

a,⁎

Jinzhong Wang , Shijiang Chen , Qizhi Tao , Ting Zhang a b

MARK

b

School of Finance, Southwestern University of Finance and Economics, Chengdu 611130, China School of Business Administration, University of Dayton, Dayton, OH 45469, United States

A R T I C L E I N F O

A BS T RAC T

JEL: G13 C13

We develop a dynamic factor model to forecast the implied volatility surface (IVS) of Shanghai Stock Exchange 50ETF options. Based on the assumption that dynamic change in IVS is mean-reverting and Markovian, we use a state space model to capture the dynamics of IVS, and set the latent factors to be the Ornstein–Uhlenbeck processes. We obtain the optimal estimations of parameters using the Kalman filter algorithm. Empirical results show that our model performs better than the traditional IVS model in terms of fitting ability and prediction performance.

Keywords: Dynamic factor model Implied volatility surface Kalman filter Ornstein–Uhlenbeck process

1. Introduction This paper's objective is to construct a dynamic factor model to forecast the implied volatility surface (IVS) of Shanghai Stock Exchange 50ETF (SSE 50ETF) options after considering the crosssectional features and dynamics of the option. The SSE 50ETF option was launched by the Shanghai Stock Exchange on February 9, 2015, the first option product in China's financial markets. Considering the size and growth of the Chinese market, we expect more option products to become available in the future. Thus, research on volatility based on Chinese market characteristics is increasingly important for risk management and portfolio optimization. IVS consists of a series of volatilities implied by options with different strike prices and maturities. Under the assumption of Black– Scholes (1973) model, IVS should be constant. In practice, however, it is not flat. It has the form of a curved surface and changes randomly over time. For example, Rubinstein (1994) and Jackwerth and Rubinstein (1996) propose that IVS has the shape of a "U" along the strike dimension, which is called an “implied volatility smile.” In the maturity dimension, implied volatility usually shows a monotonically increasing or decreasing trend, known as the implied volatility term structure. In the literature, the no-arbitrage condition plays an important role in modeling IVS. Previous studies model the economics of several stylized facts in a general equilibrium framework. For example, Alexander and Veronesi (2002) investigate investor uncertainty with respect to a regime-switching process, and find that the current state of the economy endogenously gives rise to the stochastic volatility.

⁎

Massimo and Timmermann (2003) conclude that investor learning generates asymmetric skews and systematic patterns in IVS. Due to diverse trading across the surface, different segments of IVS adjust themselves to new information with different speeds, causing the latent factors to exhibit autoregressive characteristics. The predictability of IVS is the result of investor uncertainty and learning, and indicates the existence of latent factors affecting the shape of IVS. Traders use several rules of thumb to manage the volatility surface, including the sticky strike, sticky delta, and square root of time rules, as described in Derman (1999). Daglish et al. (2007) analyze the no-arbitrage conditions for the dynamic of IVS derived from their risk-neutral process, and they examine whether the rules of thumb are consistent with the condition. These authors find that the relative sticky delta rule satisfies the no-arbitrage condition. Further, implied volatility evolves as time moves forward (Lee, 2004). It is a convention to use latent factors to represent the dynamic of the implied volatility or IVS. For instance, Chalamandaris and Tsekrekos (2010) examine whether modeling the time-varying properties can improve predictive ability. These authors show that a parsimonious VAR model can achieve a good fit of these properties, and that the VAR factor model shows superior predictive ability outside the sample relative to the benchmarks. Cont and da Fonseca (2002) model IVS as a function of several orthogonal random factors. These authors find that implied volatility exhibits positive autocorrelation and mean reversion, and are well approximated by the AR(1)/Ornstein– Uhlenbeck process. We model the dynamic features of IVS with a restricted dynamic factor model. We set the restrictive conditions to be the “smile” of

Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (S. Chen), [email protected] (Q. Tao), [email protected] (T. Zhang).

http://dx.doi.org/10.1016/j.econmod.2017.04.009 Received 26 November 2016; Received in revised form 15 February 2017; Accepted 12 April 2017 Available online 19 April 2017 0264-9993/ © 2017 Elsevier B.V. All rights reserved.

