### Citations

5052 |
Pricing of Options and Corporate Liabilities
- Black, Scholes
- 1973
(Show Context)
Citation Context ...h are marginally significant. Estimates of the loss for call options range from -0.57% of the index value for in-the-money VIX calls to -1.41% for out-of-the-money calls. The negative average returns on index and VIX options directly suggest that the market prices of volatility and volatility-of-volatility risks are negative. We then show that the cross-sectional spreads in average option returns are significantly related to the volatility and volatility-of-volatility risks. In lieu of calculating exact model betas, we compute proxies for the option exposures to the underlying risks using the Black and Scholes (1973) vega and volga. Vega represents an increase in the Black-Scholes value of the option as the implied volatility increases by 1%, and thus provides an estimate for the exposure of equity options to volatility risks, and of VIX options to volatility-of-volatility risks. Volga is the second partial derivative of the option price with respect to the volatility, which we use to measure the sensitivity of the index option price to the volatility-of-volatility risks. Vega and volga vary intuitively with the moneyness of the option in the cross-section, and help us proxy for the betas of the options t... |

1978 | A theory of the term structure of interest rates - Cox, Ingersoll, et al. - 1985 |

1799 | A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix," - Newey, West - 1987 |

1540 | Risk, return and equilibrium: empirical tests
- Fama, MacBeth
- 1973
(Show Context)
Citation Context ...options’ volatility and volatility-of-volatility betas. Panel A shows univariate and multivariate regressions of delta-hedged S&P500 option gains scaled by the index on the sensitivities of the options to volatility and volatility-of-volatility risks. Our encompassing regression for delta-hedged S&P500 options is: GAINSit,t+τ = Πit,t+τ St = λ1V EGA i t + λ2V OLGA i t + γt + i t,t+τ . (4.4) 17 Since each date includes multiple options, as in Bakshi and Kapadia (2003) we allow for a date-specific component in Πit,t+1 due to the option expirations. 10 Conceptually, our approach is related to Fama and MacBeth (1973) regressions. Instead of estimating risk betas in the first stage, due to the non-linear structure of option returns, we measure the our exposures from economically motivated proxies for the risk sensitivities.11 The results in Panel A show that both volatility and the volatility of volatility are priced in the cross-section of delta-hedged S&P500 option returns. Options more exposed to volatility and volatility-of-volatility risks have more negative expected returns. The univariate estimates for vega and volga are -0.051 and -0.007. Both t-statistics are highly significant at conventional lev... |

1511 | A closed-form solution for options with stochastic volatility with applications to bonds and currency options.
- Heston
- 1993
(Show Context)
Citation Context ...gression, and is largely unchanged in the multivariate regression. A one standard deviation increase in the current value of V V IX2 is associated with more than one-third standard deviation increase in next month realized variance of VIX. The empirical evidence suggests that fluctuations in the volatility of volatility are not directly related to the level of the volatility itself. This is consistent with our two-volatility model specification in Section 2. In many reduced form and structural models, the volatility of volatility is directly linked to the level of the volatility. For example, Heston (1993) models volatility as following a Cox, Ingersoll, and Ross (1985) square-root process. In that case, the level of volatility itself should forecast future realized volatility of volatility. The evidence in the data does not support this assumption, and calls for a richer dynamics of the volatility process, with separate movements in the volatility of volatility. 4 Evidence from Options In this section, we analyze the implications of equity and VIX option price dynamics for the pricing of volatility and volatility-of-volatility risks in the data. Our economic model suggests that the market pric... |

761 | Risks For the Long Run: A Potential Resolution of Asset Pricing Puzzles.
- Bansal, Yaron
- 2004
(Show Context)
Citation Context ... asset. As argued in Bakshi and Kapadia (2003), delta-hedged option payoffs are very useful to study volatility-related risks as they most cleanly isolate the exposures to volatility risks.3 Indeed, under a standard linear risk premium assumption, we show that the expected payoff on the delta-hedged position in equity index options consists of the risk compensations for both volatility and volatility-ofvolatility risks. For volatility options, the expected payoffs only involve the compensation for volatility-of-volatility risks. The risk compensations are given by the product of the 1See e.g. Bansal and Yaron (2004), Bloom (2009), Bansal, Kiku, Shaliastovich, and Yaron (2013), Fernandez-Villaverde and Rubio-Ramırez (2013) for the discussion of macroeconomic volatility risks, and Coval and Shumway (2001), Bakshi and Kapadia (2003), Campbell, Giglio, Polk, and Turley (2012) for market volatility risks. 2We use the terms “variance risk” and ”volatility risk” interchangeably unless otherwise specified. 3For example, unlike delta-hedged positions, zero-beta straddles analyzed in Coval and Shumway (2001) are not dynamically rebalanced and may contain a significant time-decay option premium component. 1 market... |

710 | Transform analysis and asset pricing for affine jump diffusions, - Duffie, Pan, et al. - 2000 |

