Differential Pricing Effects of Volatility on Individual Equity Options Mobina Shafaati Abstract This study analyzes the impact of volatility on the prices of individual equity options. Using the daily option quotes on individual equity options, we show that call option price is increasing with the underlying idiosyncratic volatility and decreasing with the underlying systematic volatility. Furthermore, put option price is an increasing function of both idiosyncratic and systematic volatility, while its sensitivity to the systematic component of volatility is higher than the idiosyncratic component. The novelty of this study is that to analyze the relation between volatility and option price, we account for the non-zero correlation of the underlying asset price and volatility. Thus, the pricing effect of volatility on options is also depending on the source of the volatility. Shafaati is with the E. J. Ourso College of Business, Louisiana State University. I am especially grateful to Professor Don Chance for his guidance, suggestions, and encouragement. I also thank Professor Robert Brooks and Professor Junbo Wang for helpful comments and discussions.
1 Introduction This paper investigates the relationship between individual equity option price and the volatility of the underlying stock s returns, while we account for the non-zero correlation of the underlying stock price and volatility. Unlike the commonly accepted belief that volatility of stock s returns is beneficial to the value of options, this paper argues that the underlying volatility can be either beneficial or detrimental to the value of equity options depending on the source of volatility. Although, the higher idiosyncratic volatility is beneficial to the value of both call and put option, higher systematic volatility is associated with lower (higher) value of the call (put) option. The main reason for this seemingly counterintuitive result is that in addition to the direct effect, volatility affects the option value indirectly through the underlying asset value. The positive relation between volatility and option value has been explained intuitively and analytically in previous literature. When the underlying volatility of the call option is higher, the probability of a large positive change in the underlying stock price is higher, and so is the probability of the call option maturing in the money. The higher volatility also increases the probability of a large negative change in the underlying stock price; however, the higher chance of decrease in stock price is irrelevant to the call option value, because the lower limit of the option value cannot be less than zero. Therefore, higher volatility increases the expected payoff of the option and consequently increases the value of the option. Similar reasoning can be provided for the put option, with this difference that an increase in the probability of large negative change in the underlying stock price is valuable, and positive change is irrelevant. The results of my study contribute to the literature of option pricing as it challenges the somewhat simplistic view that volatility and option value are directly related. This result is a standard paradigm that appears throughout finance. Yet this paradigm holds only under the restrictive condition that volatility has no effect on the price of the underlying. For equities, this condition is antithetic to the core principle of finance that investors are averse to risk. As noted above, however, the problem is more complex than that. It matters whether the volatility is diversifiable or not, and it also matters how stock prices are affected by volatility, taking into account all other factors that affect stock prices. My paper also contributes to the recent literature explaining the cross-sectional variations in the expected option returns. For example, Hu and Jacobs (2016) show that there is a negative (positive) cross-sectional relationship between the returns of the individual equity options for call
(put) options. Galai and Masulis ((1976)), Friewald, Wagner and Zechner ((2014)), and Hu and Jacobs ((2016)) provide the theoretical proof for this cross-sectional relation within the Black and Scholes ((1973)) option pricing paradigm. However, these studies do not distinguish between the idiosyncratic and systematic components of volatility, although they have dissimilar relationship with the underlying asset. This paper also contributes to other branch of the literature studying the relation between delta-hedged option returns and volatility of the underlying stock. Cao and Han ((2013)) posit that there is a negative relation between idiosyncratic volatility and option returns for both call and put options. They attribute this negative relation to market imperfections and constrained financial intermediaries, and explain that dealers charge higher fees on the stocks with higher idiosyncratic volatility because of their higher arbitrage costs. Goyal and Saretto ((2009)) also report a negative relationship between delta-hedged call or straddle returns and the underlying historical and implied volatility. These studies, however, are focused on the delta-hedged option returns, meaning that they do not account for the non-zero correlation between underlying asset price and volatility. This proposal is organized as follows. Section 2 explains the research hypotheses. In Section 3, we employ a Monte-Carlo simulation to test the hypotheses in a CAPM world. In Section 4, the analysis variables and data sources, as well as the proposed empirical analysis are explained, and Section 5 concludes. 2 Hypotheses Development We investigate the relation between volatility and option value, while we account for the non-zero correlation of the underlying volatility and stock price. In this section, we establish our hypotheses regarding the relationship between option value and the underlying volatility in a CAPM world. When we analyze the volatility-option value relationship in a CAPM world, we should remember that the idiosyncratic volatility is not a priced factor. As a result, the indirect effect of the change in idiosyncratic volatility, σ ivol, on option value through the underlying asset price, O t (τ,s t,σ,k,r) S t S t σ ivol, is equal to zero, because S t σ ivol is zero in CAPM world. 1 On the other hand, 1 An increase in idiosyncratic volatility increases the total volatility, but not beta. For more details, see Appendix B.
