Stock Return Autocorrelation and Individual Equity Option Prices

This study demonstrates empirically the impact of stock return autocorrelation on the prices of individual equity option. The option prices are characterized by the level and slope of implied volatility curves, and the stock return autocorrelation is measured by variance ratio and first-order serial return autocorrelation. Using a large sample of U.S. stocks, we show that there is a clear link between stock return autocorrelation and individual equity option prices: a higher stock return autocorrelation leads to a lower level of implied volatility (compared to realized volatility) and a steeper implied volatility curve. The stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. The relation between stock return autocorrelation and option price structure is more pronounced when market is volatile, especially during financial crisis. The stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. Thus, stock return autocorrelation can help differentiate the price structure across individual equity options.

It is an empirical question, as to the extent that stock return autocorrelation affects option prices. To the best of our knowledge, there exists no empirical studies exploring the relative contribution of stock return autocorrelation in option pricing. In this paper, we fill the gap by investigating the impact of stock return autocorrelation on the individual equity option price structure and demonstrate a clear link between them. Our hope is that a better understanding of the sources of option price structure will help guide us in the future as we work to improve option pricing models and option trading strategies. Following, Lo (2004), Griffin, Kelly, andNardari (2010), andCao et al. (2018), we employ two measures of stock return autocorrelation: variance ratios and first-order stock return autocorrelation.
We start by testing the random walk hypothesis for optional stocks. At the 10% (5%) significance level, the random walk assumption is rejected for about one third (one quarter) of stocks in our sample, suggesting the discrepancy between the assumption of the option pricing model and the data. Using 6,137 stocks from 01/1996 to 04/2016, we demonstrate a clear link between the stock return autocorrelation and individual equity option prices. Specifically, stocks with higher return autocorrelation exhibit a lower level of implied volatility (compared to realized volatility) and a steeper implied volatility curve. We find that the stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. We also find that the relation between stock return autocorrelation and option price structure is more pronounced when market is volatile, especially during financial crisis. The level of stock price autocorrelation can help differentiate the price structure across individual equity options.
We contribute to the extant literature which documents the impact of stock return autocorrelation on option prices empirically. Lo and Wang (1995) demonstrate that option value is a function of the absolute value of the first-order autocorrelation coefficient, with the increase in autocorrelation decreasing Black-Scholes option prices. They also suggest that predictability also affects option prices nontrivially for option pricing models with stochastic volatility or jump component. By modeling the index autocorrelation by the ARMA model and incorporating the autocorrelation into Rubinsten (1996) results, Jokivuolle (1998) shows that the autocorrelation enters into the option pricing formula by adjusting volatility and underlying index value, both of which enters into option pricing models.
between the individual equity prices and stock return autocorrelation.
The rest of the paper is organized as follows. Section 2 discusses variable definitions. Section 3 presents our empirical results including univariate soring, Fama-Macbeth regressions and robustness checks. Section 4 concludes.

Option Price Structure
Following An et al. (2014) and Yan (2011), we use the interpolated volatility surface computed by OptionMetrics to construct individual equity option price structure. OptionMetrics computes option implied volatility using binomial trees and the interpolated volatility surface is then constructed using a kernel smoothing algorithm. One advantage of using the volatility surface is that it avoids having to make potentially arbitrary decisions on which strikes or maturities to include in computing an implied call or put volatility for each stock (see An et al., 2014). Since we are looking at the monthly frequency, we use the 30-day interpolated volatility surface at the last trading day of each month following An et al. (2014). Bollen and Whaley (2004) use the Black-Scholes deltas to measure moneyness. Based on their deltas, options are placed into five moneyness categories. Yan (2011) uses options whose deltas equal to the average of the upper and lower bound of each moneyness category, as in Bollen and Whaley (2004), to define the moneyness of options. Following Bollen and Whaley (2004) and Yan (2011), we use standard options on the implied volatility surface with deltas equal to 0.5, -0.5, and -0.25 as ATM calls, ATM puts, and OTM puts on the last trading day of each month.

: Implied-Realized Volatility Spread
The options implied volatility, , for stock in month , is defined as the average implied volatility of ATM calls and ATM puts, with deltas equal to 0.5 and -0.5 on the implied volatility surface on stock at the end of month , respectively. Follow Bali and Hovakimian (2009)  This measure captures the absolute deviation of the ratio of return variances measured over 2 weeks to those measured over 1 week from 1, which is the expected value of the ratio under the random walk hypothesis. Greater deviation of the variance ratio from 1 signals higher serial return autocorrelation.
Balancing between estimation efficiency from a larger sample and the relatively shorter option maturities, we opt for a one-year (52 weeks) rolling window. Specifically, at the end of each month, we calculate variance ratio and the absolute deviation of variance ratio from 1 using return data in the past one year (52 weeks).

First-order Stock Return Autocorrelation
As an alternative measure of stock return autocorrelation, we also examine the absolute value of first-order monthly stock return autocorrelation with 3-year rolling windows, following Lo and Wang proxies for greater deviation from random walk.

Control Variables
In order to rule out the possible effects on option prices from other firm-specific characteristics and factors, we include a number of control variables in our empirical tests. Duan and Wei (2009) on the excess return of S&P 500 for the past 24 months. Following Fama and French (1992), the firm size from July of year to June of year are measured based on the market equity in June of year . Stock trading volume is defined as the total number of shares traded in the month. We adopt 12-month moving average Amihud (2002) illiquidity ratio to proxy for stock illiquidity. For each firm, we sum the trading volumes of all options that meet our requirements from each day in each month, as our measure of option liquidity. Where is the maximum number of non-overlapping -week returns.

