Extracting expected stock risk premia from option prices and the information contained in non-parametric-out-of-sample stochastic discount factors

Abstract

This paper analyzes the factor structure and cross-sectional variability of a set of expected excess returns extracted from option prices and a non-parametric and out-of-sample stochastic discount factor. We argue that the existing potential segmentation between the equity and option markets makes it advisable to avoid using only option prices to extract expected equity risk premia. This set of expected risk premia significantly forecasts future realized returns, and the first two principal components explain 94.1% of the variability of expected returns. A multi-factor model with the market, quality, funding illiquidity, the default premium and the market-wide variance risk premium as factors significantly explains the cross-sectional variability of expected excess returns. The (asymptotically) different from zero adjusted cross-sectional R-squared statistic is 83.6%.

Keywords:

• Exact expected returns
• Risk-neutral variance
• Out-of-sample stochastic discount factor
• Cross-section of expected returns

JEL Classification:

• G12
• G13

Acknowledgements

We thank Geert Bekaert, Martijn Boons, John Cochrane, Alfonso Novales, Carlo Sala and the anonymous referee for helpful comments. We also thank conference participants at the financial seminar at the Nova School of Business and Economics, the 25th Finance Forum Meeting at University Pompeu Fabra, and the 2018 World Finance Congress. Any errors are entirely our own.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See the follow up papers by Bakshi et al. (2017), Borovicka et al. (2016), Jackwerth and Menner (2020), Jensen et al. (2019), and Schneider and Trajani (2019).

2 Under no-arbitrage assumptions, Chabi-Yo and Loudis (2020) propose bounds on the conditional expected market risk premium that are functions of higher-order risk-neutral return moments. They show that their bound measures perform similarly to Martin’s (2017) measure for short forecasting horizons. In addition, also using option prices, Schneider (2019) obtains a model-free decomposition of the realized forward market return to conclude that, at short horizons, the main component of the market return comes from downside risk, while at longer horizons variance risk dominates.

3 Martin (2017) discusses several examples for alternative preference specifications and return distributions, including the traditional CAPM.

4 Kadan and Tang (2020) show the conditions under which expression (3) holds for individual stock returns.

5 See the evidence reported by Driessen et al. (2009), who employ a similar database.

6 See also Asness et al. (2020) for additional evidence supporting this argument.

7 The window from six days to 60 days corresponds to the maximum range of time to maturity we allow in the necessary interpolation to have enough options every day in the sample with 30 days to maturity.

8 Given that the ISDF values depend on the returns of the assets employed in their estimation, we use portfolios based on all factors of the FF five-factor model plus momentum in order to capture as many differential characteristics as possible.

9 Other relevant assumptions are fully discussed by Martin and Wagner (2019).

10 In addition, we employ the approach of Connor and Korajczyk (1988). Given the similarity between the results under the two estimation procedures, we do not report the results. All of them are available from the authors upon request.

11 It is important to point out that we do not fish for factors. We guide our initial selection of candidates by using the 11 most popular and successful factors that have been employed in the analysis of the cross-sectional variability of past average returns.

12 In all the cross-sectional regressions reported in the next sections, we estimate betas using the same rolling window regressions of the realized returns on the risk factors with 60 months of data.

13 It is true, however, that the market VRP losses statistical significance when we employ adjusted standard errors.

14 We perform two robustness analysis. In the first exercise, we use a different sorting procedure and construct 20 portfolios using the market betas of realized returns, as in the traditional tests of the cross-sectional pricing literature. The cross-sectional results are also highly significant, although they are not as impressive as in Table 5. The KRS R-squared value is 66.4% instead of the 83.6% reported in Table 5. In a second exercise, we repeat the analysis using the Jiang and Tian (2005) procedure to estimate the model-free implied variances. The difference with respect to Martin (2017) is that the out-of-the-money options are weighted by the inverse of the square of their strikes. Overall, the results do not change significantly relative to the ones reported in this paper. The results are available from the authors upon request.

Additional information

Funding

The authors acknowledge financial support from the Ministry of Science, Innovation and Universities through grant PGC2018-095072-B-I00. In addition, Belén Nieto and Gonzalo Rubio acknowledge financial support from Generalitat Valenciana grant Prometeo/2017/158 and the Bank of Spain, and Ana González-Urteaga acknowledges financial support from the Ministry of Science, Innovation and Universities through grants ECO2016-77631-R (AEI/FEDER.UE) and PID2019-104304GB-I00 and UPNA Research Grant for Young Researchers, Edition 2018.