Hypotheses

Having a time-series of estimated pricing kernels ready, the hypothesis regarding their shape is now motivated by Figure 3.2. This figure plots two implied volatility curves and pricing kernels stemming from different risk-neutral and subjective densities. The pricing kernels are estimated on the 28th January 1998 and represent a horizon of 23 calendar days. This particular date is not chosen because the plotted pricing kernels exhibit a particularly nice shape, e.g. many pricing kernels exhibit an even more pronounced w shape than the pricing kernel in Panel A. The reason is rather that all relevant effects can be illustrated at once using this particular day.

Risk-Neutral Right Tail Hypothesis

For the first hypothesis consider the pricing kernels from Panels A and B of Figure3.2, which both are calculated by the same subjective density pb_t, stemming from the GJR estimated on the full sample. The w-shaped pricing kernel in Panel A is calculated by a risk-neutral density, which uses all available implied volatilities (passing the filter described in Section 3.3), see Panel D of Figure 3.2. On the other hand, the pricing kernel in Panel B is calculated by a risk-neutral density, where the two most right implied volatility observations were taken out of the estimation, see Panel E. The effect on the resulting pricing kernels is considerable: The pricing kernel in Panel A is increasing at the right end, while the pricing kernel in Panel B is decreasing at the right end. As both pricing kernels stem from the same subjective density, the effect is caused by the risk-neutral density: As the implied volatility curves from Panel D and E are almost identical up to a return of aboutR_t= 1.05, the associated risk-neutral densities are almost identical up to R_t = 1.0, see Figure3.7 in Appendix 3.7.1.¹⁰ The implied volatility curve in Panel E is extrapolated by keeping the slope from the last observed implied volatilities, which were not manually

10Even though the implied volatility curves are almost identical up to Rt= 1.05, we cannot expect that corresponding risk-neutral densities are identical up toRt= 1.05 since the risk-neutral probability of a certain returnRtis given by the second derivative of the call price atRt. Hence, the risk-neutral probability for a certain return depends on the call prices in its neighborhood.

3.2. METHODOLOGY

Figure 3.2: Different Pricing Kernels and Implied Volatility Curves on January, 28, 1998.

We plot implied volatility curves and pricing kernels stemming from different risk-neutral and subjective densities. For the pricing kernels in Panels A, B, C, and F, returns are on the horizontal axis and the values of the pricing kernel are on the vertical axis. For the implied volatilities in Panels D and E, returns are on the horizontal axis and the implied volatility is on the vertical axis. Panels A, C, and F stem from the same risk-neutral density, which is derived from the implied volatility curved depicted in Panel D.

Panel B stems from the risk-neutral density, which is derived from the implied volatility curved depicted in Panel E. Panels A and B are created by using a GJR for the subjective density, and Panel C stems from a GARCH. In Panel D, all available option prices (marked by a circles with crosses) at that day are used to estimate the smooth implied volatility function (solid line). The smooth implied volatility function in Panel E on the other hand is constructed by ignoring the two most right (call) options. The dashed w-shaped pricing kernel in Panel F stems from decreasing the variance-risk-premium mechanically by increasing the spot varianceσ²_t of the GJR by 20%. Similarly, the dotted u-shaped pricing kernel in Panel F stems from increasing the variance-risk-premium.

taken out of the estimation. As a consequence, the implied volatility curve is not rising at the right end and the resulting right tail of the risk-neutral distributionqb_t is thinner, leading to a decreasing right end of the pricing kernelmbt=bqt/bpt. Since it turns out that such deep out-of-the money implied volatilities, typically stemming from call options, do not appear as frequently as their out-of-the money put counterparts, the first hypothesis is formulated as follows:

H1: Whenever deep out-of-the money calls are observed, the pricing kernel is increasing at the right end, and thus either w- or u-shaped.

Note also that the pricing kernel in Panel B could be classified as monotonically decreasing with some estimation noise, consistent with the findings ofBarone-Adesi, Engle,

3.2. METHODOLOGY and Mancini (2008). The authors use a GJR for the subjective density and fit another GJR for the risk-neutral density. Hence, their finding of mostly monotonically decreasing pricing kernels might be due to the combination of using a GJR and not having many out-of-the money calls in the sample or a loose fit to these options.¹¹

Subjective Right Tail Hypothesis

Consider the pricing kernels of Panel A and C in Figure 3.2 for illustrating the issues regarding the right tail of the subjective distribution. As both pricing kernels stem from the same risk-neutral distribution, the difference between the w-shaped pricing kernel in Panel A and the tilde shaped pricing kernel in Panel C is due to different subjective densities: Panel A is stemming from a GJR, while Panel C stems from a regular GARCH.

While the left tail of the subjective densities of the GJR and the GARCH are quite similar, see Figure 3.8 in Appendix 3.7.1, the right tail of the subjective distribution pb_t stemming from the GARCH is much fatter, making the pricing kernel mb_t=qb_t/pb_t decreasing at the right end. As a result, the pricing kernel associated with the GARCH is tilde shaped in contrast to the w-shaped GJR pricing kernel. To examine if this difference in subjective probabilities can be justified by the data, the second hypothesis is formulated as follows:

H2: Subjective densities, which imply w- or u-shaped pricing kernels, fit the realized return series better than subjective densities, which imply a pricing kernel decreasing at the right

end.