440 | The Impact of Uncertainty Shocks,”
- Bloom
- 2007
(Show Context)
Citation Context ...hi and Kapadia (2003), delta-hedged option payoffs are very useful to study volatility-related risks as they most cleanly isolate the exposures to volatility risks.3 Indeed, under a standard linear risk premium assumption, we show that the expected payoff on the delta-hedged position in equity index options consists of the risk compensations for both volatility and volatility-ofvolatility risks. For volatility options, the expected payoffs only involve the compensation for volatility-of-volatility risks. The risk compensations are given by the product of the 1See e.g. Bansal and Yaron (2004), Bloom (2009), Bansal, Kiku, Shaliastovich, and Yaron (2013), Fernandez-Villaverde and Rubio-Ramırez (2013) for the discussion of macroeconomic volatility risks, and Coval and Shumway (2001), Bakshi and Kapadia (2003), Campbell, Giglio, Polk, and Turley (2012) for market volatility risks. 2We use the terms “variance risk” and ”volatility risk” interchangeably unless otherwise specified. 3For example, unlike delta-hedged positions, zero-beta straddles analyzed in Coval and Shumway (2001) are not dynamically rebalanced and may contain a significant time-decay option premium component. 1 market price of risk... |

418 | The jump-risk premia implicit in options: Evidence from an integrated time-series study
- Pan
- 2002
(Show Context)
Citation Context ...isk premia. For robustness, we confirm that our predictability results are robust to controlling for jump risk measures such as the slope of the implied volatility curve, realized jump intensity (BarndorffNielsen and Shephard (2006) and Wright and Zhou (2009)), and risk-neutral skewness (Bakshi, Kapadia, and Madan (2003)). Hence, we argue that the VIX and VVIX have a significant impact on option returns even in the presence of stock market and volatility jumps; we leave a formal treatment of jumps for future research. Reduced-form models which highlight the role of jumps include Bates (2000), Pan (2002), and Duffie, Pan, and Singleton (2000), among others. Our paper proceeds as follows. In Section 2 we discuss our model which links expected delta-hedged equity and volatility option gains to risk compensations for volatility and volatility-of-volatility risk. In Section 3, we describe the construction of both the model-free 4 implied variance measures and high-frequency realized variance measures, and summarize their dynamics in the time-series. We show that the implied variances have a strong ability to forecast future realized variance. Section 4 provides the empirical evidence from option ... |

332 | Power and Bipower Variation with Stochastic Volatility and Jumps.
- Barndorff-Nielsen, Shephard
- 2004
(Show Context)
Citation Context ...d until 2007. We apply the same methodology and construct the index for an additional year back to 2006. The correlation between our measure of the VVIX and the published index is over 99% in the post-2007 sample. Our empirical results remain essentially unchanged if we restrict our sample to only the post-2007 period. 10 futures returns over the next 30-days, and is a model-free, forward-looking measure of the implied volatility of volatility. In addition to the implied volatilities, we can also compute the realized volatilities for the stock market and the VIX. The construction here follows Barndorff-Nielsen and Shephard (2004) using high-frequency, intraday data.7 Realized variance is defined as the sum of squared high-frequency log returns over the trading day: RVt = N∑ j=1 r2t,j . (3.3) Barndorff-Nielsen and Shephard (2004) show that RVt converges to the quadratic variation as N →∞. We follow the standard approach of considering 5 minute return intervals. A finer sampling frequency results in better asymptotic properties of the realized variance estimator, but also introduces more market microstructure noise such as the bid-ask bounce discussed in Heston, Korajczyk, and Sadka (2010). Liu, Patton, and Sheppard (20... |

309 | Post-'87 Crash Fears in S&P 500 Futures Options
- Bates
- 1998
(Show Context)
Citation Context ...t prices and risk premia. For robustness, we confirm that our predictability results are robust to controlling for jump risk measures such as the slope of the implied volatility curve, realized jump intensity (BarndorffNielsen and Shephard (2006) and Wright and Zhou (2009)), and risk-neutral skewness (Bakshi, Kapadia, and Madan (2003)). Hence, we argue that the VIX and VVIX have a significant impact on option returns even in the presence of stock market and volatility jumps; we leave a formal treatment of jumps for future research. Reduced-form models which highlight the role of jumps include Bates (2000), Pan (2002), and Duffie, Pan, and Singleton (2000), among others. Our paper proceeds as follows. In Section 2 we discuss our model which links expected delta-hedged equity and volatility option gains to risk compensations for volatility and volatility-of-volatility risk. In Section 3, we describe the construction of both the model-free 4 implied variance measures and high-frequency realized variance measures, and summarize their dynamics in the time-series. We show that the implied variances have a strong ability to forecast future realized variance. Section 4 provides the empirical evidence ... |

276 |
Intertemporal asset pricing without consumption data,
- Campbell
- 1993
(Show Context)
Citation Context ...lied volatility is equivalent to higher prices for index options) since they provide a hedge against increases in volatility, which are associated with a deteriorating investment opportunity set (see =-=Campbell, 1993-=-). The average realized variance of VIX innovations is 0.539, which translates to 73.4% annualized volatility. This is our measure of the physical volatility-of-volatility. VIX realized variance is mu... |

