Volatilities implied by price changes in the S&P 500 options and futures contracts
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DOI: 10.1007/s11156-013-0354-z
- Cite this article as:
- Hilliard, J. & Li, W. Rev Quant Finan Acc (2014) 42: 599. doi:10.1007/s11156-013-0354-z
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Abstract
We develop a new volatility measure: the volatility implied by price changes in option contracts and their underlying. We refer to this as price-change implied volatility. We compare moneyness and maturity effects of price-change and implied volatilities, and their performance in delta hedging. We find that delta hedges based on a price-change implied volatility surface outperform hedges based on the traditional implied volatility surface when applied to S&P 500 future options.
Keywords
Price-change implied volatilityImplied volatilityS&P 500 options and futures contractsDelta hedgingJEL Classification
G13C611 Background
Over the past 35 years, research in option pricing has resulted in numerous models that relax one or more assumptions of the classic Black–Scholes (BS) model. The most widely cited of these include the stochastic volatility models of Heston (1993), and Hull and White (1987); Merton’s stochastic interest rate model (1973); Merton’s (1976) and Bates’s (1991) jump diffusion models; the stochastic volatility and stochastic interest rates models of Amin and Ng (1993), and Bakshi et al. (1997); the stochastic volatility jump diffusion model of Bates (1996); and the affine jump-diffusion models of Duffie et al. (2000).
These models generally explain option prices better than the original Black–Scholes model. However, their predictive powers depend on the ability to estimate parameters accurately. Parameter estimation errors result in additional pricing errors, which can mitigate a model’s effectiveness. Although the Black–Scholes model has been shown to exhibit consistent pricing biases (Rubinstein 1985, 1994), it performs relatively well in out-of-sample tests vis-à-vis more complex models.
Traders mainly use the so-called Practitioner Black–Scholes (PBS) model to estimate implied volatility from a range of options of different maturity and moneyness. Using interpolation, they create a volatility surface relating implied volatility to moneyness and maturity. The result is a continuous surface that can be used to estimate the price of any option. When recalibrated with sufficient frequency, this method yields good results and outperforms more sophisticated models (Christoffersen and Jacobs 2004).
The key assumption in this approach is that change in option price at a very short horizon is primarily the result of change in the underlying price, while the effect of a change in volatility is negligible.
Throughout this study, we refer to the volatility estimate given by Eq. (2) as price-change implied volatility while reserving the term price-level implied volatility for the volatility obtained by matching observed price with model price (Eq. 1). Because we are interested in out-of-sample performance, we focus on the empirical evaluation of the effectiveness of this new measure.
We investigate the fundamental question of whether price-change implied volatility produces more efficient hedges than hedges based on price-level implied volatility. In other words, does the volatility implied by price changes extract information about future price changes more efficiently? Since hedges are structured to minimize the effects of price changes, intuition suggests that price-change models should be better at implying parameters than price-level models. In short, changes in an option position should be hedged to minimize the effect of changes in the underlying; therefore this suggests that the delta should be implied using price changes in both the option and the underlying.
We start by examining and comparing the statistical properties of different volatility measures. Using the S&P 500 futures contract as the underlying, we compare the time series behavior of price-change implied volatility with that of both price-level implied volatility and realized volatility. The findings indicate that price-change implied volatility has time series behavior similar to that of price-level implied volatility and the moving average of historical volatility. However, the dispersion of price-change implied volatility is higher than the dispersion of either of these more traditional measures.
In other respects, price-change implied volatility is similar to price-level implied volatility. For example, there are differences in the average price-change implied volatility between put and call options. The discrepancy between price-level implied volatility calculated from calls and puts has been documented in the financial literature and has been attributed to the differences in demand curves between calls and puts (Bollen and Whaley 2004; Gârleanu et al. 2009). In addition, moneyness and maturity effects in price-change implied volatility are similar to those found in price-level implied volatility.
To answer the question of whether price-change implied volatility produces more efficient hedges than the traditional price-level implied volatility, we compare the hedging performance of these two measures in 1-day, out-of-sample delta hedges. As in the PBS model, we create volatility surfaces for both price-change and price-level implied volatilities. These surfaces are created daily and used for creating delta hedges. We find that delta hedges based on price-change implied volatility surfaces give better results than hedges based on price-level implied volatility surfaces.
Obtaining reliable estimates of price-change implied volatility poses a new set of problems. Large intra-day datasets are necessary since observations require consecutive and equally spaced observations on the same option contract and its underlying.^{3} Moreover, several data screens may be needed since many observations do not provide useful information. For example, no information on price-change volatility is produced if consecutive option transactions are executed at the same price. There can be also wrong signs, where call (put) prices and their underlying move in different (the same) directions. The need for large datasets and data screens can be minimized if one can establish an accurate empirical relationship between price-change implied volatility and price-level implied volatility.
The rest of the paper is organized as follows. The concept of price-change implied volatility is developed in Sect. 2. Section 3 describes the data and Sect. 4 compares the statistical properties of different volatility measures and examines money and maturity effects. The effectiveness and implications of hedges based on price-change implied volatility versus price-level implied volatility are addressed in Sect. 5. Section 6 concludes the paper.
