Volatility risk premium (VRP) is the premium that compensates risk stemming from the fluctuation in volatility or jumps. In financial markets, this risk premium is commonly viewed as the price that option market makers require to provide liquidity and investors pay to hedge their tail risk. Although the abundant evidence has linked the VRP to liquidity, intermediation, and hedging demand, much less is known about the impact of an imbalance between supply and demand for options on VRP. In addition, higher volatility often leads to increased VRP. Inversely, large VRP attracts volatility investors that seek to benefit from the temporary mispricing in volatility. This gives a rise to an interesting but less understood question is about what happens afterward, in particular how a widened VRP may affect subsequent volatility. This dissertation therefore sets out to focus on two important VRP issues in financial market, including the impact of option demand pressure on VRP and the effect of trading the VRP.
In the first issue regarding the impact of option demand pressure on VRP, the results show that the level of demand for an index option plays a key role in determining the time variation in VRP. A positive (negative) demand pressure of an index option raises (decreases) the VRP, similar to the finding of Gârleanu, Pedersen, and Poteshman (2009) that a proportion of an option’s expensiveness reflects the effect of demand pressure. This indicates that the option prices include a component that compensates market-makers’ risk since market makers can not perfectly hedge their net exposure on the option positions.
In particular, the demand pressure effect on VRP is related to the risk aversion of market-makers supported by a significant and negative linkage between the effect of demand pressure and recent market-maker losses. Facing their trading losses, market makers with risk aversion ask a higher risk premium for accepting additional risk. Thus, these premiums for unhedgeable risks are all contributing more, thereby leading an increase in VRP. In addition, at the arrival of market jumps the demand pressure leads to a greater
impact on VRP for all three demand variables due to increased jump fear. The result provides evidence to support the finding of Todorov (2010) that time-varying risk aversion is driven by large, or extreme, market moves.
The second issue in this dissertation is to investigate the dynamic processes between VRP and volatility while focusing on the afterward effect of a large VRP. The bidirectional causality in the OLS regressions and the linear and nonlinear Granger causality tests are documented. This result confirms the findings in literature (Bakshi & Kapadia, 2003;
Bollerslev & Todorov, 2011; Eraker et al., 2003; Todorov, 2010) that uncertainty in volatility raises the VRP, and supports the contention that the feedback effect of VRP positively Granger causes the subsequent volatility. This finding suggests that VRP plays an important role in explaining future realized volatility: a large volatility premium could lead to greater realized volatility.
The feedback effect that the VRP nonlinearly Granger causes the three volatility components, continuous volatility, negative jump volatility, and positive jump volatility, is significant even after controlling for the higher volatility attributed to the unexpected information shocks.
In conclusion, this dissertation provides some insights into the issues of the impact of option demand on VRP and the effect of trading the VRP. The research results would provide us with empirical evidences to comprehend the importance of option demand pressure in determining VRP and the dynamic influence between volatility and volatility trading.
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APPENDIX
Appendix A: Decomposition of Realized Variance
This appendix presents the processes by which we decompose the realized volatility into three volatility components: continuous volatility, positive jump volatility, and negative jump volatility. Assume that dp(t) follows the general stochastic volatility jump diffusion process
( ) ( ) ( ) ( ) ( ) ( )
dp t =µ t dt+σ t dW t +κ t dq t , t≥0. p(t) denotes the logarithmic asset price at time t;
µ(t) is the instantaneous drift process; σ(t) is instantaneous volatility; W(t) is a Brownian motion process; κ(t) is the random jump size; and q(t) is a jump process with intensity λ(t).
Following Andersen and Bollerslev (1998) and Andersen, Benzoni, and Lund (2002), the realized variance (RV2) over the day t is defined as the sum of the squared intraday returns,
1
observed prices during the period t that issampled during the intra-period (∆).
As m↑∞, RV converges in probability to two different components, that is, the t2
integrated variance (the variation attributable to continuous process, simplified as CV ) t2
and the sum of squared jumps (the variation due to price jumps, simplified asJV ). It is t2
(BV) presented in Equation (A3), following Barndorff-Nielsen and Shephard (2004 and 2006).
