The information content of risk-neutral skewness for volatility forecasting
Introduction
Volatility is an important determinant in financial assets, and modeling and forecasting of volatility process in financial markets, such as exchange market, stock market, and bond market, have been investigated over the last two decades. Since volatility is useful information for assessing the degree and the risk of future payoffs, many extensive studies have reflected the importance of volatility for asset pricing, investment, risk management, and monetary policy making.
Earlier studies find that implied volatility has predictive power for future volatility. For example, Latane and Rendleman (1976) discover the forecasting power of weighted average of Black and Scholes (B-S) call option implied volatilities. In particular, their empirical evidence shows that the weighted average of B-S call option implied volatilities is generally a better predictor of future volatility than volatility based on the historical return data. Additionally, Jiang and Tian (2005) suggest the model-free implied volatility which does not depend on the option pricing models. Their model-free implied volatility subsumes the information from B-S implied volatility and historical volatility and is a more efficient forecast for future volatility.
In contrast to the volatility forecasting literature, much of the asset pricing literature focus on another measure extracted from option prices. Bakshi et al. (2003) show that the risk-neutral skewness can be expressed in terms of option prices. Based on the risk-neutral skewness of Bakshi et al. (2003), Dennis and Mayhew (2002) describe a negative relation between the risk-neutral skewness and systematic risk, beta, and argue that market risk is reflected in the risk-neutral skewness extracted from the option prices. Similarly, Doran et al. (2007) report evidence supporting the hypothesis that the risk-neutral skewness has strong predictive power in short-term decline of the stock market. They find that the risk premium of negative jump is embedded in the risk-neutral skewness. However, Rehman and Vilkov (2012) show that the risk-neutral skewness is positively related to future stock return. This contrasts with the empirical evidence supported by Dennis and Mayhew (2002) and Doran et al. (2007).2
Under the condition of nonnormal return innovation, Christoffersen et al. (2010) provide the valuation of European-style contingent claims and the wedge between one-day-ahead physical and risk-neutral conditional variances. Based on the setting outlined in Christoffersen et al. (2010), we can provide the wedge between the physical conditional variance and the risk-neutral skewness and find that the risk-neutral skewness improves the predictive power for future volatility. Our contribution lies in identifying the relation between physical conditional variance and the risk-neutral skewness and adopting the risk-neutral skewness in the volatility forecasting model.
Despite the usefulness of the risk-neutral skewness, there is currently no research for the volatility forecasting model incorporating the risk-neutral skewness.3 Our new Heterogeneous Autoregressive model of Realized Volatility, Implied Volatility and Skewness (HAR-RV-IV-SK) incorporating the risk-neutral skewness builds directly on the Heterogeneous Autoregressive model of Realized Volatility (HAR-RV). Among the recent literature in the volatility forecasting models, Corsi (2009) propose the HAR-RV model. In the HAR-RV model, the realized volatility is parameterized as a linear function of past realized volatilities over different horizons to capture the empirical memory persistence of volatility. In spite of not formally belonging to the class of long memory models, the HAR-RV model successfully achieves the purpose of modeling the long memory behavior of volatility in a simple and parsimonious way. Based on the HAR-RV model, Fradkin (2008) and Busch et al. (2011) suggest the Heterogeneous Autoregressive model of Realized Volatility and Implied Volatility (HAR-RV-IV), in which the implied volatility is utilized as an additional predictor of future volatility. Since the implied volatility provides incremental information to forecast future volatility, the HAR-RV-IV model improves forecasting accuracy. In this paper, we apply the risk-neutral skewness directly as additional explanatory variable to the HAR-RV model, for the enhancement of forecasting ability in the volatility forecasting model. Our empirical results confirm the advancement in volatility forecasting by the adoption of the risk-neutral skewness.
We perform robustness checks, changing from past realized volatility to the other forecasting variables, such as the continuous component and the jump component of volatility, the realized absolute value, and adding past negative returns to consider the leverage effect on the risk-neutral skewness, and continue to find an important role of the risk-neutral skewness in forecasting asset returns volatility. The information of the risk-neutral skewness seems to complement the information extracted from the implied volatility and the historical return, such that the t-statistic of the coefficient estimates on the risk-neutral skewness is held up in three regressions of the robustness checks.
The rest of the paper is organized as follows. Section 2 derives a theoretical relation between the physical conditional variance and the higher-order moments of the risk-neutral distribution to secure the legitimacy for the addition of the risk-neutral skewness. Section 3 sets up the notation of volatility measurement and the specification of the econometric models. Section 4 describes the data set and empirical results obtained. Section 5 describes the application to the other models for robust check. Section 6 concludes out study.
Section snippets
The relation between the physical volatility and the risk-neutral skewness
Since our intention is to represent the relation between the physical volatility and the risk-neutral skewness, it is possible to adopt the setting and the procedure described in Christoffersen et al. (2010).4 Let P denote the physical distribution of the states of nature. The dynamics of the stock price is described by the process {St}t = 0T. The information structure is given
Volatility measurement and model specification
In the first subsection, we review recent development in volatility measurement. Volatility measurement is important for volatility forecasting, because our purpose is mainly to compare the performance of the volatility forecasting models and improve the accuracy of the forecasts of future volatility. In addition, the decomposition of the realized volatility into a continuous component and a jump component of volatility is described, due to the assessment of the effect of the continuous
Empirical analysis
We analyze the volatility of S&P 500 index which is based on five-minute intra-daily returns for the evaluation of the predictive power of the risk-neutral skewness. Firstly, we describe the specification of S&P 500 index and S&P 500 index option prices and the filtering rule to ensure the confidence in the empirical results. After that, we measure the effect of the risk-neutral skewness on future volatility and the continuous component and the jump component of future volatility from the
Robustness check
The robustness checks in this paper are split into three sections. Firstly, we replace the realized volatility with the continuous component and the jump component of past realized volatilities. Secondly, we use the sum of absolute intra-daily returns, the realized absolute value, instead of the realized volatility. Finally, we add past negative returns to consider the leverage effect on the risk-neutral skewness.
Conclusion
This paper provides the theoretical relationship between the physical conditional variance and the higher-order moments of the risk-neutral distribution obtained from option prices. We derive this theoretical relationship from the wedge suggested in Christoffersen et al. (2010). Christoffersen et al. (2010) provide the wedge between one-day-ahead risk-neutral conditional variance and conditional moments of the physical distribution. Based on the theoretical relation, the risk-neutral skewness
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