2 Price-change implied volatility
The price-change model imposes some unique restrictions on the data that come directly from the model’s assumptions. First, the restriction of no wrong signs implies that we consider only observations such that \(\Updelta G\cdot \Updelta F>0\) for calls and \(\Updelta G\cdot \Updelta F<0\) for puts. This restriction is important in light of the findings of Bakshi et al. (2000). They examined S&P 500 options and found that prices of call (put) options often move in opposite (the same) directions from those of the underlying. They report that prices of calls and their underlying move in opposite directions between 7.2 and 16.3 % of the time, according to the sampling interval. Second, we require that the absolute change in the price of the option not be larger than the absolute change in the futures price, consistent with \(\left\vert \frac{\partial G}{\partial F}\right\vert \leq 1\) and negligible time decay.
The key question we ask in this paper is, does the price-change approach permit market participants with hedging needs to adjust their risk preferences more efficiently? Our approach to estimation is consistent with that of Christoffersen and Jacobs. They emphasize that parameters should be estimated by using in-sample metrics identical to those used for out-of-sample evaluation. Since we hedge price changes, the required volatility parameter should be estimated from these rather than from price levels. This approach aligns the estimation and evaluation procedures, and should theoretically lead to a better out-of-sample fit. While we advocate the use of price-change implied volatility in hedging application for the reasons specified above, we likewise conjecture that traditional implied volatility is appropriate parameter for price-level applications.
3 The data
Data used for this study are the S&P 500 futures options, their underlying S&P 500 futures contracts, and Libor rates (as proxy for the risk-free rate). The S&P 500 futures options and the underlying are pit-traded on the Chicago Mercantile Exchange (CME). The observations are taken from January 1998 to December 2006 from the CME’s Time and Sales database (tick data). Each option and its underlying futures contract are matched such that trading must occur within 30 s. The average gap between the trade of option and the underlying futures contract is 5 s. To be a valid observation, the option price has to be at least $0.25.
Descriptive statistics for calls
Dataset | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
All | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | ||
Number of call options | 76,544 | 15,049 | 10,876 | 9,406 | 7,663 | 9,549 | 7,686 | 5,570 | 5,766 | 4,979 | |
Number of strike prices | 186 | 79 | 77 | 80 | 103 | 99 | 77 | 52 | 46 | 58 | |
Average difference between the trade of option and future (s) | 5 | 4 | 4 | 5 | 5 | 4 | 5 | 6 | 6 | 6 | |
Maturity | |||||||||||
Range (days) | 1–238 | 1–156 | 1–238 | 1–150 | 1–167 | 1–238 | 1–162 | 1–120 | 1–160 | 1–149 | |
1–30 | 46,901 | 10,221 | 6,958 | 5,806 | 4,726 | 4,901 | 4,500 | 3,024 | 3,576 | 3,189 | |
31–60 | 20,138 | 3,242 | 2,533 | 2,261 | 1,876 | 3,377 | 2,208 | 1,855 | 1,456 | 1,330 | |
61–90 | 7,647 | 1,246 | 1,110 | 1,128 | 756 | 1,088 | 764 | 565 | 626 | 364 | |
91–120 | 1,801 | 323 | 264 | 207 | 298 | 175 | 209 | 126 | 105 | 94 | |
121–150 | 50 | 16 | 10 | 4 | 6 | 6 | 4 | 0 | 2 | 2 | |
151–180 | 5 | 1 | 0 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | |
181–210 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
211–240 | 2 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | |
241–270 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Moneyness | |||||||||||
F/K ≤ 0.925 | 8,720 | 1,081 | 1,347 | 1,788 | 1,269 | 1,799 | 1,192 | 167 | 56 | 21 | |
0.925 < F/K ≤ 0.975 | 31,711 | 6,257 | 4,774 | 4,277 | 3,377 | 3,807 | 2,950 | 2,429 | 1,823 | 2,017 | |
0.975 < F/K ≤ 1.025 | 34,519 | 7,394 | 4,495 | 3,221 | 2,892 | 3,564 | 3,295 | 2,889 | 3,849 | 2,920 | |
1.025 < F/K ≤ 1.075 | 1,194 | 256 | 199 | 109 | 102 | 218 | 206 | 56 | 32 | 16 | |
F/K ≥ 1.075 | 400 | 61 | 61 | 11 | 23 | 161 | 43 | 29 | 6 | 5 |
Descriptive statistics for puts
Dataset | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
All | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | ||
Number of put options | 101,010 | 20,495 | 14,692 | 12,365 | 10,532 | 11,422 | 9,750 | 7,588 | 6,778 | 7,388 | |
Number of strike prices | 183 | 92 | 98 | 92 | 98 | 99 | 88 | 67 | 62 | 77 | |
Average difference between the trade of option and future (s) | 5 | 4 | 4 | 4 | 5 | 4 | 5 | 6 | 6 | 6 | |
Maturity | |||||||||||
Range (days) | 1–265 | 1–204 | 1–202 | 1–265 | 1–176 | 1–181 | 1–156 | 1–185 | 1–147 | 1–174 | |