According to Barndorff-Nielsen and Shephard (2004) and Barndorff-Nielsen, Shephard, and Winkel (2006), the asymptotic convergence of BV only captures the continuous price variations even in the presence of jumps. For m↑∞, BV converges in probability to integrated volatility in Equation (A2). We thus estimate the contribution of jump to the realized variance by differencing RV2 with BV.
, , 2
This study is required to detect the arrival jumps up to intraday level. The significant intraday price jumps are identified using the nonparametric test proposed by Lee and
In the absence of jumps, Lee and Mykland (2008) address a reasonable rejection region by deriving the limiting distribution of the maximum of the statistic. This process guides us to choose the relevant threshold for the test to distinguish the presence of jumps at any testing time. The statistic is given asζ =(Lt −Cn) /Sn, where
2 log / (log log(log )) /(2 2 log )
Cn = n c− π + n c n ,c= 2 /π , and Sn =1/(c 2 log )n .
The cumulative distribution function of ζ is given as (P ζ ≤x)=exp(−e−x). Given any significance level, we can solve for x to determine the threshold for significant jumps. For example, the corresponding threshold, rejecting the null hypothesis of no jumps, is 4.60 (2.97) at 1% (5%) significance level.
Based on a significance level α, the size of the jump on day t is denoted as
1
2 2
{ }( ),
t t t t
JV I RV BV
ζ >Φ−α
= − (A5)
where I(.) is the indicator function; Φ1-αis the critical value for the (1–α) level test; and ζ is the statistic of detected jumps.
Obviously, JV2 is the excess realized variance over the continuous variance. It is zero in the absence of jumps and greater than zero otherwise. Further, jump variance is split into negative jump variance (nJV ) and positive jump variance (2 pJV ), depending on the 2 cumulative returns that correspond to the price jumps within the one-day period. If the cumulative return is negative (positive), then it is identified as a negative (positive) jump variance. By contrast, the variation contributed by continuous price process is written as
2 2 2
t t t
CV =RV −JV (A6)
In our empirical work, this study estimates JV2 using α=0.99 and computes RV2 using the 5-minute returns. In addition, following Lee and Mykland (2008), the jump test statistic L is calculated using the past 270 5-minute intraday returns.
Appendix B: The Modified Baek and Brock Test
This appendix details the modified Baek and Brock (1992)’s nonlinear Granger causality test, proposed by Hiemstra and Jones (1994). Baek and Brock (1992) developed a nonparametric statistical method for detecting nonlinear causal relationships. The nonlinear causality between time series is detected by using the correlation integral approach.
Consider two strictly stationary and weakly dependent time series Xt and Yt. LetX denote tm
the m-length lead vector of Xt, andXt LxLx− and Yt Ly−Ly are the Lx-length and Ly-length lag m-length lead vectors of Xt within a distance d of each other, given that two corresponding Lx-length lag vectors of Xt and two Ly-length lag vectors of Yt are within distance d of each other. The probability on the right-hand side of Equation (B1) is the conditional probability that two arbitrary m-length lead vectors of Xt are within a distance d of each other, conditional only on that their corresponding Lx-length lag vectors are within distance d of each other.
The test based on Equation (B1) can be restated by expressing the conditional probability in terms of the corresponding ratios of joint probabilities:
1( , , ) 3( , ) used to test the condition in Equation (B2). The correlation integral, an estimator of spatial dependence across time, is defined as a proportion of the number of observations within the distance d of each other to the total number of observations. These correlation-integral
Assume that Xt and Yt are strictly stationary, weakly dependent, and satisfy the mixing conditions as specified in Denker and Keller (1983). Under the null hypothesis that Yt does not strictly Granger cause Xt, the test statistic G is asymptotically normally distributed, according to Hiemstra and Jones (1994). That is,
1( , , ) 3( , ) 1 2