extent that more options are considered in the estimation. Therefore, given a sufficiently large cross-section of options, different loss functions will produce approximately the same risk-neutral distributions.
function of market returns. This is confirmed in Figure 3.1, where we calculate the risk-neutral and physical distributions as the lognormal densities with the correspond-ing risk-neutral and physical parameters. The distributions are plotted in the left panel, while the ratio between them, characterizing the EPK, is depicted in the right panel. As can be seen, there is no pricing kernel puzzle in this economy, as the EPK is monotoni-cally decreasing with respect to the market index, considered to be a proxy for aggregate wealth.
Our framework allows to obtain the same EPK resulting from the Black-Scholes model using only market returns in the estimation. More specifically, since CRRA is a particular case of the HARA class of utility functions, there is a minimum disper-sion risk-neutral measure (and corresponding pricing kernel) that is consistent with the CRRA representative investor. This measure is completely determined by the parameters governing the physical distribution of returns (for more details, see Chapter 1). In the parametrization considered, the risk-neutral measure is given by γ∗ = −0.8. Using the simulated underlying returns from the physical distribution, we calculate the risk-neutral measure and obtain the corresponding SPD via Breeden and Litzenberger (1978). The left panel of Figure3.2 confirms that the estimated risk-neutral distribution matches the model-implied SPD, while the right panel shows that the estimated pricing kernel pro-jected onto market returns is equal to the EPK.12 Therefore, in a Black-Scholes economy, the true SPD and EPK can be obtained without using any option data. In other words, options are redundant with respect to the underlying index.
The fact that options are redundant with respect to the index begs the question of what happens if we include options in the estimation of the SPD and the pricing kernel. To investigate that, we include the prices of five options in the estimation of the minimum dispersion risk-neutral measure with γ∗ = −0.8.13 We find that the investor sets zero weights to the options in the portfolio. In fact, the investor buys approximately the same amount of the market index as the investor trading only on the market. This confirms the redundancy of options in this economy. In the left panel of Figure 3.3, we plot the SDFs estimated using only market returns and using market returns and options, over the endogenous wealth.14 As can be seen, they are basically the same. Moreover, the right panel shows that the SDF estimated with options is also equal to the model-implied
12Note that the estimated pricing kernel is discrete and defined for each realization of state of nature as given by each simulated market return from the physical distribution.
13More specifically, we consider three out-of-the-money (OTM) puts with strike prices 0.8, 0.89 and 0.99 and two OTM calls with strikes 1.09 and 1.20. The results are robust to other specifications. Using only OTM options is interesting as OTM puts only pay off for negative realizations of the market, while OTM calls only pay off for positive realizations. This allows for a better interpretation of the optimal portfolio of the investor, while for the purposes of estimating the SPD it is equivalent to using only calls or puts for all strikes.
14We set W0= 1 in order to make it comparable with the market index, and b= 0 anda=−1/γ in order to obtain the endogenous wealth of the CRRA investor.
EPK when projected onto market returns. More than that, by comparing the two plots, we can see that the endogenous wealth is equal to the market index.
In sum, there is no pricing kernel puzzle in the Black-Scholes economy. Using our framework, we show that this is because options are redundant and the endogenous wealth of the representative investor equals the market index, as assumed by the literature on the puzzle.
3.4.2 Stochastic Volatility and Correlated Jumps Economy
The SVCJ model presents additional sources of risk and market incompleteness given by stochastic volatility and jumps, generating a more realistic economy capable of reproducing stylized facts of real option markets.15 Using option prices implied by the model, we calculate the risk-neutral distribution using the Breeden and Litzenberger (1978) formula. For the physical probabilities, we derive the kernel density with a Gaus-sian kernel from one million underlying returns sampled from the model-implied physical distribution.16 As depicted in the left panel of Figure3.4, the risk-neutral distribution is more skewed to the left, has fatter tails and is more peaked than the physical distribution.