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implied volatilities and set the term structure similar to that used by Christoffersen et al. (2012) and Christoffersen et al. (2013). The early literature on implied volatility shows that implied volatility has a strong relation with the underlying asset. The deterministic volatility function (DVF) hypothesis is proposed by Derman and Kani (1994), Dupire (1994), and Rubinstein (1994). These authors develop various versions of DVF and attempt to explain the movement of different parts of IVS. They use a binary or trinomial lattice to obtain a more accurate fitting result. Dumas et al. (1998), however, examine Standard & Poor's 500 index options data using their DVF model and find that the result for DVF is no better than an ad hoc method. We establish a series of nested factor models based on an ad hoc method for IVS as in Goncalves and Guidolin (2006); this is a convenient way to model static IVS. We propose a series of conditions on the model specification. First, following the procedure of Goncalves and Guidolin (2006) and Christoffersen et al. (2013), we use a restricted factor model that decomposes the static IVS into three separate items to represent the implied volatility level, smile, and term structure with five factors. This method is simple but effective, and it reduces the difficulty of the estimation procedure for the dynamic factor model. Second, we assume the dynamics of IVS are mean-reverting and Markovian, and model the time series of the latent factors with an Ornstein–Uhlenbeck (OU) process, as in Cortazar et al. (2004). This is because the dynamic features of the OU process are compatible with our hypothesis. An alternative dynamic factor model of IVS is the VAR-type factor model proposed by Goncalves and Guidolin (2006). Following Diebold and Li (2006), they develop a two-step procedure for forecasts of IVS and show that forecasts obtained from this procedure are consistent. But theoretically, the two-step estimate method may be subject to an estimate bias. We use a state space model to capture the dynamics of IVS, and set the latent factors to be the OU processes. As a result, we obtain the optimal estimations of parameters using the Kalman filter algorithm. Empirical results show that our model performs better than the traditional IVS model in terms of fitting ability and prediction performance. The remainder of this paper proceeds as follows. Section 2 describes the data and method, summary statistics, and preliminary analysis. Section 3 provides details of our factor dynamic model and parameter estimation procedure. Section 4 reports model estimation results and comparative results of different models. Section 5 reports a robustness check for the comparison of models. We set forth our conclusions in Section 6.

Table 1 Descriptive statistics.

Volume (%)

τ≤60

60 < τ < 180

τ≥180

Total

6.68%

4.62%

2.87%

14.29%

0.3996 8.14%

0.3635 6.09%

0.3612 5.41%

0.380225 17.40%

0.3415 8.33%

0.3327 5.90%

0.3327 6.01%

0.335684 17.81%

0.3486 8.14%

0.3338 5.90%

0.3266 5.85%

0.336274 17.40%

0.3910 15.48%

0.4008 10.82%

0.3612 13.20%

0.384312 33.09%

0.4053 46.77%

0.3937 35.81%

0.3868 17.42%

0.394128 100.00%

0.3804

0.3690

0.3605

0.3670

m ≥ 0.03 Average IV Volume (%) 0.01 < m < 0.03 Average IV Volume (%) −0.01 < m < 0.01 Average IV Volume (%) −0.03 < m < − 0.01 Average IV Volume (%) m ≤ − 0.03 Average IV Volume (%) Total Average IV

underlying asset price and the SHIBOR 1Y rate as the risk-free interest rate. Since the SSE 50ETF does not pay dividends in the sample period, the dividend yield is zero. Implied volatility is obtained by backcalculating using the Black–Scholes option pricing model. 2.2. Preliminary analysis We subdivide our data sample into 15 groups according to moneyness and date to maturity (DTM), following Goncalves and Guidolin (2006). A call option is defined as DOTM (deep out-of-the-money) if m < −0.03, OTM (out-of-the-money) if −0.03 < m < −0.01, ATM (at the money) if −0.01 < m < 0.01, ITM (in-the-money) if 0.01 < m < 0.03, and DITM (deep in-the-money) if m > 0.03. We classify put options using the same rule, with m replaced by –m. Moreover, we classify DTM into three categories: short-term options if DTM < 60 days, medium-term options if 60 < DTM < 180, and long-term options if DTM > 180 days. Table 1 reports the grouping results. Regarding DTM, short, medium and long-term option trading volumes are 46.77%, 35.81%, and 17.42%, respectively. It is not surprising to note that short-term options have the highest volume, because they have high leverage, which is preferred by traders who want to hedge short-term risk or engage in speculative trading. Regarding moneyness, trading volume concentrates on negative moneyness, which means traders prefer outof-money options. To view the characteristics of IVS more intuitively, we put all sample data into a 3-D picture and plot an IVS by local quadratic interpolation method, as shown in Fig. 1 below. It is clear that all implied volatilities in each DTM have a “smile” feature, and they gradually fade away with DTM. These are classic features of IVS.

