The Right Tail of the Risk-Neutral Distribution

The first hypothesis H1 states that whenever deep out-of-the money calls are observed, the pricing kernel is increasing at the right end and is hence either w- or u-shaped. The intuition is that if no out-of-the-money calls are observed, the implied volatility curve is not (or not sufficiently) increasing at the right end, leading to a thinner right tail of the risk-neutral density. Hypothesis H1 is examined from three perspectives.

First, we examine the influence of the moneyness of the deepest out-of-the-money call on the right end of the pricing kernel for the monthly horizon. According to our definition

16For robustness, we use several alternative specifications for identifying monthly and weekly pricing kernels as decreasing/increasing at the right end. These specifications do not change the results, see Appendix3.7.2.

3.4. EMPIRICAL PRICING KERNELS ESTIMATES Table 3.2:

The Right Tail of the Risk-Neutral Distribution

Estimation results when regressing the shape of the pricing kernel at the right end (Y_t= 1 if decreasing, Y_t = 0 if increasing) on explanatory variables. In Panel A, a monthly horizon is considered and the explanatory variable isX_t^otm-call, which measures the moneyness of the deepest out-of-the-money call option at datet. In Panel B the explanatory variable isXotm-IV-slope

t , which measures the steepness of the interpolated implied volatility curve at datetat a moneyness of 1.15. Panel C uses the same explanatory variableX_t^otm-callas Panel A but considers a weekly horizon. The second column in all panels reports the results from estimating the GJR on the full sample, while the third column reflects the results when the GJR is estimated on a rolling window. N denotes the number of observations.

Panel A:Logit(Yt) =β0+β1·X_t^otm-call+t, monthly GJR on Full Sample GJR on a Rolling Window

β₀ 9.7693^∗∗∗ 5.3859^∗∗

β1 −9.2648^∗∗∗ −4.9081^∗∗

N 331 331

Panel B: Logit(Yt) =β0+β1·Xotm-IV-slope

t +t, monthly GJR on Full Sample GJR on a Rolling Window

β₀ −0.0020 0.5144^∗∗∗

β₁ −1554.26^∗∗∗ −1898.28^∗∗∗

N 331 331

Panel C: Logit(Yt) =β0+β1·X_t^otm-call+t, weekly GJR on Full Sample GJR on a Rolling Window

β0 6.2647^∗ 6.0565^∗

β₁ −6.9051^∗∗ −6.6778^∗∗

N 475 475

of a decreasing pricing kernel, 41% of the pricing kernels are decreasing at the right end and 59% are increasing when using the GJR estimated on the full sample, see Panel A of Table 3.1. To test the significance of the influence of the deepest out-of-the-money call option, we run a Logit regression of the shape of the pricing kernel at the right end (Y_t = 1 if decreasing,Y_t= 0 if increasing) on the moneyness level of the deepest call option X_t^otm-call at date t (which passes the filter and displays a maturity of 23 calendar days).

The second column of Panel A in Table 3.2reports the coefficients when estimating the pricing kernels with the GJR calibrated on the full sample, and the third column reports the results when using a rolling window GJR instead. We see in both cases evidence for H1 as the coefficients are highly significant and the negativity of the slope coefficients βb₁

3.4. EMPIRICAL PRICING KERNELS ESTIMATES indicates that the deeper the last observed call option is out-of-the-money, the lower the probability of obtaining a pricing kernel, which decreases at the right end.

Alternatively, we can look at the validity of hypothesis H1 through the influence of out-of-the-money calls on the slope of the interpolated volatility curve at the right end.

Even if there is a deep out-of-the money call observed at date t, this does not necessarily guarantee that the implied volatility curve is increasing (steeply enough) at the right end:

since one typically observes more out-of-the-money puts than calls, more weight is given to puts. The resulting interpolated implied volatility curve might be not (or not sufficiently) increasing at the right, even though some deep out-of-the-money calls are observed. To test for such an effect, we regress the shape of the pricing kernel at the right endY_t on the slope of the interpolated implied volatility curveXotm-IV-slope

t at moneyness 1.15. The results reported in Panels B of Table 3.2again provide evidence for hypothesis H1. The coefficient on the slopeβb₁ is highly significant for the full sample GJR, as well as for the rolling window GJR.

We finally look at the weekly pricing kernels. Before examining the estimation results from Panel C of Table 3.2, some words of caution are in order since, in contrast to the monthly horizon, we do expect less significant coefficients: the global minimum of the implied volatility curve for a short horizon, such as a week, is typically closer at-the-money than the global minimum of an implied volatility curve for a monthly horizon. Hence, even if we do not observe deep out-of-the-money calls, the implied volatility already gets its kick up by average out-of-the-money calls.¹⁷ We therefore expect that the deepest out-of-the-money call will have a less significant influence on the shape of the pricing kernel at the right end, since the average calls already have inherit the “true” shape of the implied volatility to its interpolated version. The empirical results confirm this intuition.

When looking at the amount of decreasing weekly pricing kernels in Panel B of Table 3.1, only 25% instead of 41% of the pricing kernels are now decreasing at the right end.

The coefficients from regressing the shape of the pricing kernel at the right end Y_t on the moneyness level of the deepest call option X_t^otm-call in Panel C of Table 3.2 are still negative, but less significantly.

In conclusion, all three points of view confirm hypothesis H1: if no deep out-of-the money calls are observed, the pricing kernels tend to be decreasing at the right end. If we nevertheless observe such call options (or the interpolated implied volatility curve is already increasing due to the calls near at-the-money, as in the case of the weekly options)

17The average minimum of the implied volatility curve lies at a moneyness of 1.06 for the monthly options and at 1.03 for the weeklies. As the implied volatility curve has to be monotonically increasing after the first (going from low to high moneyness) minimum is obtained, we can learn from short term options how to extrapolate the implied volatility curve at the right end in a more natural way. Note that an implied volatility curve, which is not monotonically increasing after the first minimum is obtained, could allow for arbitrage.

3.4. EMPIRICAL PRICING KERNELS ESTIMATES the resulting pricing kernel will be increasing at the right end and is therefore either w-or u-shaped. From a risk-neutral density modeling perspective this suggests the use of models, which make the implied volatility curve u-shaped even though sometimes deep out-of-the-money calls are not observed. Such methods increase the amount of pricing kernels increasing at the right end, see Appendix3.7.2.