Implied volatility measures

Finally, several ways of calculating implied volatility measures are presented. The starting point isIVBlack, which is the implied volatility that can be gained from inverting the Black

(1976)-formula for the valuation of futures options.²² Although the underlying assumption of Black (1976)’s model is a constant volatility over time, which implies the same implied volatility for options with different moneyness levels, empirical evidence has shown that implied volatilities differ with the moneyness of options. The question therefore arises as to which implied volatility best represents the perceived volatility of the underlying futures contract. Most often, the volatility of at-the-money options is used because they have the greatest liquidity (Poon and Granger (2003)). I therefore also use the implied volatility of the option with a strike price nearest to that of the underlying.²³ Indeed, Black’s formula allows pricing options of the European type but the options used here are of the American type. Hence, implied volatilities are calculated with the approximation of Barone-Adesi and Whaley (1987), which allows for early exercise. To calculate the implied volatility for a specific month, I extract the implied volatility of options traded on the last trading day of the previous month with a time to maturity of 30 calendar days. Since the required time to maturity is not exactly available, I linearly interpolate the implied volatility of options that are nearest to being less than and more than 30 calendar days from maturity.

This procedure is also carried out for each of the following implied measures.

The main disadvantage of the IVBlack is its model dependence. This could lead to biased estimators if, for example, the model’s assumed price process differs from the true one.

The observable non-constant volatility in the moneyness dimension—often referred to as the volatility smile—is clear evidence against the assumed price process.²⁴ A solution of the model dependence problem is model-free implied volatilities, which only assume complete markets but make no assumptions about the price process. The principle of model-free implied volatility estimation approaches is then based on the implications of complete markets: The fair price of any derivative can be calculated by discounting the expected payoff under risk-neutral probabilities with the risk-free interest rate (risk-neutral valuation).²⁵ I follow the approach of Bakshi, Kapadia, and Madan (2003) (BKM) for calculating model-free implied volatilities,IVBKM. Unlike the IVBlack, which is obtained

22The formula is a variation of the option valuation equation of Black and Scholes (1973), which considers that entering a futures position does not require a capital investment. For the formula, this means that the dividend yield is set equal to the risk-free interest rate.

23If both a call and a put option are the nearest options, the call option is chosen.

24For an elaborate discussion on volatility smiles, see, for example, Hull (2009, p. 389-406).

25For the theoretical foundation of risk-neutral valuation, see Cox and Ross (1976).

by using the information of only one option, the implied volatility resulting from the method of BKM uses the information of all at- and out-of-the-money options available at a certain point of time with the required time to maturity. Since the BKM model requires a continuum of strike prices but only a discrete number of prices is available, I use an interpolation method based on all available strikes. Following the procedure of Jiang and Tian (2005), I apply a cubic spline with a smoothing parameter of 0.3 to the Black implied volatilities of all traded options²⁶ at a specific date with the same specific time to maturity and a flat extrapolation outside the strike range to obtain a volatility curve as a continuous function of moneyness. I then use Black’s formula to translate this curve back into a continuum of option prices. I do not assume the Black model to hold for this procedure. The formula is only used as a data transformation that makes the interpolation numerically more stable (Chang, Christoffersen, Jacobs, and Vainberg (2012)). I also apply the data filters as described by Jiang and Tian (2005).

Although the IVBKM is model independent, it could still lead to biased estimators. One possible reason is that the volatility is calculated under the risk-neutral measure. Since the probability distribution under the real-world (physical) measure can differ from this if market participants are not risk neutral, an adjustment is necessary to obtain the “real”

expected volatility.²⁷ If market participants are risk averse, they require compensation for the volatility risk.²⁸ To consider market participants’ risk preferences in the volatility estimation, I follow the method of Prokopczuk and Wese Simen (2014) and calculate the risk-adjusted model-free implied BKM volatility,IVBKMRA. The IVBKMRA is calculated by dividing the IVBKM by the average relative volatility risk premium, which is the square root of the average ratio of the implied BKM variance and the standard realized variance

26I mainly use settlement prices from the electronic market. If no electronic trading data is available (as it is the case for the earlier years), I use settlement prices from floor trading. If no settlement price is

available, I use the closing price.