2. Data and preliminary analysis 2.1. Data We use a daily dataset of European options on the SSE 50ETF from the Shanghai Stock Exchange covering the period February 9, 2015 through February 5, 2016. The dataset consists of 244 trading days and 517 options contracts (including both call and put). To better show the real level and shape of IVS and reduce the influence of market noise, we filter the data by removing options that may be redundant, inactive, or that contain data errors, following the procedure of Barone-Adesi et al. (2008). First, we remove in-themoney options and keep only out-of-the-money options by the standard of moneyness (that is, K/F). This is because out-of-the-money options are more sensitive to volatility and are traded more actively. We obtain more accurate implied volatilities. Using both option types may create a data redundancy issue. Second, we eliminate options with remaining maturity less than seven calendar days, because these options have a liquidity problem and may contain noise. If a maturity date is approaching, a minor bias in the price of options may cause a substantial issue in the implied volatilities. After the data-filtering process described above, we obtain 4855 observations. We take the SSE 50ETF daily closing prices as the

3. Model and parameter estimation 3.1. Modeling approach IVS is formulated from all options trading in the market, with different maturities and strike prices, having a complex, three-dimensional dynamic change process. It is important to model the time series of the implied volatility of every option contract, but it is difficult to estimate such a high-dimension model. However, the variations in the implied volatilities are highly connected and influenced by several common factors. By identifying the main factors that affect variations in IVS, we establish stochastic models to depict their stochastic properties and relations with each other, thus modeling the procedure 296

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Fig. 1. IVS interpolated by local quadratic method.

the sticky delta rule, we obtain the following function using Taylor expansion rule1:

of IVS in an appropriate way. Such an approach can greatly reduce the dimensionality of IVS dynamic changes. The IVS dynamic factor model is expressed by the following mathematical equations:

σˆ(t , m, τ ) = g[t , m, τ , y1(t ), ..., yp(t )]

(1)

dyi (t ) = αidt + γdω i (t ) i = 1, … p i

(2)

ln σ (m, τ ) = β1 + β2m + β3m 2 + β4τ + β5mτ + β6τ 2 + ε

(3)

where, m represents moneyness and τ represents DTM; β1 represents the level of logarithmic implied volatility; β2 and β3 respectively represent the slope and curvature of implied volatility in the moneyness dimension; β4 and β6 respectively represent the slope and curvature of implied volatility in the DTM dimension; β5 refers to the mixed relation among implied volatility, moneyness, and time to maturity; and ε denotes a random disturbance term. To avoid producing a negative number, the dependent variable is set to logarithmic form. Eq. (3) above provides the general form of a cross-sectional model of IVS. We create a series of nested models according to Eq. (3), estimate parameters of the model separately, and make predictions out of the sample. After comparing these models’ sample fitting effect and prediction performance, we choose the following model as our crosssectional model of IVS:

where σˆ(t , m, τ ) denotes implied volatility on day t for an option with DTM τ, moneyness m. ɡ(·) depicts the relation between σˆ(t , m, τ ) and common factors m and τ, {yi }i =1,...p . is the coefficient and the latent factor that controls the shape of IVS. Eq. (2) describes the stochastic processes of the latent factors. αi denotes the drifting parameter of the factor yi , and γi denotes the volatility parameter. dωi(t ) is a standard Brownian motion that drives yi to fluctuate, and dωi(t )dωj (t ) = ρij dt , where ρij denotes the correlation coefficient between yi and yj . Our dynamic factor model consists of two equations. Eq. (1) confirms the relation between the implied volatility and common factors, and Eq. (2) models the dynamic change of the latent factors. There are three core steps: the relation between the implied volatility and common factors, the dynamic model of the latent factors, and the parameter estimation method.

ln σ (m, τ ) = β1 + β2m + β3m 2 + β4τ + β5mτ + ε

(4)

In Eq. (4), as in Eq. (3), β1 refers to the level of logarithmic implied volatility; β2 and β3 respectively represent the slope and curvature of implied volatility in the moneyness dimension; β4 represents the slope of implied volatility in the DTM dimension; β5 refers to the mixed relation among implied volatility, moneyness, and DTM; and ε denotes a random disturbance term.

