Inferring information from the S&P 500, CBOE VIX, and CBOE SKEW indices

• Author: Wenjun Zhang,Xinfeng Ruan,Jiling Cao
• DOI: http://doi.org/10.1002/fut.22093
• Published date: 01 June 2020
• Date: 01 June 2020

J Futures Markets. 2020;40:945–973. wileyonlinelibrary.com/journal/fut © 2020 Wiley Periodicals, Inc.

|

945

Received: 8 September 2019

|

Accepted: 31 December 2019

DOI: 10.1002/fut.22093

RESEARCH ARTICLE

Inferring information from the S&P 500, CBOE VIX, and

CBOE SKEW indices

Jiling Cao

1

|

Xinfeng Ruan

2

|

Wenjun Zhang

1

1

Department of Mathematical Sciences,

School of Engineering, Computer and

Mathematical Sciences, Auckland

University of Technology, Auckland, New

Zealand

2

Department of Accountancy and

Finance, Otago Business School,

University of Otago, Dunedin, New

Zealand

Correspondence

Xinfeng Ruan, Department of

Accountancy and Finance, Otago

Business School, University of Otago,

Dunedin 9054, New Zealand.

Email: xinfeng.ruan@otago.ac.nz

Abstract

This paper compares the information extracted from the S&P 500, CBOE VIX, and

CBOE SKEW indices for the S&P 500 index option pricing. Based on our empirical

analysis, VIX is a very informative index for option prices. Whether adding the

SKEW or the VIX term structure can improve the option pricing performance

depends on the model we choose. Roughly speaking, the VIX term structure is

informative for some models, while the SKEW is very noisy and does not contain

much important information for option prices. This paper also extends Zhang et al.

(2017, J Futures Markets,37,211–237) into three typical affine models.

KEYWORDS

affine model, CBOE SKEW, CBOE VIX, MCMC, option pricing, SPX, term structure

JEL CLASSIFICATION

G12; G31

1

|

INTRODUCTION

The Chicago Board Options Exchange (CBOE) introduced the old CBOE Volatility Index (VXO) in 1993 to measure the

market's expectation of 30‐day volatility implied by S&P 100 Index option prices. In 2003, the CBOE launched the new

CBOE Volatility Index (VIX) by supplying a script for replicating volatility exposure with a portfolio of the S&P 500

Index (SPX) options. The new method estimates expected volatility by averaging the weighted prices of SPX puts and

calls over a wide range of strike prices. It is colloquially referred to as the fear index or the fear gauge. Addition to the

VIX, the CBOE also published the 9‐Day Volatility Index (VIX9D), the 3‐Month Volatility Index (VIX3M), and the 6‐

Month Volatility Index (VIX6M) on the SPX Index.

Documented by Jiang and Tian (2005), the model‐free volatility subsumes all information contained in the

Black–Scholes implied volatility and past realized volatility and it is a more efficient forecast for the future realized

volatility. Lin (2007) implements a generalized method of moments (GMM) to estimate the model parameters in both

physical and risk‐Řneutral probability measures by using the VIX and 5‐min‐based integrated volatilities and

documents that the VIX is very informative for VIX futures prices. To avoid the computational burden associated with

option valuation, Duan and Yeh (2010) obtain the model parameters and the latent stochastic volatility from the

maximum likelihood estimates (MLEs) under a jump‐diffusion model. This takes the advantage that the VIX is a linear

function of the latent stochastic volatility, so that it is possible to obtain the joint likelihood function of the SPX return

and the VIX. Furthermore, Kaeck and Alexander (2012), Yang and Kanniainen (2017), and Zhu and Lian (2012) use the

Markov chain Monte Carlo (MCMC) method to simultaneously estimate the model parameters in both physical and

risk‐neutral probability measures and the latent variables using the SPX and VIX data. Yang and Kanniainen (2017)

and Zhu and Lian (2012) infer information from the SPX and 30‐day VIX indices under an affine jump‐diffusion model

and non‐affine Lévy model, respectively, while Kaeck and Alexander (2012) extract the information from the SPX,

30‐day VIX and 360‐day VIX indices. All of the above studies investigate pricing performance under the different

continuous‐time models calibrated by using the same data set. In contrast to them, we explore option pricing

performance for the models calibrated by using the different data sets. Our studies raise an important question of

whether adding a new index can really improve the option pricing performance.