229 | Econometrics of testing for jumps in financial economics using bipower variation. - Barndorff-Nielsen, Shephard - 2006 |

213 |
The Relation Between Implied and Realized Volatility,”
- Christensen, Prabhala
- 1998
(Show Context)
Citation Context ...f his VVIX measure, computed using numerical integration rather than the model-free VIX construction, is lower than the average realized volatility of VIX. One of the 12 variance premium, this evidence suggests that investors dislike volatility-of-volatility risks, and the market price of volatility-of-volatility risks is negative. In addition to unconditional moments, we can also analyze the conditional dependence of volatility and volatility of volatility. Specifically, we consider the predictability of future realized variances by the VIX and VVIX, in spirit of Canina and Figlewski (1993), Christensen and Prabhala (1998), and Jiang and Tian (2005) who use option implied volatilities to predict future realized volatilities. We follow Christensen and Prabhala (1998) and Jiang and Tian (2005) and conduct our predictability regressions of future realized variances using monthly, non-overlapping samples. We follow a standard approach in the literature and consider both univariate and multivariate encompassing regressions to assess the predictability of future realized variances by the VIX and VVIX. In our main specification, the dependent variable is the realized variance (RV ) over the next month, for both the S&... |

200 | Numerical Integration of Stochastic Differential Equations - Milstein - 1995 |

190 |
The information content of implied volatility.
- Canina, Figlewski
- 1993
(Show Context)
Citation Context ...hows that the average level of his VVIX measure, computed using numerical integration rather than the model-free VIX construction, is lower than the average realized volatility of VIX. One of the 12 variance premium, this evidence suggests that investors dislike volatility-of-volatility risks, and the market price of volatility-of-volatility risks is negative. In addition to unconditional moments, we can also analyze the conditional dependence of volatility and volatility of volatility. Specifically, we consider the predictability of future realized variances by the VIX and VVIX, in spirit of Canina and Figlewski (1993), Christensen and Prabhala (1998), and Jiang and Tian (2005) who use option implied volatilities to predict future realized volatilities. We follow Christensen and Prabhala (1998) and Jiang and Tian (2005) and conduct our predictability regressions of future realized variances using monthly, non-overlapping samples. We follow a standard approach in the literature and consider both univariate and multivariate encompassing regressions to assess the predictability of future realized variances by the VIX and VVIX. In our main specification, the dependent variable is the realized variance (RV ) ove... |

162 | The realtive contribution of jumps to total price variation.
- Huang, Tauchen
- 2005
(Show Context)
Citation Context ... t+ τ) = erf τWt,t+τ − 3µt,t+τerf τVt,t+τ + 2µ3t,t+τ[ erf τVt,t+τ − µ2t,t+τ ]3/2 , (5.4) where Vt,t+τ ,Wt,t+τ , Xt,t+τ are given by the prices of the volatility, cubic, and quartic contracts. Importantly, these measures are computed model-free using the observed option prices. The details for the computations are provided in the Appendix. Finally, our third measure of jump risks is based on the high-frequency index and VIX data, rather than the option prices. It corresponds to the realized jump intensity, and 20 relies on the bipower variation methods in Barndorff-Nielsen and Shephard (2004), Huang and Tauchen (2005), and Wright and Zhou (2009). Specifically, while the realized variance defined in (3.3) captures both the continuous and jump variation, the bipower variation, defined as: BVt = π 2 ( M M − 1 ) M∑ j=2 |rt,j−1||rt,j |(5.5) measures the amount of continuous variation returns. Hence, we can use the test statistic to determine if there is a jump on any given day: Jt = RVt−BVt RVt√ θ M max 1, QVt BV 2t , (5.6) where θ = ( Π 2 )2 +π−5, and QVt is the quad-power quarticity defined in Huang and Tauchen (2005) and Barndorff-Nielsen and Shephard (2004). The test statistic is distributed as N (0, 1). We... |

145 | Expected option returns.
- Coval, Shumway
- 2001
(Show Context)
Citation Context ...Indeed, under a standard linear risk premium assumption, we show that the expected payoff on the delta-hedged position in equity index options consists of the risk compensations for both volatility and volatility-ofvolatility risks. For volatility options, the expected payoffs only involve the compensation for volatility-of-volatility risks. The risk compensations are given by the product of the 1See e.g. Bansal and Yaron (2004), Bloom (2009), Bansal, Kiku, Shaliastovich, and Yaron (2013), Fernandez-Villaverde and Rubio-Ramırez (2013) for the discussion of macroeconomic volatility risks, and Coval and Shumway (2001), Bakshi and Kapadia (2003), Campbell, Giglio, Polk, and Turley (2012) for market volatility risks. 2We use the terms “variance risk” and ”volatility risk” interchangeably unless otherwise specified. 3For example, unlike delta-hedged positions, zero-beta straddles analyzed in Coval and Shumway (2001) are not dynamically rebalanced and may contain a significant time-decay option premium component. 1 market price of risk, the risk exposure of the asset, and the time-varying quantity of each source of risk. The model thus delivers clear, testable predictions for the expected option returns and th... |