1–30 | 57,367 | 12,429 | 8,258 | 6,845 | 5,950 | 5,762 | 5,135 | 4,097 | 4,035 | 4,856 | |
31–60 | 27,801 | 4,952 | 3,918 | 3,352 | 2,619 | 4,043 | 3,034 | 2,331 | 1,800 | 1,752 | |
61–90 | 12,452 | 2,193 | 2,092 | 1,786 | 1,408 | 1,317 | 1,293 | 947 | 826 | 600 | |
91–120 | 3,227 | 853 | 408 | 367 | 545 | 272 | 286 | 206 | 115 | 175 | |
121–150 | 126 | 58 | 22 | 9 | 5 | 20 | 1 | 5 | 2 | 4 | |
151–180 | 29 | 9 | 3 | 3 | 5 | 7 | 1 | 0 | 0 | 1 | |
181–210 | 6 | 1 | 1 | 1 | 0 | 1 | 0 | 2 | 0 | 0 | |
211–240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
241–270 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | |
Moneyness | |||||||||||
F/K ≤ 0.925 | 333 | 47 | 13 | 96 | 116 | 43 | 18 | 0 | 0 | 0 | |
0.925 < F/K ≤ 0.975 | 1,358 | 285 | 119 | 198 | 316 | 254 | 69 | 58 | 16 | 43 | |
0.975 < F/K ≤ 1.025 | 29,562 | 5,798 | 3,496 | 3,049 | 2,941 | 3,263 | 2,872 | 2,848 | 2,742 | 2,553 | |
1.025 < F/K ≤ 1.075 | 29,368 | 5,859 | 4,027 | 3,547 | 2,678 | 2,754 | 2,804 | 2,502 | 2,314 | 2,883 | |
F/K ≥ 1.075 | 40,389 | 8,506 | 7,037 | 5,475 | 4,481 | 5,108 | 3,987 | 2,180 | 1,706 | 1,909 |
The risk-free rate is calculated from Libor rates based on British Bankers’ Association Data. Libor rates with overnight maturity and maturities of 1, 2 weeks, and monthly are posted daily. Libor rates that match option maturity are obtained by linear interpolation and then converted to continuously compounded rates.
Sequences of records on the same contract are required since we investigate price changes. Therefore, datasets containing records with consecutive observations for 1-, 2-, 3-, and 4-day lags are created. For each strike price, the option that trades closest to 10:00 a.m. is selected because this is generally the time of heaviest trading activity. From these contracts, only those that trade with 1-, 2-, 3-, or 4-day lags (24, 48, 72, and 92 h, respectively) are used in a particular dataset. For example, a valid observation for a 1-day lag consists of a trade on the same contract on two consecutive days (Monday and Tuesday, Tuesday and Wednesday, Wednesday and Thursday, and Thursday and Friday). The time lag between daily observations must be between 23 and 25 h for 1-day lags. To prevent overlapping data for 2-day lags, only trades on the same contract on Monday and Wednesday or Wednesday and Friday are used. Similarly, for 3- and 4-day lags, the only trades considered are those on the same contract on Monday and Thursday and Monday and Friday, respectively. In addition, we require that the absolute change in the price of the option is larger than $0.20 since very small price changes lead to “noisy” estimates. We use only options with time to expiration at least 14 days; this is a common (although ad hoc) procedure to avoid short-term maturity biases. Finally, we omit observations with price-change implied volatility larger than 0.7.
Descriptive statistics
Dataset | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Calls1 | Calls2 | Calls3 | Calls4 | Puts1 | Puts2 | Puts3 | Puts4 | Calls1H | Puts1H | ||
Number of options | 2,274 | 252 | 201 | 169 | 3,061 | 410 | 349 | 268 | 11,741 | 14,817 | |
Number of strike prices | 126 | 95 | 86 | 75 | 119 | 97 | 98 | 88 | 134 | 127 | |
Maturity | |||||||||||
Range (days) | 14–106 | 15–106 | 14–98 | 17–104 | 14–111 | 15–125 | 14–98 | 17–97 | 14–111 | 14–139 | |
1–30 | 1,143 | 64 | 41 | 32 | 1,426 | 126 | 93 | 64 | 8,138 | 8,874 | |
31–60 | 803 | 115 | 90 | 84 | 1,140 | 167 | 139 | 133 | 3,164 | 4,773 | |
61–90 | 285 | 68 | 60 | 51 | 445 | 107 | 97 | 67 | 418 | 1,032 | |
91–120 | 43 | 5 | 10 | 2 | 50 | 9 | 20 | 4 | 21 | 136 | |
121–150 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | |
151–180 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
181–210 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
211–240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
241–270 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
Moneyness | |||||||||||
F/K ≤ 0.925 | 271 | 56 | 56 | 47 | 16 | 0 | 1 | 0 | 1,419 | 0 | |
0.925 < F/K ≤ 0.975 | 1,213 | 123 | 92 | 89 | 34 | 11 | 6 | 2 | 6,481 | 3 | |
0.975 < F/K ≤ 1.025 | 770 | 72 | 53 | 32 | 591 | 77 | 54 | 37 | 3,832 | 2,422 | |
1.025 < F/K ≤ 1.075 | 19 | 1 | 0 | 1 | 957 | 108 | 90 | 72 | 8 | 5,198 | |
F/K ≥ 1.075 | 1 | 0 | 0 | 0 | 1,463 | 214 | 198 | 157 | 1 | 7,194 |
The dataset for puts with 4-day lags (Puts1) contains 3,061 observations, datasets of puts with 2-, 3-, and 4-day lags contain 410, 349, and 268 observations, respectively. Similarly, the majority of put options are out-of-the-money or at-the-money options. The most common are options with maturities up to 3 months.