To capture the curve of implied volatilities with a shape of “smirk”or “skew,”implied by the SPX option prices, the

CBOE launched the CBOE Skew Index (SKEW) in February 2011. The index value typically reflects the trading activity

of portfolio managers hedging tail risk with options, to protect portfolios from a large, sudden decline in the market

(i.e., a black swan event or market crash). Similar to the VIX, the SKEW measures the perceived tail risk of the

distribution of SPX returns over a 30‐day horizon by using the model‐free method in Bakshi, Kapadia, and Madan

(2003). Zhang, Zhen, Sun, and Zhao (2017) provide an exact formula for the skewness of stock returns implied in the

Heston (1993) model and separately use the SPX return, the CBOE VIX, and SKEW term structures to calibrate the

model. Unfortunately, they do not study the option pricing performance after adding SKEW information. Recently, Liu

and van der Heijden (2016) find that the estimation errors of true skewness by using the CBOE SKEW method are very

large. In this paper, we further investigate whether adding the SKEW can improve models' option pricing performance.

In this paper, we consider three typical models. The first model is the stochastic volatility model with contemporaneous

jumps in returns and volatility (SVCJ), which is the most popular affine model in the literature, for example, Bakshi, Cao,

and Chen (1997); Broadie, Chernov, and Johannes (2007); Da Fonseca and Ignatieva (2019); Duan and Yeh (2010); Eraker

(2004); Eraker, Johannes, and Polson (2003); Kaeck, Rodrigues, and Seeger (2017); Lin and Chang (2010); Neuberger (2012);

Neumann, Prokopczuk, and Simen (2016); Ruan and Zhang (2018); Zhu and Lian (2011, 2012); and others. Bakshi et al.

(1997), Broadie et al. (2007), Eraker (2004), and Neumann et al. (2016) document that the SVCJ model is good enough to fit

options and returns data simultaneously. According to the empirical observation in Bates (2006), that is, more jumps occur

during more volatile periods, we adopt the second model from Aït‐Sahalia, Karaman, and Mancini (2015) and Bates (2006).

The second model, the SVCJ model with stochastic jump intensity, is labeled as “SCVJI.”The jump intensity is a linear

function of the spot variance. The SCVJI model is very popular in asset pricing, for example, Drechsler (2013), Drechsler and

Yaron (2010), Eraker and Shaliastovich (2008), and Jin (2014). Recently, Du and Luo (2019) find that jump propagation (i.e.,

the phenomenon in which the strike of one jump substantially raises the probability for more to follow) dominates the jump

risks from the joint time series of SPX and its options and use a two‐factor Hawkes jump‐diffusion model to capture jump

propagation. Therefore, the last model we consider is the SVCJ model with Hawkes process (SVCJH). Du and Luo (2019)

further document that the SVCJH can explain well the pronounced smirk pattern in option implied volatility, which is

modeled as the SKEW in this paper.

As Kaeck and Alexander (2012) have documented that the SVCJ model with the stochastic long‐term level in the

volatility (i.e., two‐factor SVCJ model, labeled as J2‐A2 in Kaeck & Alexander, 2012) has larger out‐of‐sample option

pricing errors than the SVCJ model (labeled as J2‐A1 in Kaeck & Alexander, 2012), in this paper, we do not consider the

two‐factor SVCJ model. In Kaeck and Alexander (2012) and Yang and Kanniainen (2017), even though the models are

non‐affine, the close‐form solution of the VIX can be obtained. However, it is difficult to get an explicit SKEW formula

under the non‐affine models, as the SKEW formula relies on the moment generating function (MGF) of the log SPX

return, which can be derived explicitly under the affine models. In order to get a close‐form solution for the SKEW, we

focus on the affine models in Duffie, Pan, and Singleton (2000) rather than the non‐affine models.

In line with Kaeck and Alexander (2012), Yang and Kanniainen (2017), and Zhu and Lian (2012), we use the MCMC

method to calibrate the models as the MCMC method has sampling properties superior to other methods documented in the

literature. Comparing with the MLEs in Duan and Yeh (2010), the MCMC method can handle more complex data, which are

no longer a linear function of latent variables, like option data in Eraker (2004) and SKEW data in this paper. It allows us

directly get the latent variables through the MCMC procedure. Furthermore, Jacquier, Polson, and Rossi (2002) show that the

MCMC method works better than the MLEs in terms of estimating the parameter of stochastic volatility models.

This paper is the first to investigate whether adding the VIX term structure and SKEW data to a model can improve model's

option pricing performance, compared with Duan and Yeh (2010), Kaeck and Alexander (2012), Lin (2007), Yang and

Kanniainen (2017), and Zhu and Lian (2012). The answer depends on the model we choose. Roughly speaking, the VIX is a

very important index for option prices, while the SKEW is very noisy for option prices. Adding the VIX term structure

information to a model can improve the option pricing performance only for some models. This paper also extends the Heston

(1993) model for the CBOE SKEW in Zhang et al. (2017) and further confirms the observation in Liu and van der Heijden

(2016), that is, the estimation errors of true skewness by using the CBOE SKEW method are very large.

The remainder of our article is organized as follows. Section 2 presents the framework. Section 3 introduces the data.

The details of the model calibration are shown in Section 4 and the empirical results are given in Section 5. Section 6

946

|

CAO ET AL.