Option Implied Volatility as a Factor in Portfolio Models

Ian Kaplan

Independent Study, Spring 2013

Abstract

Options implied volatility can be viewed as an indicator of informed opinion about the actual future volatility of the stock or index. A number of papers have speculated that this information can be used as a factor to predict future returns.

This paper discusses computational experiments that use options volatility factors that were described in a paper by Baltussen et al [1]. This paper by Baltussen et al applies options factors to a large portfolio of stocks.

In this paper the options volatility factors are derived from S&P 500 options (options ticker SPX). The power of these options factors to predict S&P 500 returns is investigated.

Linear models were constructed to evaluate the power of the options factors are predictors of future returns. One of these factors, volatility skew, is a reasonably strong factor when it comes to explaining current returns. As predictors for future returns, this options factors is extremely weak and could not be used in practice. The other options factors are even weaker, both for current returns and future returns.

Option Implied Volatility as a Factor in Stock Returns

One reason that market inefficiencies exist is because information is not always widely available. Unlike daily stock price information, which can be easily downloaded from sites like Yahoo Finance, options information generally must be purchased. Option information is also more complicated to analyze than stock price information. These characteristics lend credibility to the idea that options information could be under-utilized in the market.

An Experiment

The paper by Baltussen et al uses options implied volatility information for a relatively large portfolio of stocks (and the related options).  These stocks are drawn from the S&P 500/Citigroup Broad Market Index equities. The authors used options data from OptionMetric and stock price information from CRSP.  

Baltussen et al constructed several different portfolios in their study, consisting of between 100 and 1250 stocks. Using information from the options factors, they took a long position in some stocks and a short position in other stocks.

A large database of options data, of the type used by Baltussen et al is costly. Perhaps an even higher barrier to application is the complexity of processing such a large database to extract the options factors.  The fact that the authors carried out such a large analysis is an impressive accomplishment.

The costs related to exploiting options data serve as a barrier of entry and hold out the possibility that the information encapsulated in factors constructed from options implied volatility may not be fully reflected in asset prices.

The experiments reported in this paper use S&P 500 options data (SPX) and the daily data from the S&P 500 index, since I do not have access to the large OptionMetrics database used by Baltussen et al (one of my classmates kindly shared the S&P 500 options data).

In this paper I explore whether the factors reported by Baltussen et al, calculated from the SPX options, can be used as factors to predict S&P 500 returns.

Option Implied Volatility Factors

In this paper, three options implied volatility factors from Baltussen et al where investigated. These are:

1. 1.Out-of-the-money options skew

   $({\mathit{SKEW}}^{\mathit{OTM}})$

    

    
2. 2.Realized vs. Implied Volatility

   $(\mathit{RVIV})$

   .

    

   . 
3. 3.At-the-money volatility skew

   $({\mathit{SKEW}}^{\mathit{ATM}})$

    

   . 

Baltussen et al report a fourth option factor, the change in the at-the-money skew. This factor is problematic because it does not fit with the other options factors (there are 51 values, instead of 52) so this factor was omitted from the study.

Options Period

Following Baltussen et al, the options factors were calculated from options that where within 10 to 40 trading days of expiration. The more recent options (2011 vs. 2005) include Saturday, since some options expire on Saturday, although there is no options trading after market close on Friday. Option quotes with zero volume or zero open interest were filtered out.  

At-the-money options

An option is defined as at-the-money (ATM) if the ratio of the strike price to the underlying stock price is between 0.95 and 1.05, for both calls and puts.

Out-of-the-money options

A put is defined as out-of-the-money (OTM) if the ratio between its strike price and the stock price is lower than 0.95 but higher than 0.80.

Weekly Time Period

The implied volatility values for the options are averaged over a period of five trading days.  Baltussen et al comment that this reduces the noise in the volatility value. The time period is relative to the option quote date.

Out-of-the-money skew

$({\mathit{SKEW}}^{\mathit{OTM}})$formula

According to Baltussen et al, OTM option skew may reflect worries in the market about negative price movements. When market participants have concerns about negative price movements of an asset, they are more likely to purchase put options to hedge their long positions. The demand for put options leads to a higher put option price and increased implied volatility. Stocks with a higher OTM skew would, as a result, under-perform compared to stocks with less skew.

The out-of-the-money volatility skew is calculated as follows:

${\mathit{SKEW}}_{i,t}^{\mathit{OTM}}={\mathit{IV}}_{i,t}^{\mathit{OTM}-\mathit{put}}-{\mathit{IV}}_{i,t}^{\mathit{ATM}-\mathit{call}}$

where

${\mathit{IV}}_{i,t}^{\mathit{OTM}-\mathit{put}}$

is the implied volatility of out-of-the-money put options and

${\mathit{IV}}_{i,t}^{\mathit{ATM}-\mathit{call}}$

is the implied volatility of at-the-money call options.