Variance-Risk-Premium Hypotheses

As the empirical sections3.4.1 and 3.4.2 show, we find evidence for both hypotheses, H1 and H2, indicating that the empirical pricing kernel is either w- or u-shaped. The next hypotheses therefore aim at explaining why at times w-shaped and at times u-shaped pricing kernels emerge.

A major point of criticism, which arises when taking a model estimated by past returns for obtaining the subjective density, is that the subjective volatility might be misspecified;

see for example Beare and Dossani (2016) or Sala (2016). In the context of the GJR, this can be interpreted as the variance of the simulated density being either too high or too low. In order to understand the effect of a misspecified volatility, we simulate the subjective GJR density by decreasing the initial spot volatility bσ_t on the 28th January

11Barone-Adesi, Engle, and Mancini (2008) minimize the mean squared error between market and model prices. As call prices are decreasing in moneyness, less weight is given to out-of-the-money calls.

Moreover, when using a GJR for the subjective density and the fast and stable method for the risk-neutral density with eliminating all call options exceeding a moneyness of 1.05, several pricing kernels turn out to be monotonically decreasing.

3.2. METHODOLOGY 1998 by 20%, leading to the dotted pricing kernel in Panel F. Similarly, the dashed pricing kernel in Panel F is generated by increasingσb_t by 20%. The decrease of the spot volatility makes the dotted pricing kernel more u-shaped and the hump at-the-money disappears.

Increasing the spot volatility on the other hand makes the hump of the dashed pricing kernel at-the-money more pronounced. Overall, one observes that a misspecified volatility could influence the shape of the pricing kernel substantially. Yet, pricing kernels associated with a completely unbiased volatility (risk-premia) are either more u-shaped or have a more pronounced hump at-the-money. Nevertheless, they are never monotonically decreasing.

But what if the subjective and risk-neutral volatilities are not misspecified and pricing kernels are at times either u- or w-shaped?¹² Then, the difference in volatilities should explain the remaining time variation in empirical pricing kernels. The difference between the risk-neutral volatility squared and the subjective volatility squared is known as the variance-risk-premium (VRP), see e.g. Bollerslev, Tauchen, and Zhou(2009). Our findings suggest that in times of high uncertainty (VRP is high) the pricing kernel is rather u-shaped, and in times of low uncertainty (VRP is low) the pricing kernel is w-shaped.

However, prior to formulating the hypothesis relating these observations, one has to check if the subjective volatilities are not misspecified. The first part of the third hypothesis is therefore:

H3a: The subjective densities estimated by historical returns do not systematically over-or underestimate the future realized variance.

After having assured that the time-series models used in this study provide reasonable forecasts for the future physical variance, the second part of the third hypothesis can be stated:

H3b: The variance-risk-premium is driving the shape of the pricing kernel. The pricing kernel is u-shaped when the variance-risk-premium is high and w-shaped when the

variance-risk-premium is low.

To avoid endogeneity problems, the variance-risk-premium is calculated by high-frequency returns and the VIX, seeBollerslev, Tauchen, and Zhou (2009).¹³

12One observes the inverse effect when changing the volatility of the risk-neutral density: When shifting up the whole implied volatility curve, the pricing kernel becomes more u-shaped, similar to the dotted pricing kernel in Panel F. When shifting down the whole implied volatility curve the hump of the pricing kernel at-the-money becomes more pronounced, similar to the dashed pricing kernel in Panel F. Both, risk-neutral and subjective volatility, is hence able to turn a u-shaped pricing kernel into a w-shaped one, and vice versa.

13Defining the variance-risk-premium at date t as the difference between the variance implied byqb_t (coming from the fast and stable method) andbp_t(coming from the GJR) makes all results even stronger.

3.3. DATA

3.3 Data

The data on the European S&P 500 index options range from January 1988 to August 2015. The options from 1988 to 1995 come from the Berkeley Options Database, while the data from 1996 to 2015 are provided by OptionMetrics. The databases also contain the corresponding dividends and risk-free rates. The usual filters are applied to the option data insofar as options with moneyness exceeding 1.2 and below 0.8 are excluded from the sample, as well as option prices violating general no-arbitrage constraints and quotes without volume. Consistent with the literature, only out-of-the-money calls and puts are considered. To avoid statistical issues in terms of autocorrelation, a monthly horizon is chosen in the base procedure in order to deal with non-overlapping densities and returns.

This typically leads to a maturity of 23 calendar days and picking the Wednesday option data, which is standard in the literature. For robustness, and as they are gaining more and more attention in recent literature, weekly options are considered as well. They typically have a maturity of 9 calendar days.¹⁴ We end up using 331 cross-sections of monthly options and 472 cross-sections of weeklies. The time-series of the S&P 500 returns are obtained from Datastream and cover the period from January 1985 to September 2015.

The return series starts in 1985 and therefore 3 years prior to the option data so that the full sample GJR and GARCH estimates also capture the beliefs of investors at the starting date of the options in 1988. For calculating the VRP, the VIX is obtained from Datastream and the daily realized variance from 2000 to 2015, computed out of intraday returns, is downloaded from the Oxford-Man Institute of Quantitative Finance.¹⁵