This generates the U-shaped EPK in the right panel of the figure. Under the assump-tions of Ait-Sahalia and Lo (2000) and Jackwerth (2000), the violation of monotonicity with respect to wealth (as proxied by the market index) indicates that the representative investor is locally risk-seeking in the regions where the EPK is increasing. Therefore, the pricing kernel puzzle exists in the SVCJ economy.
We start by investigating the implications of using only market returns on the estimation of the SPD and the pricing kernel. Figure3.5 reports the results for the min-imum dispersion risk-neutral measure (and pricing kernel) with γ = −1. The estimated SPD is clearly not a good approximation to the model-implied SPD, indicating that it is necessary to include options in the estimation of the SPD. Consequently, the projection of the minimum dispersion SDF onto market returns is also very different from the EPK, as can be seen in the right panel of the figure. In particular, it is monotonically decreasing with respect to the market index. This is because the index is perfectly correlated with the endogenous wealth of the investor trading only on the market.
We proceed by including a set of eleven option prices in the estimation of the minimum dispersion SPD and pricing kernel with γ = −1.17 The results are reported in Figure3.6. From the left panel, we can see that the estimated SPD closely matches the
15The market generated by the SVCJ model is incomplete even with dynamic trading.
16We follow Jackwerth (2000) in setting the kernel bandwidth to h = 1.8σ/n1/5, where σ is the standard deviation of the sample returns andnis the number of observations.
17More specifically, we consider five OTM puts with strike prices 0.8, 0.84, 0.88, 0.92, 0.96 and six OTM calls with strikes 1.01, 1.04, 1.08, 1.12, 1.16 and 1.19. The results are robust to other specifications.
model-implied SPD. Likewise, the projection onto market returns of the estimated SDF approximates the model-implied EPK. This confirms that options are non-redundant with respect to the index in the SVCJ economy.
Our framework highlights that the estimation of the pricing kernel using option prices is inherently related to the investor whose investment opportunities include the market index and the index options. In other words, the market index is only a good proxy for wealth if options are redundant, as in the Black-Scholes economy. Otherwise, the shape of the EPK should not be evidence in favor or against economic theory. This can be seen in Figure 3.7, where we plot the SDF estimated with market returns and option prices over endogenous wealth in the left panel, and its projection onto market returns in the right panel.18 The U-shaped projection is rationalized by a monotonically decreasing pricing kernel with respect to the endogenous wealth coming from the optimal portfolio of the investor. That is, the investor is risk-averse.
To help visualize the connection between the two plots in Figure 3.7, each color represents the same set of realizations of states of nature. As can be seen, the EPK is U-shaped because extreme negative and positive realizations of the market are associated to negative realizations of the optimal portfolio, leading to smaller wealth. In contrast, moderate positive market returns are the ones associated to larger endogenous wealth.
This illustrates how the assumption of perfect correlation between the market index and wealth can be misspecified. More than that, it indicates that the marginal investor in the index and index options sells protection against large movements in the index. This can also be seen by analyzing the investor portfolio weights. In our estimation, the investor sells the OTM put with strike 0.92 and the OTM call with strike 1.12. Due to these positions, the investor incurs in substantial losses for extreme negative and positive returns on the market.
In summary, we show that the U-shaped EPK is not puzzling because it is the projection of a pricing kernel that is monotonically decreasing with respect to the en-dogenous wealth that considers all available investment opportunities. In particular, the assumption that the index is perfectly correlated with aggregate wealth is misspecified and inconsistent with the estimation procedure of the EPK. The U-shaped pattern of the EPK arises due to the fact that options are non-redundant in the presence of volatility and jumps risk premia, where the marginal investor in the index and index options sells protection against large movements in the index.
18We set W0 = 1 in order to make it comparable with the market index, andb = 0 and a = 1 in order to obtain the endogenous wealth of an investor with logarithmic utility. However, the results are qualitatively the same for any other parametrization.