3.1 Volatility time series
The S&P 500 futures options are American-style options. To account for the possibility of early exercise, we calculate volatilities using the binomial tree.^{5} We compare these results with volatilities implied from the Black model. For our dataset, the results are almost identical. Volatility estimates are obtained using the bisection method.
Price-change and price-level implied volatilities during different periods for calls
Period | Volatility | Mean | SD | Minimum | Maximum |
---|---|---|---|---|---|
All | Price-change implied volatility for American options, 1-day lag | 0.1717 | 0.0830 | 0.0198 | 0.5442 |
January 1, 1998–December 31, 2006 | Price-change implied volatility for European options, 1-day lag | 0.1719 | 0.0831 | 0.0198 | 0.5327 |
Price-change implied volatility, 2-day lag | 0.1709 | 0.0955 | 0.0261 | 0.6309 | |
Price-change implied volatility, 3-day lag | 0.1755 | 0.1029 | 0.0408 | 0.6826 | |
Price-change implied volatility, 4-day lag | 0.1854 | 0.0985 | 0.0526 | 0.6362 | |
Price-level implied volatility | 0.1837 | 0.0543 | 0.0788 | 0.3986 | |
Historical volatility | 0.2197 | 0.0782 | 0.5134 | ||
Period 1 | Price-change implied volatility for American options, 1-day lag | 0.1829 | 0.0805 | 0.0256 | 0.5236 |
January1, 1998–August 31, 2000 | Price-change implied volatility for European options, 1-day lag | 0.1832 | 0.0809 | 0.0256 | 0.5212 |
Price-level implied volatility | 0.1972 | 0.0370 | 0.1266 | 0.3986 | |
Historical volatility | 0.2414 | 0.1150 | 0.4762 | ||
Period 2 | Price-change implied volatility for American options, 1-day lag | 0.2016 | 0.0786 | 0.0291 | 0.5442 |
September 1, 2000–March 6, 2003 | Price-change implied volatility for European options, 1-day lag | 0.2018 | 0.0785 | 0.0291 | 0.5327 |
Price-Level implied volatility | 0.2201 | 0.0467 | 0.1454 | 0.3947 | |
Historical volatility | 0.2826 | 0.1208 | 0.5134 | ||
Period 3 | Price-change implied volatility for American options, 1-day lag | 0.1250 | 0.0701 | 0.0198 | 0.4973 |
March 7, 2003–December 31, 2006 | Price-change implied volatility for European options, 1-day lag | 0.1250 | 0.0699 | 0.0198 | 0.4912 |
Price-level implied volatility | 0.1270 | 0.0326 | 0.0788 | 0.2417 | |
Historical volatility | 0.1426 | 0.0782 | 0.3106 | ||
High-volatility period | Price-change implied volatility for American options, 1-day lag | 0.1902 | 0.0803 | 0.0256 | 0.5442 |
January 1, 1998 to August 31, 2003 | Price-change implied volatility for European options, 1-day lag | 0.1905 | 0.0805 | 0.0256 | 0.5327 |
Price-level implied volatility | 0.2056 | 0.0423 | 0.1266 | 0.3986 | |
Historical volatility | 0.2589 | 0.1150 | 0.5134 | ||
Low-volatility period | Price-change implied volatility for American options, 1-day lag | 0.1140 | 0.0620 | 0.0198 | 0.4973 |
September 1, 2003–December 31, 2006 | Price-change implied volatility for European options, 1-day lag | 0.1140 | 0.0618 | 0.0198 | 0.4912 |
Price-level implied volatility | 0.1155 | 0.0210 | 0.0788 | 0.1938 | |
Historical volatility | 0.1283 | 0.0782 | 0.1892 |
Price-change and price-level implied volatilities during different periods for puts
Period | Volatility | Mean | SD | Minimum | Maximum |
---|---|---|---|---|---|
All | Price-change implied volatility for American options, 1-day lag | 0.2860 | 0.1341 | 0.0250 | 0.6975 |
January 1, 1998–December 31, 2006 | Price-change implied volatility for European options, 1-day lag | 0.2862 | 0.1342 | 0.0259 | 0.7056 |
Price-change implied volatility, 2-day lag | 0.2772 | 0.1338 | 0.0257 | 0.6915 | |
Price-change implied volatility, 3-day lag | 0.2816 | 0.1342 | 0.0349 | 0.6901 | |
Price-change implied volatility, 4-day lag | 0.2790 | 0.1154 | 0.0512 | 0.6867 | |
Price-Level implied volatility | 0.2632 | 0.0906 | 0.0929 | 0.6283 | |
Historical volatility | 0.2197 | 0.0782 | 0.5134 | ||
Period 1 | Price-change implied volatility for American options, 1-day lag | 0.3243 | 0.1348 | 0.0271 | 0.6964 |
January1, 1998–August 31, 2000 | Price-change implied volatility for European options, 1-day lag | 0.3247 | 0.1351 | 0.0278 | 0.7056 |
Price-Level implied volatility | 0.2953 | 0.0807 | 0.1605 | 0.6283 | |
Historical volatility | 0.2414 | 0.1150 | 0.4762 | ||
Period 2 | Price-change implied volatility for American options. 1-day lag | 0.3188 | 0.1286 | 0.0250 | 0.6975 |
September 1, 2000–March 6, 2003 | Price-change implied volatility for European options, 1-day lag | 0.3189 | 0.1286 | 0.0259 | 0.6945 |