138 | Stock Return Characteristics, Skew Laws,
- Bakshi, Kapadia, et al.
- 2003
(Show Context)
Citation Context ...2002), Eraker, Johannes, and Polson (2003)), while the fact that the implied volatility curve for VIX options slopes upwards (call options are more expensive than put options on average) is consistent with the positive volatility jumps (Drechsler and Yaron (2011) and Eraker and Shaliastovich (2008), among others). In this sense, these slope measures help capture the variation in the market and volatility jumps in the economy. Our second jump measure incorporates the whole cross-section of option prices, beyond just the slope of the smile. It is based on the model-free risk-neutral skewness of Bakshi et al. (2003): SKEW (t, t+ τ) = erf τWt,t+τ − 3µt,t+τerf τVt,t+τ + 2µ3t,t+τ[ erf τVt,t+τ − µ2t,t+τ ]3/2 , (5.4) where Vt,t+τ ,Wt,t+τ , Xt,t+τ are given by the prices of the volatility, cubic, and quartic contracts. Importantly, these measures are computed model-free using the observed option prices. The details for the computations are provided in the Appendix. Finally, our third measure of jump risks is based on the high-frequency index and VIX data, rather than the option prices. It corresponds to the realized jump intensity, and 20 relies on the bipower variation methods in Barndorff-Nielsen and Shephar... |

123 | Expected Stock Returns and Variance Risk Premium.” Review of Financial Studies
- BOLLERSLEV, TAUCHEN, et al.
- 2009
(Show Context)
Citation Context ... a structural approach, Bollerslev, Tauchen, and Zhou (2009) consider a version of the Bansal and Yaron (2004) long-run risks model which features recursive utility and fluctuations in the volatility and volatility of volatility of the aggregate consumption process. They show that in equilibrium, investors require compensation for the exposure to volatility and volatility-of-volatility risks. With preference for early resolution of uncertainty, the market prices of the two risks are negative. As a result, the variance risk premium is positive on average, and can predict future equity returns. Bollerslev et al. (2009) and Drechsler and Yaron (2011) show that the calibrated version of such a model can account for the key features of equity markets and the variance premium in the data. Our empirical results in the paper are consistent with the economic intuition in these models and complement the empirical evidence in these studies. Finally, it is worth noting that in our paper we abstract from jumps in equity returns, and focus on diffusive volatilities as the main drivers of asset prices and risk premia. For robustness, we confirm that our predictability results are robust to controlling for jump risk meas... |

122 | Delta-Hedged Gains and the Negative Market Volatility Risk Premium,”
- Bakshi, Kapadia
- 2003
(Show Context)
Citation Context ... index and VIX option returns, above and beyond volatility risks. The evidence in the data is consistent with a no-arbitrage model which features time-varying market volatility and volatility-of-volatility factors which are priced by the investors. In particular, volatility and volatility of volatility have negative market prices of risk, so that investors dislike increases in volatility and volatility of volatility, and demand a risk compensation for the exposure to these risks. Our no-arbitrage model follows and extends the one-factor stochastic volatility specification of equity returns in Bakshi and Kapadia (2003). Specifically, we introduce a separate time-varying volatility-of-volatility risk factor which drives the conditional variance of the variance of market returns.2 Both factors are priced in our model. We use the model to characterize the payoffs to delta-hedged equity and volatility options. The zero-cost, deltahedged positions represent the gains on a long position in the option, continuously hedged by an offsetting short position in the underlying asset. As argued in Bakshi and Kapadia (2003), delta-hedged option payoffs are very useful to study volatility-related risks as they most cleanly... |

122 | Towards a Theory of Volatility Trading,”
- Carr, Madan
- 1998
(Show Context)
Citation Context ...otably, while delta-hedged equity options are exposed to both the volatility Vt and volatility-of-volatility risks ηt (see equation (2.10)), delta-hedged VIX strategies are exposed only to the volatility-of-volatility risks. This helps us identify the relative importance of the two risks in the data. 9 3 Variance Measures 3.1 Construction of Variance Measures The VIX index is a model-free, forward-looking measure of implied volatility in the U.S. stock market, published by the Chicago Board Options Exchange (CBOE). The square of the VIX index is defined as in Equation (2.12) where τ = 30365 . Carr and Madan (1998), Britten-Jones and Neuberger (2000) and Jiang and Tian (2005) show that the VIX can be computed from the prices of call and put options with the same maturity at different strike prices: V IX2t = 2erf τ τ [∫ S∗t 0 1 K2 Pt(K)dK + ∫ ∞ S∗t 1 K2 Ct(K)dK ] , (3.1) where K is the strike price, Ct and Pt are the put and call prices, S ∗ t is the fair forward price of the S&P500 index, and rf is the risk-free rate. The VIX index published by the CBOE is discretized, truncated, and interpolated across the two nearest maturities to achieve a constant 30-day maturity.5 Jiang and Tian (2005) show through... |