the indirect effect of change in systematic risk on option value is not zero, because systematic risk is a priced factor. Since the direct effect of change in volatility on option price, Black- Scholes-Merton (BSM) vega, depends on the total underlying volatility, both idiosyncratic and systematic risks have a positive direct effect on option price. Using a simple one factor model in Equation 1, we can derive Equation 2 explaining the relationship between total volatility of the stock, idiosyncratic volatility, and systematic volatility. In the following equations, R s represents the return of the stock s, R m is the market index return, α s is the abnormal return, ε s is the residual, σ s is the total volatility of the stock s returns, σ m is the volatility of the market return, σ ε is the idiosyncratic volatility, and ρ s,m is the correlation between stock and market returns. Where: R s = α s + β s R m + ε s (1) σ s 2 = β s 2 σ m 2 + σ ε 2 (2) β s = ρ s,mσ s σ m (3) Figure A illustrates the relation between different types of volatilities based on Equation 2 and by using Pythagorean theorem: Figure A The correlation between stock and market returns is equal to sin(θ) in Figure A, since sin(θ) = β s σ m σ s = ρ s,m. 2 The first case is when the total volatility changes only because of the change in idiosyncratic risk, as shown in Figure B. In this case, the indirect pricing effect of change in volatility is zero, because the idiosyncratic risk is not a priced factor. So the only effect is coming from vega. The pricing effect of change in idiosyncratic volatility on call and put option value is proposed in the Proposition 1. 2 sin(θ) = opposite hypotenuse.
Figure B PROPOSITION 1: In a CAPM world, the pricing effect of an increase in idiosyncratic volatility for both call and put options includes only the direct effect, and is equal to the BSM vega. The indirect pricing effect of change in idiosyncratic volatility is zero. The next case happens when only the systematic risk changes and the idiosyncratic risk remains unchanged, as shown in Figure C. When the idiosyncratic risk remains constant, the higher systematic risk is associated with a higher correlation between stock and market return, which is equal to the arcsine of θ. 3 Figure C Considering the positive relation between systematic risk and expected stock return in the CAPM world and based on the present-value model of stock prices, an increase in systematic risk is associated with a lower stock price. Thus, when the underlying volatility of a call option increases due to higher systematic risk, it will indirectly affect the value of the option through the underlying asset price, in addition to its direct effect captured by the BSM vega. The aggregate effect of the increase in volatility on call option value depends on both direct and 3 Arcsine is the inverse of sine function, such that arcsin(sin(θ)) = θ.
indirect effects. Hence, the BSM vega overestimates the volatility-call option value relation as it does not account for the negative indirect effect. Proposition 2 explains the relation between volatility and call option value. PROPOSITION 2: In a CAPM world, assuming that there is a negative relation between volatility and stock value, an increase in the systematic volatility of the underlying stock has a positive direct effect, and a negative indirect effect on call option value through the underlying asset. The negative indirect effect of an increase in the underlying volatility dominates the positive direct effect, resulting in a lower call option value. Similar justification can be provided for put option. The only difference is that the lower price of the underlying asset associated with higher systematic volatility is beneficial to the value of put option, as it makes the option more in-the-money. Thus, higher volatility has positive direct and indirect effect on put option price. Therefore, the BSM vega underestimates the volatilityput option value relationship as it does not account for the positive indirect effect. Proposition 3 explains the relation between volatility and put option value. PROPOSITION 3: In a CAPM world, assuming that there is a negative relation between volatility and stock value, an increase in the systematic volatility of the underlying stock has a positive direct effect, and a positive indirect effect on put option value through the underlying asset. The positive indirect effect adds to the positive direct effect. Therefore, put option value is an increasing function of the systematic volatility. Consequently, the sensitivity of put option value to the underlying systematic volatility is higher than the BSM vega. 3 Monte-Carlo Simulation In this section we employ the Monte-Carlo simulation to analyze the change in the option value when the underlying volatility changes, from time t to time t+dt. The starting values of the simulation are: risk-free rate of 0.03, expected market return of 0.1, correlation of market and stock returns equal to 0.55, stock return volatility of 0.3, market return volatility of 0.2, dt equal to 1 day, dividend of $2.5 with the growth rate of 0.04, moneyness of 1, and time to maturity of 1 year. Using these starting values, we find the stock price at time t by discounting the future cash flows, and the option price at time t based on the BSM model. As we move from time t to t+dt, volatility changes by σ, affecting the expected stock return used in present-value model
of stock prices. Then we simulate the stock price at time t+dt, assuming the stock price follows a Brownian motion process, which is one of the underlying assumptions of the BSM model. In order to do so, we generate 20,000 Brownian motion parameters. Next, we find the option values associated with the new values of volatility and stock price, and take the mean value as the option price at time t+dt. Finally, we calculate the change in option value from time t to time t+dt, C and P, associated with a volatility change equal to σ. In Section 2, it is explained that an increase in the idiosyncratic portion of the volatility does not change the underlying stock price in a CAPM world. Thus, there is no indirect effect of the change in idiosyncratic portion of the volatility, and the total effect would be equal to the positive direct effect. Panels A and B of Figure 1 show the simulation results for call and put option respectively. They show that the value of both call and put options are increasing functions of idiosyncratic volatility, and BSM vega measures the relation between idiosyncratic volatility and option value correctly. The result