Price-level implied volatility | 0.3054 | 0.0778 | 0.1640 | 0.6176 | |
Historical volatility | 0.2826 | 0.1208 | 0.5134 | ||
Period 3 | Price-change implied volatility for American options, 1-day lag | 0.2010 | 0.0937 | 0.0262 | 0.6583 |
March 7, 2003–December 31, 2006 | Price-change implied volatility for European options, 1-day lag | 0.2011 | 0.0937 | 0.0262 | 0.6589 |
Price-level implied volatility | 0.1785 | 0.0484 | 0.0929 | 0.3669 | |
Historical volatility | 0.1426 | 0.0782 | 0.3106 | ||
High-volatility period | Price-change implied volatility for American options, 1-day lag | 0.3167 | 0.1313 | 0.0250 | 0.6975 |
January 1, 1998–August 31, 2003 | Price-change implied volatility for European options, 1-day lag | 0.3169 | 0.1314 | 0.0259 | 0.7056 |
Price-level implied volatility | 0.2951 | 0.0793 | 0.1605 | 0.6283 | |
Historical volatility | 0.2589 | 0.1150 | 0.5134 | ||
Low-volatility period | Price-change implied volatility for American options, 1-day lag | 0.1937 | 0.0944 | 0.0285 | 0.6583 |
September 1, 2003–December 31, 2006 | Price-change implied volatility for European options, 1-day lag | 0.1938 | 0.0945 | 0.0281 | 0.6589 |
Price-level implied volatility | 0.1675 | 0.0420 | 0.0929 | 0.3669 | |
Historical volatility | 0.1283 | 0.0782 | 0.1892 |
Several observations can be made with respect to volatility averages. For the entire sample, as well as for each subperiod, the sample average volatility for calls is smallest for price-change implied volatility, followed by price-level implied volatility and historical volatility, respectively (Table 4). The results are reversed for puts (Table 5), with sample average volatility being smallest for historical volatility, followed by price-level implied volatility and price-change implied volatility. Lag length appears to have very little effect on price-change implied volatility estimates.
Under the assumptions of the model, volatilities calculated from prices of call and put options should be the same. Clearly, this is not the case. The average price-level implied volatility for call options for the entire period is 0.1837, while the corresponding average price-level implied volatility for put options is 0.2632. Calculation of the normalized difference in sample means gives an ad hoc t score of 39.9.^{6} This trend is consistent over all periods, without regard to the behavior of the S&P 500 index. Similar differences have been previously documented in the financial literature by Bollen and Whaley (2004), among others. The explanation of this inequality is based on different demand curves for calls and puts. Puts are largely demanded by institutional investors for insurance purposes, especially after the crash of October 1987 (Fleming 1998; Rubinstein 1994). This demand may bid up prices of put options and thus result in higher implied volatilities.^{7} Gârleanu et al. (2009) document that the investors create large demand for S&P 500 index options, especially for out-of-the-money puts. Having the net short position in the out-of-the-money index options, the index option dealers cannot fully hedge their exposure and must be compensated for bearing this risk. Therefore, the demand pressure of investors results in higher option prices and in higher implied volatilities estimates from these out-of-the-money options. In our sample, 79 % of puts are out- and far-out-of-the-money puts. In fact, far-out-of-the-money puts represent 48 % of all observations. Their average price-level implied volatility is 0.3202, compared to the averages across all moneyness levels of 0.2632. Out- and far-out-of-the money calls also constitute the majority (65 %) of observations, but relatively weaker investor demand apparently has a smaller effect on price-level implied volatility for calls.
The differences between volatilities estimated from call and put options are even more pronounced for price-change implied volatility. Thus, for the entire period, the difference between price-change implied volatility estimated from puts and calls is 0.1143, compared to 0.0795 for price-level implied volatility. This suggests that the market imperfections causing differences in implied volatilities may be more important for price changes than price levels. The difference between volatilities is present for the entire sample. This difference, however, tends to be larger during a downturn in the market for calls and vice versa for puts.
