Forecasting market index volatility using Ross-recovered distributions

Abstract

The Ross recovery theorem shows that option data can reveal the market’s true (physical) expectations. We adapt this approach to international index options data (S&P, FTSE, CAC, SMI, and DAX) to improve volatility forecasting. We separate implied volatility into Ross-recovered expected volatility and a risk preference proxy. We investigate the performance of these variables, constructed domestically or globally, to forecast realized volatility as well as index excess returns. The results show evidence of significantly improved forecasts and yield new insights on the international dynamics of risk expectations and preferences. Across indexes, models using Ross-recovered, value-weighted global measures of risk preferences perform best. The findings suggest that the recovery theorem is empirically useful.

Keywords:

• Ross recovery
• Risk-neutral
• Volatility
• Forecast
• Options
• International
• Physical distribution

Acknowledgement

The authors gratefully acknowledge financial support from SSHRC, FRQSC, the Fonds de recherche AMF-GIRIF, the Chaire de recherche Industrielle-Alliance, and the Salles des Marchés FSA Jean-Turmel et Carmand-Normand. We thank conference participants at the Optionmetrics annual conference, French Finance Association (AFFI) meetings, INFINITI International Finance conference, Canadian Economic Science meetings [Société canadienne de science économique], Trondheim Business School Finance & Banking Conference, and in workshops at the Norwegian University of Science & Technology and Aix-en-Provence. Any errors are our own.

Disclosure statement

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

Notes

1 For the USA, in effect, RNV is the VIX.

2 Jensen et al. (2019) show that Ross-recovered volatility predicts future realized volatility, but their analysis is only for the US market.

3 A state of nature could be defined by any number of variables. For example, state i could be defined as a state where the S&P500 is at 2200, market volatility is at 20%, and the US economy is in a recession. A change in any of those variables would mean that we are now in a different state of nature. Empirically however, we are limited by the kind of contingent claims that are traded on the market. Thus, we implicitly assume that the possible future states of nature are completely defined by the strike prices of an asset.

4 Moneyness is the strike relative to the level of the index that day (mn = strike / level of index).

5 Our robustness analysis suggests that using 21 points in moneyness is an appropriate tradeoff given our goal of measuring variance of the recovered distribution. For other purposes, one could obtain a finer distribution of recovered expectations with the help of some additional manipulations. Once the pricing kernel for a coarse grid of moneyness is obtained, one can fill in its gap by fitting it to a curve (e.g., a spline) and then use it to transform a fine risk-neutral distribution into an equally fine probability distribution of recovered expectations.

6 Using 100 maturities over 2 years corresponds to a step of about 1 week. This level of resolution is needed in order to fully capture the information when options are available at weekly maturities.

7 Note that using VIX instead of RNV for the U.S. S&P makes no difference in the results.

8 As an indication, over the full period the average weights for the S&P500, FTSE, CAC, SMI, and DAX are respectively 69.4%, 14.3%, 6.0%, 4.8%, and 5.5%.

9 E.g., REV = Ross-recovered expected volatility at time t minus past realized volatility from t-h to t.

10 The correlation coefficients between the volatilities and their equally-weighted global counterpart in that case are 0.74 and 0.79 (RNV, REV) and 0.58 for the risk preferences (REV-RNV).

11 The BIC measure for the DAX is the exception.

12 For brevity, 4-month horizon results on excess returns forecasts are omitted from the paper, but available upon request.

13 Given the heavy weight of the S&P500, we could argue that the global measures are close to the domestic US measures, but with additional international information.

Additional information

Funding

This work was supported by Fonds de Recherche du Québec-Société et Culture [grant number 01]; Chaire de recherche FSA Industrielle-Alliance [grant number 01]; Fonds AMF-GIRIF [grant number 01]; Social Sciences and Humanities Research Council of Canada [grant number 435].