101 |
Option prices, implied price processes and stochastic volatility.
- Britten-Jones, Neuberger
- 2000
(Show Context)
Citation Context ...ged equity options are exposed to both the volatility Vt and volatility-of-volatility risks ηt (see equation (2.10)), delta-hedged VIX strategies are exposed only to the volatility-of-volatility risks. This helps us identify the relative importance of the two risks in the data. 9 3 Variance Measures 3.1 Construction of Variance Measures The VIX index is a model-free, forward-looking measure of implied volatility in the U.S. stock market, published by the Chicago Board Options Exchange (CBOE). The square of the VIX index is defined as in Equation (2.12) where τ = 30365 . Carr and Madan (1998), Britten-Jones and Neuberger (2000) and Jiang and Tian (2005) show that the VIX can be computed from the prices of call and put options with the same maturity at different strike prices: V IX2t = 2erf τ τ [∫ S∗t 0 1 K2 Pt(K)dK + ∫ ∞ S∗t 1 K2 Ct(K)dK ] , (3.1) where K is the strike price, Ct and Pt are the put and call prices, S ∗ t is the fair forward price of the S&P500 index, and rf is the risk-free rate. The VIX index published by the CBOE is discretized, truncated, and interpolated across the two nearest maturities to achieve a constant 30-day maturity.5 Jiang and Tian (2005) show through simulation analysis that the approx... |

89 |
The model-free implied volatility and its information content.
- Jiang, Tian
- 2005
(Show Context)
Citation Context ...the volatility Vt and volatility-of-volatility risks ηt (see equation (2.10)), delta-hedged VIX strategies are exposed only to the volatility-of-volatility risks. This helps us identify the relative importance of the two risks in the data. 9 3 Variance Measures 3.1 Construction of Variance Measures The VIX index is a model-free, forward-looking measure of implied volatility in the U.S. stock market, published by the Chicago Board Options Exchange (CBOE). The square of the VIX index is defined as in Equation (2.12) where τ = 30365 . Carr and Madan (1998), Britten-Jones and Neuberger (2000) and Jiang and Tian (2005) show that the VIX can be computed from the prices of call and put options with the same maturity at different strike prices: V IX2t = 2erf τ τ [∫ S∗t 0 1 K2 Pt(K)dK + ∫ ∞ S∗t 1 K2 Ct(K)dK ] , (3.1) where K is the strike price, Ct and Pt are the put and call prices, S ∗ t is the fair forward price of the S&P500 index, and rf is the risk-free rate. The VIX index published by the CBOE is discretized, truncated, and interpolated across the two nearest maturities to achieve a constant 30-day maturity.5 Jiang and Tian (2005) show through simulation analysis that the approximations used in the VIX i... |

78 | A guide to volatility and variance swaps., - Derman, Demeter, et al. - 1999 |

66 | The impact of jumps in returns and volatility, - Eraker, Johannes, et al. - 2003 |

56 | What’s vol got to do with it,
- Drechsler, Yaron
- 2011
(Show Context)
Citation Context ...rslev, Tauchen, and Zhou (2009) consider a version of the Bansal and Yaron (2004) long-run risks model which features recursive utility and fluctuations in the volatility and volatility of volatility of the aggregate consumption process. They show that in equilibrium, investors require compensation for the exposure to volatility and volatility-of-volatility risks. With preference for early resolution of uncertainty, the market prices of the two risks are negative. As a result, the variance risk premium is positive on average, and can predict future equity returns. Bollerslev et al. (2009) and Drechsler and Yaron (2011) show that the calibrated version of such a model can account for the key features of equity markets and the variance premium in the data. Our empirical results in the paper are consistent with the economic intuition in these models and complement the empirical evidence in these studies. Finally, it is worth noting that in our paper we abstract from jumps in equity returns, and focus on diffusive volatilities as the main drivers of asset prices and risk premia. For robustness, we confirm that our predictability results are robust to controlling for jump risk measures such as the slope of the i... |

53 | A tale of two indices - Carr, Wu - 2006 |

46 | An equilibrium guide to designing affine pricing models. Mathematical Finance Forthcoming.
- Eraker, Shaliatovich
- 2007
(Show Context)
Citation Context ...osest to 0.9, and for VIX options as a call option with a moneyness closest to 1.1. In both cases, the ATM option has moneyness of 1. These slopes are positive for both index and VIX options. Positive slope of the index volatility smile is consistent with the notion of negative jumps in market returns (see e.g. Bates (2000), Pan (2002), Eraker, Johannes, and Polson (2003)), while the fact that the implied volatility curve for VIX options slopes upwards (call options are more expensive than put options on average) is consistent with the positive volatility jumps (Drechsler and Yaron (2011) and Eraker and Shaliastovich (2008), among others). In this sense, these slope measures help capture the variation in the market and volatility jumps in the economy. Our second jump measure incorporates the whole cross-section of option prices, beyond just the slope of the smile. It is based on the model-free risk-neutral skewness of Bakshi et al. (2003): SKEW (t, t+ τ) = erf τWt,t+τ − 3µt,t+τerf τVt,t+τ + 2µ3t,t+τ[ erf τVt,t+τ − µ2t,t+τ ]3/2 , (5.4) where Vt,t+τ ,Wt,t+τ , Xt,t+τ are given by the prices of the volatility, cubic, and quartic contracts. Importantly, these measures are computed model-free using the observed option... |