of simulation supports Proposition 1. On the other hand, an increase in the systematic volatility has positive direct effect on both call and put option values, negative indirect effect on call option value, and positive indirect effect on put option value. Therefore, the BSM vega, which accounts for the direct effect only, overestimates the relation between systematic volatility and call option value and underestimate the relation between systematic volatility and put option value. Panels A and B of Figure 2 illustrate the output of the simulation for call and put options, respectively. The figures compare the change in option values associated with each level of change in systematic volatility estimated by BSM vega and the real change in option value. Panel A shows that not only BSM vega overestimates the relation between systematic volatility and call option value, but the relation is actually reverse in a CAPM world. Thus, an increase in the systematic volatility of the underlying is associated with lower call option value. Panel B shows that under similar assumptions there is a positive relation between put option value and volatility, and the total relation is larger in absolute value than BSM vega. These results support both Proposition 2 and 3. 4 Empirical Tests 4.1 Data and Variables We use data from both the equity option market and stock market. The data on individual equity options is obtained from the OptionMetrics IvyDB US database. The fields we use
include daily closing bid and ask quotes, trading volume and open interest of each option, implied volatility, and the option s greeks computed by OptionMetrics. The data for individual stocks, including the split-adjusted stock returns, stock prices, and trading volume, is coming from the Center for Research in Security Prices (CRSP) database. Other firm-level data can also be obtained from COMPUSTAT. Further, we obtain the daily and monthly Fama-French factor returns and risk-free rates from Kennedth French s data library. We use monthly data over the last 30 months preceding the option holding period to estimate the total volatility, which is equal to the standard deviation of the underlying asset s returns. Idiosyncratic and systematic volatility are driven using market model, Fama-French three-factor, and Fama-French-Cahart four-factor model. We further check the robustness of our results using the idiosyncratic and systematic volatility calculated by EGARCH model, explained by Fu ((2009)) and Guo, Kassa and Ferguson ((2014)). 4.2 Methodology In order to test the research hypotheses, we use the portfolio formation approach and Fama- MacBeth approach. The portfolio formation approach is used to analyze the relation between total volatility and option price, regardless of the source of volatility. To exercise the portfolio formation method, we sort options into portfolios, independently triple-sorted on moneyness, time-to-maturity, and volatility, for both call and put options. The equally-weighted daily returns of the portfolios are calculated. Within each moneyness and time-to-maturity class, a spread portfolio is formed which includes a long position on the highest volatility and a short position on the lowest volatility. Further, we compare the spread among different class of moneyness and time-to-maturity to analyze the relation between volatility and option price within each class. We also use Fama-MacBeth model to study the relation between volatility and option price, while we discriminate the sources of volatility. The endogenous variable is the daily return of either call or put option. The option return is regressed on idiosyncratic volatility, systematic volatility, moneyness, and time-to-maturity. We also control for the illiquidity of stocks and options, following Amihud ((2002)) and Christoffersen, Goyenko, Jacobs and Karoui ((2015)) respectively.
5 Conclusion Volatility is one of the most critical factors in option pricing, and previous studies have analyzed its relation with option prices. These studies, however, are mainly focused on the relation between volatility and the value of a delta hedged option portfolio, and thus do not account for the non-zero correlation of the stock price and volatility. While these studies represent a positive relation between volatility and option price, their statement is true only when all other variables, including the underlying stock price remain constant. This paper investigates the effect of a change in volatility on option price, inclusive of both direct and indirect effects. Using the Monte-Carlo simulation, we show that the negative indirect effect of an increase in systematic volatility on call option dominates the positive direct effect, resulting in a negative total effect on the call option. Therefore, unlike the conclusion of previous studies, call option price can be a decreasing function of volatility. For put option, the positive indirect effect of an increase in systematic volatility on put option adds to the positive direct effect, which results in a larger pricing effect than what the traditional BSM vega represents. On the other hand, BSM vega is a more accurate proxy for the sensitivity of option price with respect to volatility when all the change in volatility is diversifiable. This study contributes to the literature of option pricing by emphasizing on the indirect effect of volatility on the option price through the underlying asset, and also by evaluating the different pricing effects of volatility on options depending on the source of volatility, systematic and idiosyncratic volatility.
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Figure 1. Change in option value versus the change in idiosyncratic volatility. Panel A (B) exhibits the change in call (put) option price versus the change in underlying volatility, when only idiosyncratic portion of volatility can vary. The blue line is the actual change in price including both direct and indirect effect of volatility change. The red line is the expected change in option calculated as the product of BSM vega and volatility change. Panel A Panel B
Figure 2. Change in option value versus the change in systematic volatility. The Panel A (B) exhibits the change in call (put) option value versus the change in systematic volatility when the idiosyncratic portion of volatility remains unchanged. The blue line is the real change in the option value including both direct and indirect effect of volatility change. The red line is the expected change in option calculated as the product of BSM vega and volatility change. Panel A Panel B