Our results are consistent with those of Carr and Wu, even though our dataset and volatility measures are not identical. Our dataset extends from January 1998 to September 2006, and we estimate put and call volatilities using the Black model, corresponding approximately to the VXO.^{8} Using daily data, we find the S&P 500 has annual realized volatility of 0.1823 for this period. We regress price-level implied volatility on maturities between 20 and 40 days, and find a price-level implied volatility estimate at 30 days of 0.1742 for calls (n = 1078) and 0.2510 (n = 1408) for puts. The weighted average price-level implied volatility is 0.2177. The corresponding estimates for price-change implied volatility are 0.1669 for calls and 0.2803 for puts, with weighted average 0.2311. Thus, when weighted, both price-level implied volatility (0.2177) and price-change implied volatility (0.2311) are greater than realized volatility (0.1823).^{9}
4 Money and maturity considerations
Regression coefficients for price-change implied volatility for calls
Period | Dates | Number of records | a | b | c | d | e | f | Adj. r^{2} |
---|---|---|---|---|---|---|---|---|---|
Panel A: calls with 1-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 2,274 | 12.61*** | −5.42*** | 1.41** | −25.09*** | 12.65*** | 5.19*** | 0.0506 |
Period 1 | January1, 1998–August 31, 2000 | 910 | 8349*** | −1.55 | 0.44 | −17.41*** | 9.11*** | 1.52 | 0.0236 |
Period 2 | September 1, 2000–March 6, 2003 | 699 | 8.04*** | −2.25 | 0.88 | −15.96*** | 8.14*** | 1.92 | 0.0366 |
Period 3 | March 7, 2003–December 31, 2006 | 665 | 25.12*** | −2.63 | −1.43** | −51.53*** | 26.51*** | 3.20 | 0.0631 |
High-volatility period | January 1, 1998–August 31, 2003 | 1,722 | 8.66*** | −2.59** | 0.90 | −17.39*** | 8.93*** | 2.39* | 0.0250 |
Low-volatility period | September 1, 2003–December 31, 2006 | 552 | 40.84*** | −5.66** | −0.88 | −83.77*** | 43.04*** | 6.14** | 0.0644 |
Panel B: calls with 2-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 252 | 12.94*** | −6.53 | 2.14 | −26.28*** | 13.53*** | 6.34 | 0.0554 |
Panel C: calls with 3-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 201 | 7.89** | −7.82** | 1.55 | −15.06* | 7.30* | 7.91** | 0.0529 |
Panel D: calls with 4-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 169 | 9.29** | −4.02 | 0.16 | −19.43** | 10.34** | 4.31 | 0.0751 |
Regression coefficients for price-change implied volatility for puts
Period | Dates | Number of records | a | b | c | d | e | f | Adj. r^{2} |
---|---|---|---|---|---|---|---|---|---|
Panel A: puts with 1-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 3,061 | −0.39 | 3.68*** | 1.20* | 0.11 | 0.50** | −3.78*** | 0.2867 |
Period 1 | January1, 1998–August 31, 2000 | 1,310 | −1.21*** | 1.57*** | 0.60 | 1.82*** | −0.35** | −1.85*** | 0.2973 |
Period 2 | September 1, 2000–March 6, 2003 | 837 | 0.56 | 2.87*** | −0.29 | −1.30* | 0.98*** | −2.67*** | 0.2471 |
Period 3 | March 7, 2003–December 31, 2006 | 914 | 3.01*** | 6.46*** | 0.86 | −6.50*** | 3.62*** | −6.14*** | 0.1554 |
High-volatility period | January 1, 1998–August 31, 2003 | 2,297 | −0.33 | 2.83*** | 0.77 | 0.21 | 0.37** | −2.94*** | 0.2721 |
Low-volatility period | September 1, 2003–December 31, 2006 | 764 | 4.69*** | 8.49*** | 0.85 | −9.77*** | 5.21*** | −8.06*** | 0.1219 |
Panel B: puts with 2-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 410 | −0.52* | 4.03*** | 1.61 | 0.32 | 0.38* | −4.03*** | 0.2795 |
Panel C: puts with 3-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 349 | −0.97*** | 3.05*** | 3.13 | 1.16*** | 0.01 | −3.64*** | 0.3984 |
Panel D: puts with 4-day lags | |||||||||
All | January 1, 1998–December 31, 2006 | 268 | −0.75** | 1.79** | 2.42 | 0.97* | 0.008 | −2.40*** | 0.3917 |
Regression coefficients for price-level implied volatility
Period | Dates | Number of records | a | b | c | d | e | f | Adj. r^{2} |
---|---|---|---|---|---|---|---|---|---|
Panel A: calls | |||||||||
All | January 1, 1998–December 31, 2006 | 2,274 | 11.96*** | −8.36*** | 2.65*** | −23.26*** | 11.49*** | 7.95*** | 0.1327 |
Period 1 | January1, 1998–August 31, 2000 | 910 | 6.13*** | −3.87*** | 1.58*** | −12.10*** | 6.18*** | 3.61*** | 0.0664 |
Period 2 | September 1, 2000–March 6, 2003 | 699 | 8.63*** | −5.46*** | 1.73*** | −16.92*** | 8.51*** | 5.21*** | 0.0936 |
Period 3 | March 7, 2003–December 31, 2006 | 665 | 22.61*** | −5.28*** | −0.06 | −4573*** | 23.23*** | 5.56*** | 0.2254 |