45 |
Overreactions in the options market.
- Stein
- 1989
(Show Context)
Citation Context ...sitivity of option price to the volatility of volatility, we compute the Black and Scholes (1973) second partial derivative of the option price with respect to the volatility, which is known in “volga” for “volatility gamma”. Volga is calculated as: ∂2C ∂σ2 = S √ τ 2π e− d21 2 ( d1d2 σ ) = ∂C ∂σ ( d1d2 σ ) , (4.3) d2 = d1 − σ √ τ . Figure 4 shows the plot of volga as a function of the moneyness of the option. Volga is positive, and exhibits twin peaks with a valley around at-the-money. Atthe-money options are essentially pure bets on volatility, and are approximately linear in volatility (see Stein (1989)). Therefore, the volga is the lowest for at-the-money options. Deep-out-of-the-money options and deep-in-the-money options do not have much sensitivity to volatility of volatility either, since for the former it is a pure directional bet, and for the latter the option value is almost entirely comprised of intrinsic value. Options that are somewhat away from at-the-money are most exposed to volatility-of-volatility risks. Table 4 shows our cross-sectional evidence from the regressions of average option returns on our proxies of options’ volatility and volatility-of-volatility betas. Panel A sh... |

36 | When is time continuous?,
- Bertsimas, Kogan, et al.
- 2000
(Show Context)
Citation Context ...ime counterparts to the continuously-rebalanced delta-hedged gains in Equations (2.7) and (2.17): Πt,t+τ = Ct+τ − Ct︸ ︷︷ ︸ option gain/loss − N−1∑ n=0 ∆tn ( Stn+1 − Stn ) ︸ ︷︷ ︸ delta hedging gain/loss + N−1∑ n=0 rf (∆tnStn − Ct) τ N︸ ︷︷ ︸ risk-free rate , Π∗t,t+τ = C ∗ t+τ − C∗t︸ ︷︷ ︸ option gain/loss − N−1∑ n=0 ∆tn ( Ftn+1 − Ftn ) ︸ ︷︷ ︸ delta hedging gain/loss − N−1∑ n=0 rfC ∗ t τ N︸ ︷︷ ︸ risk-free rate . (4.1) ∆tn indicates option delta, e.g. ∆tn = ∂Ctn ∂Stn , and N is the number of trading days in the month. This discrete delta-hedging scheme is also used in Bakshi and Kapadia (2003) and Bertsimas et al. (2000). At the close of each option expiration, we look at the prices of all options with non-zero open interest and non-zero trading volume. We take a long position in the option, and hedge the ∆ each day according to the Black-Scholes model and hedge the ∆ risk, with the net investment earning the risk-free interest rate appropriately.9 To minimize the effect of recording errors, we discard options that have implied volatilities below the 1st percentile or above the 99th percentile. All options have exactly one calendar month to maturity; S&P500 options expire on the third Friday of every month, w... |

34 | An intertemporal CAPM with stochastic volatility, - Campbell, Giglio, et al. - 2012 |

30 | Volatility jumps.
- Todorov, Tauchen
- 2011
(Show Context)
Citation Context ...can use the test statistic to determine if there is a jump on any given day: Jt = RVt−BVt RVt√ θ M max 1, QVt BV 2t , (5.6) where θ = ( Π 2 )2 +π−5, and QVt is the quad-power quarticity defined in Huang and Tauchen (2005) and Barndorff-Nielsen and Shephard (2004). The test statistic is distributed as N (0, 1). We flag the day as having a jump if the probability exceeds 99.9% both for index returns and for the VIX. These cut-offs imply an average frequency of jumps of once every two months for the index, and about three jumps a month for the VIX. This is broadly consistent with the findings of Tauchen and Todorov (2011), who find that VIX jumps tend to happen much more frequently than S&P500 jumps. Over a month, we sum up all the days where we have a jump, and we define our jump intensity measure on a monthly level as: RJ = 1 T T−1∑ i=0 Jt+i, where T is the number of trading days in the month. We use the jump statistics to document the robustness of the link between the volatility and volatility-of-volatility factors and options gains. We consider a regression: GAINSit,t+τ = β0 + β1V IX 2 t + β2V V IX 2 t + +β3JUMPt + γGAINS i t−τ + u i + it+τ (5.7) where JUMPt is one of the above jump risk proxies. We use ... |

20 | Volatility dynamics for the S&P 500: Evidence from realized volatility, daily returns, and option prices. - Christoffersen, Jacobs, et al. - 2010 |

16 | futures and other derivatives. - Options - 2002 |

14 | What’s Vol Got to Do With It? Review of Financial Studies. - Drechsler, Yaron - 2011 |

