Pricing Kernels with Stochastic Skewness and Volatility Risk

Abstract

In this online appendix, I first provide a brief comparison of the small noise expansion series and the Taylor expansion series. I then use separably Taylor expansion series and the small noise expansion series to derive the pricing kernel in a one-market model with two dates, and the pricing kernel in a two-market model with three dates. I thereafter, investigate whether one obtains a pricing kernel that depends on the market co-skewness and the market volatility in a long run risk model and also in a model where the representative investor chooses his optimal allocation in the presence of the market equity and the variance swap contracts. The remaining part of the appendix contains additional tables and figures used in the paper.

How Does the Market Variance Risk Premium Vary Over Time? Evidence from S&P 500 Variance Swap Investment Returns

“…Previous studies measure VRP by employing synthetic VS rates (e.g., Bollerslev et al, 2009, 2012, Carr and Wu, 2009, Fan et al, 2013, Neumann and Skiadopoulos, 2013. 4 In line with the theoretical results of 2 We distinguish between stock market conditions and economic conditions in line with anecdotal evidence which suggests that the state of the stock market and that of the economy may be disconnected (e.g., a booming stock market may coincide with a poor economic state).…”

“…A number of models predict that VRP should be driven by factors related to these four conditions (e.g., Bakshi and Madan, 2006, Eraker, 2008, Bollerslev et al, 2009, Bekaert and Engstrom, 2010, Drechsler and Yaron, 2011, Chabi-Yo, 2012, and Feunou et al, 2013. Table 4 provides a list of the drivers of VRP and the way they affect it.…”

“…Eraker (2008), Bollerslev et al (2009), Bekaert and Engstrom (2010) and Drechsler and Yaron (2011) models imply that stock and macroeconomic factors correlated with the volatility and the volatility of volatility of the aggregate consumption growth should also predict VRP. Furthermore, Bakshi and Madan (2006), Chabi-Yo (2012) and Feunou et al (2013) models predict that VRP is expected to be predicted by factors nested within the (1) -(4) setting.…”

“…Identifying the predictors of VRP's time variation enhances our understanding of the predictability of the total equity risk premium which includes the equity risk premium (arising from continuous fluctuations in the price of the risky asset) and VRP (Bollerslev et al, 2009, Chabi-Yo, 2012. Exploring whether the market VRP is predictable is also of importance to market participants who trade variance.…”

How does the market variance risk premium vary over time? Evidence from S&P 500 variance swap investment returns

“…Eraker (2008), Bollerslev et al (2009), Bekaert and Engstrom (2010) and Drechsler and Yaron (2011) models imply that stock and macroeconomic factors correlated with the volatility and the volatility of volatility of the aggregate consumption growth should also predict VRP. Furthermore, Bakshi and Madan (2006), Chabi-Yo (2012) and Feunou et al (2013) models predict that VRP is expected to be predicted by factors nested within the (1) -(4) setting. 2 In addition to exploring the predictability of the market VRP, we make three more contributions.…”