Summary Statistics
For each stock in our sample, we calculate the score using all the daily data. Table 2 report the percentage of stocks for which the random walk hypothesis is rejected, for from 2 to 16, with weekly stock returns from January 1996 to April 2016. At $10% significance level, the random walk (stock return is unpredictable) hypothesis is rejected for about one third of stocks in our sample. At 5% significance level, the random walk hypothesis is rejected for about one quarter of stocks in our sample.
The discrepancy between the data and the assumption of the option pricing model and the data indicates that the measure of stock return autocorrelation may help explain the individual equity option price structure.

Fama-MacBeth Regressions
In this part, we use the Fama-MacBeth (see Fama and Macbeth, 1973) regressions to examine the cross-sectional relation between stock return autocorrelation and the option price structure with or without control variables. Table 3 and 4 present the Fama-MacBeth regression results of and on , respectively, from January 1996 to April 2016. Column (1) of Table 3 and   Table 4 report the univariate regression results of and on . Column (2)-(5) report the regression results of and on with control variables, ( ) and option liquidity ( ).

Univariate Sorting
To gauge the magnitude of impact of stock return autocorrelation on option price structure, we perform univariate sorting. On the last trading day of each month, stocks are ranked, in ascending order according to , into quintiles, and five portfolios are formed by equally weighting the stocks within that quintile. We then record the implied and realized volatility spread and the slope of implied volatility curves. Repeating these steps for every month in the sample period of January 1996 to April 2016 generates the time series of monthly options price structure for the five quintiles. We then calculate the time-series average of the monthly portfolio option prices and report them in Table 5.
Each quintile portfolio has 370 stocks per month, on average. Portfolio Q1 contains stocks with lowest , which is only 0.03 for Q1, indicating that the variance ratio for stocks in Q1 is very close to 1.
Portfolio Q5 contains stocks with highest of 1.19. and Q1 is -0.9%, statistically and economically significant. The spread between Q5 and Q1 is -0.21% with a -statistic of 3.1. The univariate regression results are consistent with the Fama-MacBeth regressions. The impact that the stock returns autocorrelation has on the level of implied volatility curve is stronger than that on the slope of the implied volatility curve. It is also noticeable that the patterns of monthly and are somewhat flat from Q1 from Q4.
To make sure that our results are not driven by outliers, we eliminate the stocks whose is in the top or bottom 1% for each month and the results remain quantitatively similar.

Robustness Check
We also perform several robustness checks. We start by replicating the Fama-MacBeth regressions using alternative measure of stock return autocorrelation, absolute value of first-order stock return autocorrelation and find that our main empirical results remain: stocks with strong return autocorrelation exhibit a lower level of implied volatility (compared to realized volatility) and a steeper implied volatility curve.
We To check if our main results are driven by relatively small stocks (Note 1) in our sample, we repeat the univariate sorting within small, median and large stocks in our sample. We find that the relationship between and stock return autocorrelation is most significant for smaller stocks in our sample. As in Panel A of Figure 1, the spread between Q5 and Q1 increases from relatively small firms to big firms. For example, the spread sorting by is -1.90, and -0.60 for small and large stocks respectively. The measure of stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. The spread between top and bottom portfolio sorting by is statistically significantly for medium-size and large stocks in our sample.
We then perform the univariate sorting before, during and after financial crisis. Our main conclusions hold in all subsamples, with the results being most significant during financial crisis, followed by before crisis for and after crisis for the , as shown in Panel B of Figure 1. For example, the difference of between Q5 and Q1 sorting by is -1.01, -1.53, and -0.56 before, during and after crisis, with -statistics of -7.84, -3.45 and -1.80, respectively. The difference of between Q5 and Q1 are -0.09, -1.02, and -0.23 before, during and after crisis, with -statistics of -0.79, -2.67, and -1.93 respectively.
It is also interesting to see if the results differ under different market conditions, especially when market is volatile. We then separate our data into two subsamples by median VIX of our entire sample and repeat the univariate soring for both samples. We plot the and of the long-short stock portfolio formed on for the two subsamples in Panel C of Figure 1. We find that the relation between stock return autocorrelation and implied-realized volatility spread is much stronger when market is volatile. The measure of stock return autocorrelation is more important in explaining the level of implied volatility curve when the market is volatile. The relations between the measure of stock return autocorrelation and implied volatility slope does not differ when market is volatile or not.

Conclusions
The asset returns are assumed to be distributed independently of each other, in various option pricing models. However, the evidence of persistence autocorrelation in asset returns of both the short-term and long-term period contradicts the assumption made in the option pricing models. Following theses researches, literature documents that stock return autocorrelation enters into the option pricing formula through adjustments in volatility and/or expected asset price. It is an empirical question, as to the extent that stock return autocorrelation affects option prices. To the best of our knowledge, there exists no empirical studies exploring the relative contribution of stock return autocorrelation in option pricing. In this paper, we fill the gap by investigating the impact of stock return autocorrelation on the individual equity option price structure and demonstrate a clear link between them. With the increases of serial correlation of stock returns, the option implied volatility becomes smaller compared to realized volatility and the implied volatility curve becomes more negatively skewed. The relation between stock return autocorrelation and option price structure is more pronounced when market is volatile, especially during financial crisis. The stock return autocorrelation is more important in explaining the level of implied volatility curve for relatively small stocks. Our hope is that a better understanding of the sources of option price structure will help guide us in the future as we work to improve option pricing models and option trading strategies.