9 |
Volatility components: The term structure dynamics of vix futures,
- Lu, Zhu
- 2010
(Show Context)
Citation Context ...ions. To help us focus on the volatility-related risks, we consider dynamic delta-hedging strategies where a long position in option is dynamically hedged by taking an offsetting position in the underlying. Delta-hedged strategies are also used in Bertsimas, Kogan, and Lo (2000), Cao and Han (2013) and Frazzini and Pedersen (2012), and are a standard risk management technique of option traders in the financial industry (Hull (2011)). In an earlier study, Coval and Shumway (2001) considers the returns on zero-beta straddles to identify volatility risk sensitive assets. Zhang and Zhu (2006) and Lu and Zhu (2010) highlight the nature and importance of volatility risks by analyzing the pricing of VIX futures. Also notably our analysis suggests that variance dynamics are richer than that of the square-root process typically considered in the literature - these findings are consistent with the results of Christoffersen, Jacobs, and Mimouni (2010) and Branger, Kraftschik, and Volkert (2014). In a structural approach, Bollerslev, Tauchen, and Zhou (2009) consider a version of the Bansal and Yaron (2004) long-run risks model which features recursive utility and fluctuations in the volatility and volatility... |

9 | Macroeconomics and volatility: Data, models, and estimation. - Fernandez-Villaverde, Rubio-Ramirez - 2010 |

9 |
Options, Futures and Other Derivatives, Prentice Hall, Upper Saddle River,
- Hull
- 2006
(Show Context)
Citation Context ...nsider the implications of volatility risk for equity index option markets. We extend their 3 approach to include volatility-of-volatility risk, and bring evidence from VIX options. To help us focus on the volatility-related risks, we consider dynamic delta-hedging strategies where a long position in option is dynamically hedged by taking an offsetting position in the underlying. Delta-hedged strategies are also used in Bertsimas, Kogan, and Lo (2000), Cao and Han (2013) and Frazzini and Pedersen (2012), and are a standard risk management technique of option traders in the financial industry (Hull (2011)). In an earlier study, Coval and Shumway (2001) considers the returns on zero-beta straddles to identify volatility risk sensitive assets. Zhang and Zhu (2006) and Lu and Zhu (2010) highlight the nature and importance of volatility risks by analyzing the pricing of VIX futures. Also notably our analysis suggests that variance dynamics are richer than that of the square-root process typically considered in the literature - these findings are consistent with the results of Christoffersen, Jacobs, and Mimouni (2010) and Branger, Kraftschik, and Volkert (2014). In a structural approach, Bollersl... |

9 |
Vix futures,
- Zhang, Zhu
- 2006
(Show Context)
Citation Context ...ing evidence from VIX options. To help us focus on the volatility-related risks, we consider dynamic delta-hedging strategies where a long position in option is dynamically hedged by taking an offsetting position in the underlying. Delta-hedged strategies are also used in Bertsimas, Kogan, and Lo (2000), Cao and Han (2013) and Frazzini and Pedersen (2012), and are a standard risk management technique of option traders in the financial industry (Hull (2011)). In an earlier study, Coval and Shumway (2001) considers the returns on zero-beta straddles to identify volatility risk sensitive assets. Zhang and Zhu (2006) and Lu and Zhu (2010) highlight the nature and importance of volatility risks by analyzing the pricing of VIX futures. Also notably our analysis suggests that variance dynamics are richer than that of the square-root process typically considered in the literature - these findings are consistent with the results of Christoffersen, Jacobs, and Mimouni (2010) and Branger, Kraftschik, and Volkert (2014). In a structural approach, Bollerslev, Tauchen, and Zhou (2009) consider a version of the Bansal and Yaron (2004) long-run risks model which features recursive utility and fluctuations in the vol... |

6 | Risk Premia and Realized Jump Risk - Wright, Zhou |

5 | Simple variance swaps. Working Paper - Martin - 2013 |

4 | Intraday patterns in the crosssection of stock returns, - Heston, Korajczyk, et al. - 2010 |

4 | Does anything beat 5-minute rv? a comparison of realized measures across multiple asset classes, working paper. - Liu, Patton, et al. - 2013 |

4 |
A Tale of Two Option Markets: Pricing Kernels and Volatility Risk,” working paper, Chicago Booth Research Paper No 12-10 - Fama-Miller Working Paper.
- Song, Xiu
- 2012
(Show Context)
Citation Context ...on returns for each moneyness bin. To examine the contribution of both risks for the time-variation in expected index option payoffs, we consider the following regression: GAINSit,t+τ = Πit,t+τ St = β0 + β1V IX 2 t + β2V V IX 2 t + γGAINS i t−τ + u i + it+τ . (4.5) where we include fixed effects ui to account for the heterogeneity in the sensitivity of options in different moneyness bins to the underlying risks. We regress the delta-hedged option gain 10Muravyev (2012) highlights options market anomalies around expirations and relates it to inventory management concerns of options dealers. 11Song and Xiu (2013) demonstrate an alternative method of estimating risk sensitivities nonparametrically using local linear regression methods. 18 scaled by the index from expiration to expiration on the value of the VIX and VVIX indices at the end of the earlier expiration; in other words, we run one-month ahead predictive regressions of delta-hedged option returns on the VIX and VVIX. We include lagged gains to adjust for serial correlation in the residuals, following Bakshi and Kapadia (2003). Panel A of Table 5 shows the regression results for the index options. The univariate regression of delta-hedged S&P5... |