3.2. Model specification According to our modeling approach, we first construct a factor model (or a cross-sectional model) that deciphers the daily relation between the implied volatility and latent factors. We then model the dynamic change of the latent factors. Finally, we connect these two models and form a state space model that successfully captures the dynamic changes in IVS.

3.2.2. Modeling time series of the latent factors At a fixed point in time, the coefficient β = (β1, β2 , β3, β4 , β5)′ in model (4) is certain. However, variations in IVS may lead to changes in these coefficients. According to Eq. (2), m and τ are both scalars, so the coefficients β are our latent factors, which control changes in IVS. We determine the patterns of IVS changes by examining the timevarying characteristics of the latent factors β. Goncalves and Guidolin (2006) adopt a VAR model to fit the β time series feature. However, we use an OU process to fit β, for two reasons. First, we assume the dynamic change of IVS to be mean-reverting and Markovian. The OU process is a stationary Gauss–Markov process, so using the OU process yields better fitting results and predictions of β. Second, the mathematical form of the OU process makes it easier for our dynamic factor model to be converted to a state space model. We then obtain optimal estimation results by applying a Kalman filter algorithm. Thus, we

3.2.1. Modeling the cross-section of IVS To reveal the cross-sectional properties of IVS, we choose factors found to exhibit more economic significance. Derman (1999) and Daglish et al. (2007) propose three commonly used rules to summarize the relation among implied volatility, strike price, and DTM: the sticky strike rule, the sticky delta rule, and the stationary square root of time rule. These rules describe static relations and have strong economic implications. The relation between implied volatility and moneyness is more significant and stable than that between implied volatility and strike price, so we build a cross-sectional model of IVS based on the sticky delta rule, which suggests that implied volatility is related to moneyness other than strike price. Eq. (1) depicts the static relations between implied volatility and the factors at a particular point in time. This can be viewed as a crosssectional model of IVS. Since the implied volatility is always in line with

1 See Goncalves and Guidolin (2006) for a detailed description of the cross-sectional model.

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where ε denotes white noise with variance of δ.

Table 2 Estimated results of the state space model. Parameters

Mean

St. Dev.

Parameters

Mean

St. Dev.

k1

16.0216

k2

5.4483

1.0959

γ4

1.9322

0.0941

3.2942

γ5

8.8428

k3 k4

26.6168 0.3153

1.1871

4.1305 3.2715

k5

13.3997

ρ12

0.0403

0.0616

5.6136

ρ13

−0.0761

0.0366

ρ14

−0.8481

0.0120

θ1

−1.8539

0.0935

ρ15

0.0283

0.1066

θ2

−2.7770

0.2086

ρ23

−0.0561

0.0288

θ3

−0.6942

0.5454

ρ24

0.0570

0.0772

θ4

−0.0233

0.0579

ρ25

−0.4971

0.0230

θ5

2.6525

0.3588

ρ34

0.0014

0.0327

ρ35

0.5492

0.0551

ρ45

0.0046

0.1398

δ

0.0053

0.0360

γ1

0.9340

0.0296

γ2

8.4487

0.3550

γ3

22.7842

1.5622

3.3.2. The state equation The state equation is used to describe the evolution of the state variables. We put them into the form of the OU processes to depict the mean-inverting and Markovian features of the factor. The following state equation is transformed from Eq. (6):

⎡ β (t ) ⎤ ⎡ θ (1 − e−k1Δt ) ⎤ ⎡ β (t − 1) ⎤ ⎡ ξ (t ) ⎤ ⎥ ⎡ −k1Δt ⎥ ⎢ 1 ⎥ ⎢ 1 ⎥ ⎢ 1 0 ⎤⎢ 1 e ⋮ ⋮ ⋮ = + ⋱ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥+ ⋮ ⎢ ⎢ − k Δt 5 ⎣ 0 ⎦⎢ β (t − 1)⎥ ⎢⎢ ξ (t )⎥⎥ e ⎢⎣ β5(t )⎥⎦ ⎢⎣ θ5(1 − e−k5Δt )⎥⎦ ⎣ 5 ⎦ ⎣ 5 ⎦

(10)

ξ(t i ) Fti−1~N (0, Q) ⎡ γγ ρ ⎤ i j ij (1 − e−(ki + kj )Δt )⎥ Q=⎢ ⎢⎣ ki + kj ⎥⎦ i, j =1,...,5 There are 26 parameters to estimate in the state space model.