27The volatility under the risk-neutral measure equals the volatility under the physical under certain assumptions, for example, if the price process is assumed to follow a geometric Brownian motion. Since no assumption about the price process is made for the IVBKM estimator, differences between the measures are probable.

28For studies investigating the volatility risk premium, see, for example, Carr and Wu (2009).

of the previous 18 months:²⁹

IVBKMRA(T) = IV BKM_T

√RA_T , (2.11)

whereby RA_T is the risk adjustment for the horizon T, which is defined as

RA_T = 1

The idea is that the IVBKM represents the forecast variance under the risk-neutral measure, while the ex post SRV is a proxy for the physical measure.

As already described in Section 2.5.1 , the absolute deviation is a more intuitive measure of volatility. While standard GARCH models are not designed to capture this volatility definition, implied estimates of the absolute deviation are straightforward. Contrary to the RAD, it is not possible to determine the implied monthly average return. Instead, I directly model the expected return over a month with an AR(1) process fitted to the 60 previous monthly returns and then calculate the implied absolute deviation from this return. The concept of the implied absolute deviation estimator, IAD, is based on the same idea as the IVBKM. Starting from the assumption of complete markets, portfolios are built at the beginning of a month with a payoff at the end of the month that reflects the expected deviation from the expected return. Therefore, European put and call options with a strike price equal to the expected price are necessary. Formally,

IAD(T) =e^rτ[C_t(τ, K) +P_t(τ, K) ], (2.13) where τ is the options’ time to maturity, K equals the expected price at the end of the horizonT (T =t+τ) at time t,r denotes the risk-free interest rate for the horizon T, and C_t(τ, K) and P_t(τ, K) are the prices of a call and a put option with time to maturity τ and strike price K at timet, respectively.

29Prokopczuk and Wese Simen (2014) use a shorter period but explain that an 18-month estimation window leads to similar results. Before June 1998, the IVBKM could not be calculated for every month;

therefore I use the 18-month window to obtain more observations and, thereby, more robust estimates.

Corresponding to the implied volatility estimator based on the method of Bakshi, Kapadia, and Madan (2003), I again use a risk adjustment to obtain implied absolute deviations under the physical measure,IADRA:

IADRA(T) = IAD_T

√RA_T, (2.14)

where RA_T is the risk adjustment for the horizon T, which is defined as

RA_T = 1 18

T−1

X

i=T−19

IAD²_i

RAD²_i. (2.15)

Table 2.2 summarizes the volatility measures described above.

Table 2.2: Overview of volatility estimators

Panel A: Realized volatility (RV) estimators Estimator

Based on SD • Standard RV (SRV)

• RV with autocorrelation correction (RVAC)

• RV based on the adjusted absolute deviation (RVAAD)

• RV based on adjusted residuals (RVAR)

• RV developed by Yang and Zhang (RVYZ) Based on AD • Realized absolute deviation (RAD)

Panel B: GARCH model-based volatility estimators Estimator

Based on SD • GARCH(1,1), monthly data (GARCHm)

• GARCH(1,1), daily data (GARCHd)

• GJR-GARCH(1,1), monthly data (GJRGARCHm)

• GJR-GARCH(1,1), daily data (GJRGARCHd)

Panel C: Implied volatility (IV) estimators Estimator

Based on SD • IV based on inversion of Black’s pricing formula (IVBlack)

• IV based on approach of BKM (IVBKM)

• IV based on approach of BKM, adjusted for risk aversion (IVBKMRA)

Based on AD • Implied absolute deviation (IAD)

• Implied absolute deviation, adjusted for risk aversion (IADRA)

Note: This table provides an overview of the different volatility measures used. Each panel distinguishes between measures that define volatility as the standard deviation (SD) or the absolute deviation (AD) of returns.