Realized (historical) vs. Implied Volatility (RVIV)

The RVIV factor is calculated as follows:

${\mathit{RVIV}}_{i,t}={\mathit{RV}}_{i,t}-{\mathit{IV}}_{i,t}^{\mathit{ATM}}$

The realized volatility

${\mathit{RV}}_{i,t}$

 

is the volatility over the past twenty trading days for the underlying asset (in this case the S&P 500). The

${\mathit{IV}}_{i,t}^{\mathit{ATM}}$

 

is the implied volatility of the ATM call and ATM put options averaged over week t.

According to Baltussen et al this factor is though to capture the volatility risk of the underlying asset. Stocks with a high RVIV are expected to under-perform those with a low value.

At-the-money Implied Volatility Skew (

${\mathit{SKEW}}^{\mathit{ATM}}$formula

 

)

An increase in the ATM IV skew is thought to reflect a pessimistic outlook for the underlying asset. Stocks with a high at-the-money skew are expected to under perform stocks with low skew.  The

 

${\mathit{SKEW}}^{\mathit{ATM}}$

 

factor is calculated as follows:

${\mathit{SKEW}}_{i,t}^{\mathit{ATM}}={\mathit{IV}}_{i,t}^{\mathit{ATM}-\mathit{put}}-{\mathit{IV}}_{i,t}^{\mathit{ATM}-\mathit{call}}$

where

${\mathit{IV}}_{i,t}^{\mathit{ATM}-\mathit{put}}$

 

is the implied volatility of the at-the-money put option (averaged over a five day trading period) and

 

${\mathit{IV}}_{i,t}^{\mathit{ATM}-\mathit{call}}$

 

is the 1-week average for the ATM call option.

Weekly Trading

Baltussen et al use the options factors to rebalance their portfolio on a weekly basis. The same approach is followed in the experiments described here.

Calculating the Factors

R code was written to process the options data and calculate the options factors.

If the options factors were used in actual trading, the factors would be calculated as week t, to predict the returns one week in the future (week t+1).

In evaluating whether the options factors are, in fact, predictive of future returns, a historical model was constructed where the options factors at week t were used to predict the actual weekly return at week t+1.

Weekly returns were calculating as follows:

$R_{t+1}=\frac{(P_{t+1}-P_t)}{P_t}$

where

$P_t$

is the S&P 500 price and

$P_{t+1}$

is the price five trading days later (again, to simulate actual trading).

A linear model was used to evaluate whether the options factors had predictive power, relative to the returns at week t+1. The data for the regression has the following format:

 

Table 1 Linear Regression Data

 

The weekRet column consists of the weekly returns at time t+1. The skew, rviv, and skewATM columns consist of the options factors calculated for week t.

In this experiment options data for two years, 2005 and 2011, was investigated. The results for both years was very similar and I expect that other years would not differ substantially.

Regression Result

The linear regression results, regressing the 52 factors (calculated at time t) against 52 weekly returns (calculated at time t+1) is shown below.

 

 

Table 2 Linear Regression for 2005 Options Data

 

$R^2=0.056524$

 

(2005)

Table 3 Linear Regression for 2011 Options Data

 

$R^2=0.036985$

(2011)

The p-values and the standard errors suggest that the regression is unreliable. This conclusion is amplified by the

$R^2$

values, which are very low.

According to the model, the options factors have a negative relationship to the returns, so they should be negative regression

 

$\beta$

values. This is only the case for the

${\mathit{SKEW}}^{\mathit{OTM}}$

value.

Residual Analysis

The plots below show the standardized residuals vs. the fitted values and the QQ-plot of the standardized residuals.

One interpretation of these plots would be that the linear model does a good job of describing the relationship between the option factors and the future returns, since the residual values have close to a Gaussian distribution.

This interpretation would be incorrect, however. As it turns out, the relationship between the options factors and the future returns is highly random, so the residuals are random as well.

Figure 1 (2005) Residual Analysis - Weekly Return ~ skew + rviv + skewATM

 

Figure 2 (2011) Residual Analysis Weekly Return ~ skew + rviv + skewATM

 

Weekly Returns vs. Individual Options Factors

This section shows weekly returns plotted against each of the options factors, for 2005 and 2011. The regression plots mirror the regression summary statistics: the regression is highly unreliable.