3 | Volatility, the macroeconomy, and asset prices, forthcoming in - Bansal, Kiku, et al. - 2013 |

3 |
Cross-section of option returns and idiosyncratic stock volatility,
- Cao, Han
- 2013
(Show Context)
Citation Context ...on markets, and have negative market prices of risks. Related Literature. Our paper is most closely related to Bakshi and Kapadia (2003) who consider the implications of volatility risk for equity index option markets. We extend their 3 approach to include volatility-of-volatility risk, and bring evidence from VIX options. To help us focus on the volatility-related risks, we consider dynamic delta-hedging strategies where a long position in option is dynamically hedged by taking an offsetting position in the underlying. Delta-hedged strategies are also used in Bertsimas, Kogan, and Lo (2000), Cao and Han (2013) and Frazzini and Pedersen (2012), and are a standard risk management technique of option traders in the financial industry (Hull (2011)). In an earlier study, Coval and Shumway (2001) considers the returns on zero-beta straddles to identify volatility risk sensitive assets. Zhang and Zhu (2006) and Lu and Zhu (2010) highlight the nature and importance of volatility risks by analyzing the pricing of VIX futures. Also notably our analysis suggests that variance dynamics are richer than that of the square-root process typically considered in the literature - these findings are consistent with th... |

3 |
Bond risk premia and realized jump risk,
- Wright, Zhou
- 2009
(Show Context)
Citation Context ...y markets and the variance premium in the data. Our empirical results in the paper are consistent with the economic intuition in these models and complement the empirical evidence in these studies. Finally, it is worth noting that in our paper we abstract from jumps in equity returns, and focus on diffusive volatilities as the main drivers of asset prices and risk premia. For robustness, we confirm that our predictability results are robust to controlling for jump risk measures such as the slope of the implied volatility curve, realized jump intensity (BarndorffNielsen and Shephard (2006) and Wright and Zhou (2009)), and risk-neutral skewness (Bakshi, Kapadia, and Madan (2003)). Hence, we argue that the VIX and VVIX have a significant impact on option returns even in the presence of stock market and volatility jumps; we leave a formal treatment of jumps for future research. Reduced-form models which highlight the role of jumps include Bates (2000), Pan (2002), and Duffie, Pan, and Singleton (2000), among others. Our paper proceeds as follows. In Section 2 we discuss our model which links expected delta-hedged equity and volatility option gains to risk compensations for volatility and volatility-of-volat... |

2 |
Expected vix option returns, working paper.
- Song
- 2013
(Show Context)
Citation Context ...d volatility is also high during other times of economic distress and uncertainty, such as May 2010 (Eurozone debt crisis and flash crash), and August 2011 (U.S. debt ceiling crisis). The VVIX largely follows the same pattern. During normal times, the VVIX is above the VIX realized variance, although during times of extreme distress we see the realized variance of VIX can exceed the VVIX. The average level of the VVIX (87%) is greater than the average level of the VIX realized volatility (73.4%), so that the volatility-of-volatility premium is also positive.8 Similar to our discussion of the 8Song (2013) shows that the average level of his VVIX measure, computed using numerical integration rather than the model-free VIX construction, is lower than the average realized volatility of VIX. One of the 12 variance premium, this evidence suggests that investors dislike volatility-of-volatility risks, and the market price of volatility-of-volatility risks is negative. In addition to unconditional moments, we can also analyze the conditional dependence of volatility and volatility of volatility. Specifically, we consider the predictability of future realized variances by the VIX and VVIX, in spirit o... |

1 | The tail in the volatility index, working paper - Du, Kapadia - 2012 |

1 |
Order flow and expected option returns, working paper.
- Muravyev
- 2012
(Show Context)
Citation Context ... the same bins as we used for average returns in Table 3, and average the scaled gains within each bin, so that we have a time-series of option returns for each moneyness bin. To examine the contribution of both risks for the time-variation in expected index option payoffs, we consider the following regression: GAINSit,t+τ = Πit,t+τ St = β0 + β1V IX 2 t + β2V V IX 2 t + γGAINS i t−τ + u i + it+τ . (4.5) where we include fixed effects ui to account for the heterogeneity in the sensitivity of options in different moneyness bins to the underlying risks. We regress the delta-hedged option gain 10Muravyev (2012) highlights options market anomalies around expirations and relates it to inventory management concerns of options dealers. 11Song and Xiu (2013) demonstrate an alternative method of estimating risk sensitivities nonparametrically using local linear regression methods. 18 scaled by the index from expiration to expiration on the value of the VIX and VVIX indices at the end of the earlier expiration; in other words, we run one-month ahead predictive regressions of delta-hedged option returns on the VIX and VVIX. We include lagged gains to adjust for serial correlation in the residuals, following... |

1 | Dacheng Xiu, 2013, A tale of two option markets: Pricing kernels and volatility risk, working paper - Song |

1 | The fine structure of variance: Consistent pricing of vix derivatives, working paper. - Branger, Kraftschik, et al. - 2014 |