## Abstract

A recent strand in the literature has investigated the relationship between idiosyncratic risk and future stock returns. Although several authors have found significant predictive power of idiosyncratic volatility, the magnitude and direction of the dependence is still being debated. Using a sample of all S&P 100 constituents, we identify positive risk premia for option-implied idiosyncratic risk. Depending on the model used to identify unsystematic risk, we observe a statistically and economically significant average annual premium of 1.72 percent. To investigate whether this impact is driven by the definition of idiosyncratic risk, we extend the pricing kernel by implied skewness. Using a double-sorting procedure, we show that the compensation of unsystematic risk is mainly driven by firms with high positive implied skewness.

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## Notes

- 1.
- 2.
See, e.g., Vanden (2006) and the references therein.

- 3.
See, e.g., Bakshi and Madan (2006), Bollerslev et al. (2009), and the references therein on volatility and variance spreads. This collection is by no means exhaustive. Furthermore, see Driessen et al. (2009), Krishnan et al. (2009), and Buss and Vilkov (2011) for empirically observable correlation risk premia.

- 4.
A great deal of literature compares the predictive power of implied and historical measures for volatility. Early research in this field found that estimators based on implied volatility typically outperform estimates based on historical volatility. See, e.g., Latané and Rendleman (1976), Schmalensee and Trippi (1978), and Beckers (1980). Contrary to these findings, Canina and Figlewski (1993), Day and Lewis (1992), and Lamoureux and Lastrapes (1993) deny predictive power of implied volatility. More recent work, for example, includes Blair et al. (2001), Lehar et al. (2001), Jiang and Tian (2005), and DeMiguel et al. (2011). Buss and Vilkov (2011), among others, find a high predictive power of intraday return data to estimate volatility. Others find a reasonable empirical fit for component models, such as Zhu (2009).

- 5.
See Buss and Vilkov (2011).

- 6.
Empirically, this was found by Krishnan et al. (2009).

- 7.
Buss and Vilkov (2011) show that the correlation MIDAS matrix estimator is positive semi-definite if and only if the correlation matrix under the physical measure is positive semi-definite and

*ρ*_{ t }≤0. - 8.
- 9.
The outline of this section is based on Meucci (2009).

- 10.
Examples of studies using Gaussian Random Matrix Theory for financial applications are Laloux et al. (2000), Drożdż et al. (2001), Plerou et al. (2002), and Malevergne and Sornette (2004). The robustness of the eigenvalue analysis is investigated by Rajkovic (2000), as well as by Sharifi et al. (2004).

- 11.
- 12.
For details on the estimation, see Meucci (2009).

- 13.
This definition of implied skewness might seems a bit crude at first sight. However, more sophisticated measures, such as Bakshi et al. (2003), typically require liquid option quotes for far out-of-the-money strike prices. These are usually unavailable for equity options.

- 14.
See also Pojarliev and Polasek (2003).

- 15.
Our results are robust to forming value-weighted portfolios.

- 16.
- 17.
- 18.
See Burda et al. (2011).

- 19.
See Kollo and Ruul (2003).

## References

Ang, A., Hodrick, R., Xing, Y., Zhang, X.: The cross-section of volatility and expected returns. J. Finance

**61**(1), 259–299 (2006)Ang, A., Hodrick, R., Xing, Y., Zhang, X.: High idiosyncratic volatility and low returns: international and further US evidence. J. Financ. Econ.

**91**(1), 1–23 (2009)Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab.

**33**(5), 1643–1697 (2005)Bakshi, G., Madan, D.: A theory of volatility spreads. Manag. Sci.

**52**(12), 1945–1956 (2006)Bakshi, G., Kapadia, N., Madan, D.: Stock return characteristics, skew laws, and the differential pricing of individual equity options. Rev. Financ. Stud.

**16**(1), 101–143 (2003)Barberis, N., Huang, M., Santos, T.: Prospect theory and asset prices. Q. J. Econ.

**116**(1), 1–53 (2001)Beckers, S.: The constant elasticity of variance model and its implications for option pricing. J. Finance

**35**(3), 661–673 (1980)Blair, B., Poon, S.-H., Taylor, S.: Forecasting S&P 100 volatility: the incremental information content of implied volatilities and high-frequency index returns. J. Econom.

**105**(1), 5–26 (2001)Bollerslev, T., Tauchen, G., Zhou, H.: Expected stock returns and variance risk premia. Rev. Financ. Stud.

**22**(11), 4463–4492 (2009)Burda, Z., Jarosz, A., Nowak, M., Jurkiewicz, J., Papp, G.: Applying free random variables to random matrix analysis of financial data. Part I: the gaussian case. Quant. Finance

**11**(7), 1103–1124 (2011)Buss, A., Vilkov, G.: Option-implied correlation and factor betas revisited. Working Paper (2011)

Canina, L., Figlewski, S.: The informational content of implied volatility. Rev. Financ. Stud.

**6**(3), 659–681 (1993)Carhart, M.: On the persistence of mutual fund performance. J. Finance

**52**(1), 57–82 (1997)Day, T., Lewis, C.: Stock market volatility and the information content of stock index options. J. Econom.

**52**(1–2), 267–287 (1992)DeMiguel, V., Garlappi, L., Uppal, R.: Optimal versus naive diversification: how efficient is the 1/N portfolio strategy? Rev. Financ. Stud.