High-volatility period | January 1, 1998–August 31, 2003 | 1,722 | 8.24*** | −5.24*** | 2.02*** | −16.17*** | 8.15*** | 4.90*** | 0.0871 |
Low-volatility period | September 1, 2003–December 31, 2006 | 552 | 16.39*** | −4.90*** | 0.43 | −32.93*** | 16.64*** | 4.99*** | 0.0923 |
Panel B: puts | |||||||||
All | January 1, 1998–December 31, 2006 | 3,061 | −0.26 | 4.03*** | 3.14*** | −0.15 | 0.63*** | −4.54*** | 0.5434 |
Period 1 | January1, 1998–August 31, 2000 | 1,310 | −0.87*** | 2.73*** | 2.89*** | 1.11*** | 0.003 | −3.35*** | 0.7014 |
Period 2 | September 1, 2000–March 6, 2003 | 837 | 0.74** | 2.94*** | 1.61*** | −1.57*** | 1.09*** | −3.16*** | 0.4928 |
Period 3 | March 7, 2003–December 31, 2006 | 914 | −0.47 | 3.48*** | 2.57*** | 0.036 | 0.57** | −3.92*** | 0.6512 |
High-volatility period | January 1, 1998–August 31, 2003 | 2,297 | −0.07 | 3.51*** | 2.71*** | −0.28 | 0.60*** | −3.95*** | 0.5880 |
Low-volatility period | September 1, 2003–December 31, 2006 | 764 | 0.45 | 4.62*** | 1.84*** | −1.76*** | 1.45*** | −4.87*** | 0.7052 |
Moneyness and maturity effects in call options are strongly significant at the 99 % level for price-level implied volatility for all periods studied. More specifically, note in Table 8, panel A, that for all periods the coefficients of moneyness are significant and negative, while the coefficients of moneyness squared are significant and consistently positive. Moneyness and maturity effects on price-level implied volatility for puts are more ambiguous. The maturity effect is most notable, and is consistently positive and significant, for all subperiods (Table 8, panel B).
Price-change implied volatility for calls also shows a significant dependence on moneyness but does not consistently and significantly depend on maturity (Table 6). Price-change implied volatility for puts shows consistently significant dependence on maturity through all periods, but the moneyness effect is weaker (Table 7). Price-change implied volatility yields lower R^{2}s than price-level implied volatility in these regressions for both calls and puts.
5 Hedging using price-change and implied volatility
Does price-change implied volatility capture additional useful information implicit in option prices? How does this new measure affect risk management? In this section, we investigate these questions. In particular, we compare the hedging performance of both price-change and price-level implied volatility in 1-day, out-of-sample delta hedges.
We implement and evaluate a hedging strategy as follows: First, we use the Black model and nonlinear least squares to estimate the \(\user2{\theta }^{+}\)vector consistent with Eq. (12) for several forms of the loss function. Given the estimated \(\user2{\theta }^{+}\) vector, we arrive at the fitted value σ, for options of all moneyness and maturity, from Eq. (8) by setting the error term to zero. For any option from our sample, the hedge ratio is computed using the fitted σ value, and a hedging portfolio is formed according to Eq. (9). On the next day, the hedging portfolio is updated by dV, using Eq. (10) and the portfolio error is computed as \(e=dG^{\ast }+dV\) . The performance of the hedging strategy is evaluated as the portfolio error aggregated across all options in our sample. Different choices of the loss function clearly lead to different estimates of the \(\user2{\theta }^{+}\)vector and thus different fitted values for σ. We are interested in how these different choices of the loss function match up in their performance of the hedging strategy.
Estimating the \(\user2{\theta }\)-vector requires at least six observations in order to provide estimates of the six parameters. This requirement is not restrictive for price-level implied volatility since options on the S&P 500 futures contracts are heavily traded. However, observations for estimating price-change implied volatility require that the same contract be traded on two consecutive days. The datasets used in previous sections^{12} do not provide enough observations under this criterion to ensure a statistically acceptable number of observations. To overcome this deficiency, in this section the trades of options are not restricted to the time of highest liquidity but instead include all contracts traded within a 23- to 25-h window. The statistics for the resulting datasets (Calls1H and Puts1H) are given in Table 3.