Ψ = {k1, k 2, k 3, k4, k5, θ1, θ2, θ3, θ4, θ5, γ1, γ2, γ3, γ4, γ5, ρ12 , ρ13 , ρ14 , ρ15 , ρ23 , ρ24 , ρ25 , ρ34 , ρ35 , ρ45 , δ}

(11)

propose the following model to fit β:

dβi(t ) = ki(θi − β i (t ))dt + γdω i (t ) i

3.3.3. Maximum likelihood estimation The maximum likelihood estimation of the state space model is described by Harvey (1989). Let zt t−1 denote the conditional expectations of zt , and Ft denote the conditional variance of zt conditional on information available at time t−1. Then, zt t−1 and Ft can be obtained recursively by application of the Kalman filter. The prediction error decomposition of the log-likelihood function takes the form:

(5)

where, ki represents the revision rate of the ith factor, θi represents the long-run average, and γi represents the factor's volatility. The following is the closed-form solution of Eq. (5):

β (t ) = θ (1 − e−kΔt ) + e−kΔt ⋅β (t − 1) + γ

1 (1 − e−2kΔt ) ⋅z(t ) z(t )~N (0, 1) 2k (6)

l (z1, ...zt ; Ψ ) = l (z1; Ψ ) −

3.2.3. Dynamic factors system We already have the cross-sectional model of IVS, Eq. (4), and the time series model of the latent factors, Eq. (5). We then combine them to obtain the following dynamic factors system to depict properties of IVS:

⎧ ln σ (m, τ ) = β + β m + β m 2 + β τ + β mτ + ε 1 2 3 4 5 ⎪ ⎨ dβ i (t ) = ki(θi − βi(t ))dt + γdω i (t ) i ⎪ ⎪ dωi(t )ωj (t ) = ρij dt i , j = 1, ...5 ⎩

t =2

−

1 2

T

∑ ωt′ log Ft−1ωt t =2

where l (z1; Ψ ) is the unconditional log-likelihood function for zt and ωt = zt − zt t−1 is the tth prediction error, k is the dimension of zt , and Ψ summarizes the set of parameters to be estimated by maximizing (12). 4. Empirical results 4.1. Estimation results of the model

(7) Table 2 provides the main estimation results of the state space model. The estimated parameters are significant except for a few correlation coefficients close to zero. ki represents the revision rate of the ith factor. The fourth factor has a minimum value, meaning it has significant influence from its lagged terms. θi represents the factor's long-run average. The average level of factors 1 to 4 is negative, which reflects the factors’ influence on IVS. γi represents the volatility of a factor. The estimated value of the first factor is lowest, implying that this factor is the most stable. In addition, some correlation coefficients ρij are close to zero, indicating that these factors have a weak relation with each other, except for a strong negative relation between the first factor and the fourth, and the second with the fifth. Fig. 2 plots the estimated latent factors of the dynamic factor model. The first factor depicts the level of the volatility surface over time. This is because the other factor loadings are averaged in all cases. The second factor seems to be stationary across the sample period, though the value varies. This kind of variation is a special case for the dynamic factor model. Because of the identification restrictions that we place on the factors, the factor loadings have smaller initial entries in the second and third columns. Fig. 3 shows the term structure and smile results for IVS. Based on the factor loading results, the term structure is divided into six moneyness groups, and the smile is divided into four maturity groups. The restricted models do not allow large variations in the term

(8)

We now convert the model into a state space model and use the Kalman filtering algorithm for parameter estimation. This procedure eliminates the distortion problem of estimation results, which may occur in a “two-step” procedure. Meanwhile, the Kalman filter algorithm can also apply to the unbalanced panel data. 3.3. Parameter estimation 3.3.1. Measurement equation We take the cross-sectional model as the measurement equation. At time t, the number of observations is q, and the coefficient β is the latent process. Thus, the measurement equation can be expressed as follows:

⎡ ln σt (m1,τ1)⎤ ⎡1 m1 m12 τ1 m1τ1 ⎤⎤⎡ β1(t )⎤ ⎡ ε1(t ) ⎤ ⎥⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⋮ ⋮ ⎥⎥⎢ ⋮ ⎥ + ⎢ ⋮ ⎥ ⋮ ⎢ ⎥ = ⎢⋮ ⋮ ⋮ ⎥ 2 ⎢ ⎥ ⎢⎣ ln σt (mq,τq ) ⎥⎦ ⎣1 mq mq τq mqτq ⎦ ⎢⎣ βq(t ) ⎥⎦ ⎢⎣ εq(t )⎥⎦ ⎦

T

∑ log Ft

(12)

where the third equation models the relation of the latent factors. The equation system intuitively reflects the change process of IVS, and is easily converted to the state space model as follows:

⎧ z (t ) = H (t )x (t ) + ε (t ) ⎨ ⎩ x (t ) = c + Ax (t − 1) + ξ(t )

Tk 1 log(2π ) − 2 2

(9) 298

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Fig. 2. Estimated factors.

In the second step, we fit the time series of the factors β with a VAR model as follows:

structure and smile of IVS. This could explain the poorer performance of the restricted factor models. Finally, Fig. 4 plots the fitting results of our dynamic factor model. We create six figures from the 24 groups of data. The top two figures show the fitting result of the center of the surface, and the bottom four show the corners. Generally speaking, the fitting result in the center of the surface appears very good. But results in the corners appear somewhat problematic, as the residuals do not look like white noise. We note that problems fitting the corners of the surface may be more serious for the restricted factor model.

p

βˆ t =c +

∑ Φjβˆ t −j+ut

ut ~N (0, Ω ) (13)

j =1

We first set the initial lag order of the VAR to p=12. Then, the final lag order is set to p=4 by information criterion. In order to better show the estimation effectiveness of the state space model and VAR model, we also establish a random walk model, which is equivalent to setting the lag order of the VAR model to 1:

4.2. Comparison and analysis of models

βˆ t =c + Φ1βˆ t −1+ut

Let us examine whether our approach is effective by comparing our model with other models. Following Goncalves and Guidolin (2006) two-step estimation method, we first estimate model (4) and report the statistical results of the coefficients β in Table 3. The time series diagrams are shown in Fig. 5. We subject time series of β to ADF tests. The results show that factors β are all rejected unit root hypothesis at the 1% confidence level. Therefore, they are all stationary time series, implying that we are able to establish the time series model directly. As shown in Fig. 2, there is substantial volatility clustering for the five factors and they tend to comove together. Therefore, we calculate the correlation coefficients for each and for the autoregressive orders. Results are reported in Table 4 below. Comparing the contents of Table 4 with the estimation results of the state space model (Table 2), two important findings emerge. First, there are very strong correlations between β1 and β4 , and β2 β5. This is consistent with the analysis results reported in Section 4.1 and Fig. 2. Second, ki indicates the mean-reverting velocity parameter of the state factors. A smaller value for this parameter means that the corresponding state factor is affected by its lag more significantly. The values for k 2 and k5 are the two lowest, indicating that they are reverting to their mean very slowly, due to the large impact of their lag items. This characteristic is consistent with the state factor autoregressive orders reported in Table 4.

This model is called a “random walk model,” because βˆt equals βˆt−1 plus an independent and identical random disturbing term. Table 5 documents the dynamic features of the factors. All factors seem to be persistent, since the diagonal values of the variance matrix are very small. The off-diagonals are close to zero. There is a strong correlation between the innovations of the factors. For the VAR model, this correlation is −0.87 between the first and second factors, 0.63 between the first and third factors, and −0.59 between the second and third factors. To compare the fitting and forecast ability of the three models, we divide our data into two groups. The first group is used in-sample to fit the models and the second is used to forecast out of sample. The criteria used are RMSE and RRMSE. We conduct a one step ahead of the prediction, and obtain a predicted IVS each day. VAR model and random walk model are the benchmarks for comparison. Table 6 shows the results from comparing the three models by the standards of RMSE, RRMSE in the sample, and out of the sample. We note that the state space model has a better fitting result and predictive capability than the VAR model and random walk model, and its RMSE out of sample is slightly larger than that in the sample, by 3.5%. These results are more stable than the other two models. Note that the results for the random walk model are the worst among the three models, which means the latent factors of the IVS model are not random walk variables. The state space model and VAR models both

Fig. 3. Term structure and smile of IVS.