**22**(5), 1915–1953 (2009)DeMiguel, V., Plyakha, Y., Uppal, R., Vilkov, G.: Improving portfolio selection using Option-Implied volatility and skewness. Working Paper (2011)

Diavatopoulos, D., Doran, J., Peterson, D.: The information content in implied idiosyncratic volatility and the Cross-Section of stock returns: evidence from the option markets. J. Futures Mark.

**28**(11), 1013–1039 (2008)Driessen, J., Maenhout, P., Vilkov, G.: The price of correlation risk: evidence from equity options. J. Finance

**64**(3), 1377–1406 (2009)Drożdż, S., Kwapień, J., Grümmer, F., Ruf, F., Speth, J.: Quantifying the dynamics of financial correlations. Physica A

**299**(1–2), 144–153 (2001)Fama, E., French, K.: Common risk factors in the returns on stocks and bonds. J. Financ. Econ.

**33**(1), 3–56 (1993)Fama, E., MacBeth, J.: Risk, return, and equilibrium: empirical tests. J. Polit. Econ.

**81**(3), 607–636 (1973)Focardi, S., Fabozzi, F.: The Mathematics of Financial Modeling & Investment Management. Wiley, New York (2009)

Fu, F.: Idiosyncratic risk and the cross-section of expected stock returns. J. Financ. Econ.

**91**(1), 24–37 (2009)Geman, S.: A limit theorem for the norm of random matrices. Ann. Probab.

**8**(2), 252–261 (1980)Goetzmann, W., Kumar, A.: Equity portfolio diversification. Rev. Finance

**12**(3), 433–463 (2008)Harvey, C., Siddique, A.: Conditional skewness in asset pricing tests. J. Finance

**55**(3), 1263–1295 (2000)Jiang, G., Tian, Y.: The model-free implied volatility and its information content. Rev. Financ. Stud.

**18**(4), 1305–1342 (2005)Kollo, T.O., Ruul, K.: Approximations to the distribution of the sample correlation matrix. J. Multivar. Anal.

**85**(2), 318–334 (2003)Kraus, A., Litzenberger, R.: Skewness preference and the valuation of risk assets. J. Finance

**31**(4), 1085–1100 (1976)Krishnan, C., Petkova, R., Ritchken, P.: Correlation risk. J. Empir. Finance

**16**(3), 353–367 (2009)Laloux, L., Cizeau, P., Bouchard, J.-P., Potters, M.: Noise dressing of financial correlation matrices. Phys. Rev. Lett.

**83**(7), 1467–1470 (1999)Laloux, L., Cizeau, P., Potters, M., Bouchard, J.-P.: Random matrix theory and financial correlations. Int. J. Theor. Appl. Finance

**3**(3), 391–397 (2000)Lamoureux, C., Lastrapes, W.: Forecasting stock-return variance: toward an understanding of stochastic implied volatilities. Rev. Financ. Stud.

**6**(2), 293–326 (1993)Latané, H., Rendleman, R.: Standard deviations of stock price ratios implied in option prices. J. Finance

**31**(2), 369–381 (1976)Ledoit, O., Wolf, M.: Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. J. Empir. Finance

**10**(5), 603–621 (2003)Ledoit, O., Wolf, M.: A well-conditioned estimator for large-dimensional covariance matrices. J. Multivar. Anal.

**88**(2), 365–411 (2004)Lehar, A., Scheicher, M., Strobl, G.: Trade versus time series based volatility forecasts: evidence from the austrian stock market. Financ. Mark. Portf. Manag.

**15**(4), 500–515 (2001)Lehmann, B.: Residual risk revisited. J. Econom.

**45**(1–2), 71–97 (1990)Lintner, J.: The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat.

**47**(1), 13–37 (1965)Malevergne, Y., Sornette, D.: Collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices. Physica A

**331**(3–4), 660–668 (2004)Marčenko, V., Pastur, L.: Distribution of eigenvalues for some sets of random matrices. Math. USSR Sb.

**1**(4), 457–483 (1967)Merton, R.: A simple model of capital market equilibrium with incomplete information. J. Finance

**42**(3), 483–510 (1987)Meucci, A.: Managing diversification. Risk

**May**, 74–79 (2009)Plerou, V., Gopikrishnan, P., Rosenow, B., Nunes Amaral, L., Guhr, T., Stanley, H.: Random matrix approach to cross correlations in financial data. Phys. Rev. E

**65**(6), 066126 (2002)Pojarliev, M., Polasek, W.: Portfolio construction by volatility forecasts: does the covariance structure matter? Financ. Mark. Portf. Manag.

**17**(1), 103–116 (2003)Rajkovic, M.: Extracting meaningful information from financial data. Physica A

**287**(3–4), 383–395 (2000)Schmalensee, R., Trippi, R.: Common stock volatility expectations implied by option premia. J. Finance

**33**(1), 129–147 (1978)Sharifi, S., Crane, M., Shamaie, A., Ruskin, H.: Random matrix theory for portfolio optimization: a stability approach. Physica A

**335**(3–4), 629–643 (2004)Vanden, J.: Option coskewness and capital asset pricing. Rev. Financ. Stud.

**19**(4), 1279–1320 (2006)Zhu, J.: Pricing volatility of stock returns with volatile and persistent components. Financ. Mark. Portf. Manag.

**23**(3), 243–269 (2009)

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### Cite this article

Süss, S. The pricing of idiosyncratic risk: evidence from the implied volatility distribution.
*Financ Mark Portf Manag* **26, **247–267 (2012). https://doi.org/10.1007/s11408-012-0183-4

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### Keywords

- Idiosyncratic risk
- Implied volatility
- Implied skewness
- Principal portfolios
- Random matrix theory

### JEL Classification

- G12