Volatility Risk Premium

VRP represents the premium associated with uncertainty in volatility and is often measured by the difference between the statistical and risk-neutral expectations of the forward variation in the asset return. To measure VRP faced by liquidity providers, this study follows Bollerslev and Todorov (2011) and Todorov (2010) and define VRP over the next τ trade days as the risk-neutral volatility less the expected realized volatility, quantified as

[ , ] [ , ]

1/ . ^Q( ) 1/ . ^P( ),

t t t t t t t

VRP = τ E σ _+τ − τ E σ _+τ (1)

where E^Q(.) and E^P(.) indicate the expectations under risk-neutral and statistical measures, respectively.⁷

2.1.1. Estimate of risk-neutral volatility

The risk-neutral volatility at the first term in Equation (1) is calculated directly from option prices. As demonstrated in Bakshi and Madan (2000), Britten-Jones and Neuberger

7 Note that our VRP measure in Equation (1) is opposite to the definition of Bollerslev and Todorov (2011) and Todorov (2011). They calculate the VRP paid by hedgers, which is negative on average, whereas we

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(2000), and Jiang and Tian (2005), this risk-neutral volatility is equal to option implied volatility. In this study, the approach proposed by Jiang and Tian (2005, 2007) is adopted to compute the implied volatilities of call and put options directly from option prices. This method corrects the inherent methodological problem in the most widely used Black–

Scholes (1973) model for deriving the option-implied volatility, which assumes that the underlying asset’s return follows a lognormal distribution that is virtually found to be too fat-tailed to be lognormal.

Britten-Jones and Neuberger (2000) derive a model-free measure of implied volatility under the diffusion asset price process, and Jiang and Tian (2005) further extend their result to the case of jump diffusion. The model-free implied variance is defined as an integral of option prices over an infinite range of exercise prices, denoted as

∫

measure, K is the exercise price, τ denotes the time to maturity, F0 and C^F( ,τ K) are the forward asset and option prices.

However, in reality, options are trades in the marketplace only over a finite range of exercise prices. The limited availability of discontinuous exercise prices may lead to truncation and discretization errors in the numerical integration for the model-free implied volatility.⁸ To resolve the problem, Jiang and Tian (2005, 2007) develop an interpolation–

extrapolation scheme to reduce the influence of truncation and discretization errors.⁹

8 The truncation error results from disregarding exercise prices beyond the range of the listed exercise prices in the marketplace, and the discretization error arises from the discontinuous exercise prices. In general, the truncation error is negligible while the truncation points are more than two standard deviations from the forward asset price (Jiang & Tian, 2005).

9 The steps are specified as follows. At first, a wider range of exercise prices relative to available exercise prices is set up by given left and right truncation points K_min and K_max. Next, to obtain the not-traded option prices between these two truncation points, a cubic splines method is used to interpolate the Black–Scholes implied volatilities per the ∆K price interval between available exercise prices. Finally, the extracted implied volatilities are translated into option prices by using the Black–Scholes model, and the implied volatility is further computed from these option prices. In addition, for options with exercise prices beyond the available range in option market, Jiang and Tian suggest that the slope of the extrapolated segment (on both sides) should be adjusted to match the corresponding slope of the interior segment at the minimum or maximum available exercise price.

Following the approach of Jiang and Tian (2005, 2007), the model-free implied prices, in which K_min and K_max are referred as left and right truncation points, respectively.

In our empirical work, option and asset prices are used instead of forward prices to calculate the implied volatility. Under the assumption of deterministic interest rate, the forward option price and forward asset price at time t are respectively represent as C^F( ,τ K)=C( ,τ K) / ( , )B t τ and

/ ( , )

t t

F =S B t τ , in which S is spot price, _t C( ,τ K)is the option price, and B t( , )τ is the time t price of a zero-coupon bound that pays $1 at time τ.

To avoid the bid–ask bounce problem, the midpoint of the quote rather than the transaction price is used to compute the implied volatility (Bakshi, Cao, and Chen, 1997, 2000). As

10 The strike price intervals of TXO stipulated by the TAIFEX are grouped into three categories. First, when a strike price is below 3,000 points, the strike price intervals are 50 points in the nearby month and the next two calendar months and 100 point intervals for all months in excess of two months in distance. Second, when a strike price is between 3,000 and 10,000 points, the strike price intervals are 100 points in the nearby month and the next two calendar months, and 200-point intervals for far months. Third, when a strike price is over 10,000 points, the strike price intervals are 200 points in the nearby month and the next two calendar months,

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first calculated every five-minute interval and then averaged across intervals in a day. The five-minute implied volatility of the call (and put) is backed out from call (put) prices by using Equation (2).

2.1.2. Estimate of expected realized volatility

The expected realized volatility at the second term in Equation (1) is estimated using a VecHAR model constructed on the volatility components of model-free realized volatility.

Andersen, Bollerslev, and Diebold (2007) find that the forecasting to the future realized volatility improves significantly when using continuous volatility (CV) and jump volatility (JV) decomposed from realized volatility as separate regressors. They show that volatility components provide better forecasting than realized volatility itself because of the distinct features associated with the CV series and JV series: CV is strongly serially correlated while JV is less persistent and far less predictable than CV. The different features for the two components indicate separate roles in the forecast of realized volatility. In addition, Barndorff-Nielsen and Shephard (2001), Bollerslev, Kretschmer, Pigorsch, and Tauchen (2009), and Todorov and Tauchen (2006, 2011) find that the future volatility increases more following negative price jumps.

Following the Bollerslev and Todorov (2011), the daily close-to-close realized volatility is decomposed into four parts with different characteristics: overnight volatility (NV), CV, nJV, and pJV. In brief, the daily realized volatility is first calculated using 5-minute intraday returns and then decomposed it into four volatility components. This process produces four daily series, one for each volatility component, CV, nJV, pJV, and NV.

The VecHAR model in Equation (3) proposed by Busch, Christensen, and Nielsen

(2011)¹¹ is applied to forecast the one-period ahead of volatility components. The model follows Andersen et al. (2007) to contain daily, weekly, and monthly volatility measures in the VecHAR forecasting specifications. The expected realized volatility with a 22-day horizon is the square root of the relevant forecasts’ sum for the volatility components. The VecHAR model uses the four-dimensional vector Zt, consisting of the CV, nJV, pJV, and

where Z_t–1, Z_t–5, and Z_t–22, respectively, denote the vector of lagged daily, weekly, and monthly volatility components. B0 is a vector of the intercept term. B1, B5, and B22 are matrices for the regression coefficients, in which the first column, second column, third column, and fourth column in each matrix correspond to the parameters of the four volatility components, respectively. The model uses the past 800 days for the estimation of parameters B₀, B₁, B₅, and B22. The one-period-ahead volatility component vector is obtained using the estimated parameters and the past 22 days’ volatility components. To produce daily-frequency forecasting, this study rolls forward daily, using the same window length (800 days) for every forecasting.