299

ut ~N (0, Ω )

(14)

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Fig. 4. Fit for dynamic factor model. Table 3 Statistics for common factors.

Table 4 Correlation coefficient and AR order of factors.

Factors

Mean

St. Dev.

Min.

Max.

βˆ1

−1.0263

0.3149

−1.6568

0.7172

βˆ2

−2.5546

4.1661

−18.8015

7.2283

βˆ3

8.9080

52.3744

−122.0369

βˆ4

−0.1564

0.3081

−1.6375

βˆ5

−0.8898

7.2355

−20.5567

31.6857

Adjusted β 2 RMSE

0.7455

0.0900

0.5028

0.9484

0.0288

0.0165

0.0050

0.0902

β1

β2

β3

β4

β1

1.0000

0.0357

−0.1728

−0.5813

0.1978

β2

0.0357

1.0000

−0.0758

0.0692

−0.3543

178.1783

β3

−0.1728

−0.0758

1.0000

0.0729

0.4474

0.7652

β4

−0.5813

0.0692

0.0729

1.0000

0.0287

β5

0.1978

−0.3543

0.4474

0.0287

1.0000

AR order

1

2

0

1

2

Fig. 5. Time series diagrams of the five state factors from February 9, 2015 through February 5, 2016.

300

β5

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Table 5 Factor dynamics of the two-step model. ф

μ

∑ ( × 10−4)

β1, t −1

β2, t −1

β3, t −1

β4, t −1

β1, t

β2, t

β3, t

β4, t

β1, t

0.196

0.995

0.028

−0.005

−0.001

1.059

−0.453

0.051

0.026

β2, t

0.003

−0.002

0.971

0.056

0.031

−0.453

0.212

−0.022

0.017

β3, t

0.008

0.001

0.001

0.967

0.991

0.051

−0.022

0.005

0.002

β4, t

0.005

0.001

0.001

−0.001

0.994

0.033

0.011

−0.001

0.001

6. Conclusions

Table 6 Comparison of the three models.

In sample Out of sample

RMSE RRMSE RMSE RRMSE

State Space Model

VAR Model

Random Walk Model

0.0713 6.71% 0.0738 7.03%

0.0745 7.60% 0.0844 8.67%

0.0813 8.27% 0.0901 9.32%

State Space Model

VAR Model

Random Walk Model

0.0649 6.12% 0.0670 6.41%

0.0698 7.11% 0.0796 8.13%

0.0822 8.38% 0.0895 9.24%

This paper develops a dynamic factor approach to model the IVS of the SSE 50ETF options. The model design assumes that the dynamic change of IVS is mean-reverting and Markovian. We model the dynamics of IVS by estimating a state space model in which the latent factors are set to be OU processes. Moreover, we take a cross-sectional model as the measurement equation to fit the IVS daily. The setting of the state space model allows us to obtain the optimal estimations of parameters based on historical information via Kalman filter algorithm. The empirical results show that our model is better in terms of both fitting ability and prediction performance. Further, a moving window approach improves our results.

Table 7 Results from the moving window approach.

In sample Out of sample

RMSE RRMSE RMSE RRMSE

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capture the variation features. The VAR model is dominant in comparison with the random walk model. In sum, the state space model is more suitable to model the features and dynamics of IVS. 5. Robustness This section presents additional results on the performance of three models inside and outside the sample to check the robustness of our previous results. In previous sections, we divide the sample period into two subperiods: the first is used to obtain the parameters of three models and the second for checking the predictive abilities. Now, we use a moving window approach for a robustness check. We first choose window sizes for the models by using the RMSE as our loss-function— the smaller the RMSE the better. We then compute the RMSE for different windows sized between 2 and 100 days. We choose 71 days for the state space model, 43 days for the VAR model, and 11 days for the random walk model. We then fit models and forecast the next day's IVS by the window sample every day. Table 7 presents the results from this approach. Note that the results from this moving window approach support the previous conclusion; that is, the state space model is superior to the VAR model and the VAR model is superior to the random walk model. Further, this approach improves the results from the state space model (by about 9%) and the VAR model (by about 6%), but not the random walk model, comparing to the previous results. We believe this is due to the IVS fluctuating for a certain time, as shown from the time series of the five state factors in Fig. 5.

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