Figure 3 (2005) S&P 500 Weekly Returns vs. SPX Volatility Skew

 

 

Figure 4: (2011) S&P Weekly Returns vs. SPX Volatility Skew

 

Figure 5 (2005)  Weekly Returns vs. (Actual volatility – SPX at-the-money Implied Volatility)

 

 

Figure 6 (2011)  Weekly Returns vs. (Actual volatility – SPX at-the-money Implied Volatility)

 

 

Figure 7  (2005) ATM Put implied volatility - ATM Call implied volatility

 

 

Figure 8 (2011) ATM Put implied volatility - ATM Call implied volatility

 

Modeling the Predictors with the Current Return

The linear model summary statistics for the options factors regressed against the weekly returns calculated for the same week are shown below.

All three of the regression

$\beta$

values are negative, which corresponds to the way that  Baltussen et al suggest these factors behave.

The standard error and the p-values suggest that the out-of-the-money skew

$({\mathit{SKEW}}^{\mathit{OTM}})$

and the realized vs. implied volatility factors are not reliable, at-the-money skew appears to be highly reliable.

 

 

Table 4  2011 option factors regressed against the current week returns

 

$R^2=0.325964$

The

$R^2$

value also suggests a much more reliable regression.  The regression plot for the current weekly returns vs. the ATM volatility skew is shown below.

Since about 30% of the current return behavior is described by the ATM volatility skew, this might be a useful factor in risk analysis. Outside of risk analysis, where we are trying to estimate the risk “today”, a factor for today's return is not very useful.

An Experiment with Wavelet Smoothed Values

Perhaps one reason that the regression between the volatility predictors and the future return has so much error is that there is a lot of (Gaussian) noise in the return.  This experiment investigates whether reducing the amount of Gaussian noise in the return time series will improve the linear regression.

Wavelet thresholding was used to smooth the return time series at time t.

The (historical) future return at time

$t+1$

is the calculated from the wavelet smoothed price at time t, with the future price at time

 

$t+1\mathrm{:}{R}_{t+1}=\frac{(P_{t+1}-{\wp }_t)}{{\wp }_t}$

where

${\wp }_t$

is the wavelet smoothed price at time t.

Calculating the returns using the smoothed price at time t turns out to be a very bad idea. The summary statistics for the linear model is shown below. Here the

$\beta$

values are negative (which fits the conceptual model) but the error is very large and the p-values show that the regression values are unreliable.

 

Note that the p-value for the ATM skew (skewATM) as gone from 0.2 to 0.8.  The R2 value has gotten even worse:

$R^2=0.019006\mathit{vs.}{R}^2=0.036985$

 

when the regression is done against the standard returns.

Discussion

In a retrospective re-reading of Baltussen et al's paper the flaws in their analysis are more stark than they were on the first few readings.

The S&P 500 options implied volatility factors appear to have no value when it comes to predicting future returns. A variety of other factors, like momentum, have stronger predictive value.

The strongest factor in the paper by Baltussen et al is the ATM skew factor. The ATM skew factor may have utility in a risk model since about 30% of the current return behavior of the S&P 500 can be ascribed to this factor. Unfortunately, as a factor in an “alpha” model of future returns, this factor has no value.

The experiments reported in this paper use S&P 500 options and underlying values. In contrast,  Baltussen et al examine large portfolios of individual stocks. The two sets of results are not directly comparable. However, the fact that the S&P 500 options factors fail so dramatically to predict future returns raises questions about the results presented by Baltussen et al.

Baltussen et al also analyze the options factors in linear models. Their

$R^2$

 

errors are very low (e.g., nearer to zero than to 1), ranging form 0.5% to 1.1% for the three options factors discussed here.  These results are, again, difficult to compare to the results in this paper, since Baltussen et al are doing cross sectional regression of the factors across the asset returns. But the low

 

$R^2$

 

value suggests a linear model that is highly unreliable. How factors with such low predictive power can produce the excess returns reported in this paper is unclear.

By using a large portfolio of stocks that is rebalanced weekly, Baltussen et al obscure the factors that contribute to the excess return they claim is delivered by the options factors. For example, the excess return could simply be due to the high portfolio turn over.  They do compare the options factors to several of the Fama-French factors. The Fama-French factors have higher predictive power, suggesting that the options factors would not be a good choice for an alpha model.

2Chalamandaris, Georgios; Rampolis, Leonidas S; Exploring the role of realized return distribution in the formation of the implied volatility smile, Jornal of Banking and Finance, 2012 (Vol 36, No. 4)

1Baltussen, Guido;  van der Grient, Bart; de Groot, Wilma; Hennink, Erik and Zhou, Weili; Exploiting Option Information in the Equity Market, Financial Analysts Journal, Vol 68, Number 4, July/August 2012