One-day, out-of-sample delta hedges for call options
Volatility measure | Number of records | RMSE | AE | Average error | Min error | Max error |
---|---|---|---|---|---|---|
Panel A: volatility surfaces based on MSE loss function | ||||||
Panel A1: $-based loss function | ||||||
Price-level implied volatility | 6,366 | 1.4285 | 0.9810 | −0.1191 | −6.8920 | 12.9439 |
Price-change implied volatility | 6,351 | 1.2301 | 0.8109 | 0.0102 | −6.6837 | 12.0989 |
Implied Hedging volatility | 6,264 | 1.3321 | 0.8719 | −0.0149 | −10.8748 | 11.2041 |
Panel A2: %-based loss function | ||||||
Price-level implied volatility | 6,366 | 1.3511 | 0.9110 | −0.0439 | −6.7652 | 12.7465 |
Price-change implied volatility | 6,351 | 1.3724 | 0.8680 | −0.0004 | −6.784 | 14.4684 |
Implied hedging volatility (H-calls) | 6,264 | 1.3767 | 0.8814 | −0.0106 | −12.4908 | 12.2681 |
Panel B: volatility surfaces based on AE loss function | ||||||
Panel B1: $-based loss function | ||||||
Price-level implied volatility | 6,366 | 1.3934 | 0.9453 | −0.0466 | −6.8295 | 11.2833 |
Price-change implied volatility | 6,364 | 1.2347 | 0.8114 | −0.0051 | −6.6568 | 10.8933 |
Implied hedging volatility | 6,366 | 1.2474 | 0.8342 | −0.0330 | 7.2627 | 11.1094 |
Panel B2: %-based loss function | ||||||
Price-level implied volatility | 6,366 | 1.3444 | 0.9165 | −0.0540 | −6.7447 | 11.6819 |
Price-change implied volatility | 6,364 | 1.3001 | 0.8367 | 0.0077 | −6.7514 | 13.6093 |
Implied hedging volatility (H-calls) | 6,366 | 1.3461 | 0.8667 | 0.0079 | 6.941 | 12.0494 |
Panel C: fitted price-change volatility | ||||||
Panel C1: $-based loss function | ||||||
Fitted price-change implied volatility | 6,339 | 1.2101 | 0.8035 | 0.0011 | −7.3309 | 10.9487 |
Panel C2: %-based loss function | ||||||
Fitted price-change implied volatility | 6,339 | 1.3477 | 0.8572 | 0.0012 | −10.9728 | 13.2609 |
Panel D: contract volatilities | ||||||
Price-level implied volatility | 6,366 | 1.3222 | 0.8992 | 0.0432 | −6.3610 | 10.8809 |
Price-change implied volatility (H-calls) | 6,263 | 1.6215 | 1.0124 | 0.0014 | −9.6739 | 21.0734 |
Panel E: unhedged position | ||||||
Unhedged position | 6,351 | 3.9720 | 2.7449 | −0.4097 | −28.0000 | 37.0000 |
One-day, out-of-sample delta hedges for put options
Volatility measure | Number of records | RMSE | AE | Average error | Min error | Max error |
---|---|---|---|---|---|---|
Panel A: volatility surfaces based on MSE loss function | ||||||
Panel A1: $-based loss function | ||||||
Price-level implied volatility | 5,465 | 1.4460 | 0.9587 | −0.0292 | −6.3620 | 13.6521 |
Price-change implied volatility | 5,521 | 0.9133 | 0.6389 | −0.2000 | −5.5023 | 8.0130 |
Implied hedging volatility | 5,427 | 0.9855 | 0.6872 | −0.1655 | −5.5023 | 7.5001 |
Panel A2: %-based loss function | ||||||
Price-level implied volatility | 5,465 | 1.3128 | 0.8859 | −0.0327 | −3.9932 | 13.2823 |
Price-change implied volatility | 5,521 | 1.0759 | 0.7325 | −0.2031 | −5.5023 | 9.9707 |
Implied hedging volatility (H-calls) | 5,427 | 1.2882 | 0.8140 | −0.1300 | −5.5023 | 18.8074 |
Panel B: volatility surfaces based on AE loss function | ||||||
Panel B1: $-based loss function | ||||||
Price-level implied volatility | 5,138 | 1.3040 | 0.8661 | −0.0746 | −4.4888 | 11.9001 |
Price-change implied volatility | 5,138 | 0.9811 | 0.6798 | −0.2349 | −10.5293 | 6.8871 |
Implied hedging volatility | 5,138 | 1.0103 | 0.7098 | −0.2201 | −5.7975 | 7.9834 |
Panel B2: %-based loss function | ||||||
Price-level implied volatility | 5,138 | 1.1994 | 0.8321 | −0.1067 | −4.3642 | 10.2458 |
Price-change implied volatility | 5,138 | 1.1925 | 0.7454 | −0.2019 | −10.5297 | 19.4128 |
Implied hedging volatility (H-calls) | 5,138 | 1.2288 | 0.7873 | −0.1700 | −5.7393 | 19.4146 |
Panel C: fitted price-change volatility | ||||||
Panel C1: $-based loss function | ||||||
Fitted price-change implied volatility | 5,609 | 0.9154 | 0.6368 | −0.1863 | −4.2387 | 8.6344 |
Panel C2: %-based loss function | ||||||
Fitted price-change implied volatility | 5,609 | 1.1364 | 0.7437 | −0.1630 | −5.3142 | 13.1232 |
Panel D: contract volatilities | ||||||
Price-level implied volatility | 5,465 | 0.8872 | 0.6151 | −0.1616 | −3.9682 | 8.8271 |
Price-change implied volatility (H-calls) | 5,465 | 1.1092 | 0.7531 | −0.1846 | −5.5080 | 11.4527 |
Panel E: unhedged position | ||||||
Unhedged position | 5,465 | 3.2133 | 2.2139 | −0.3627 | −17.5000 | 23.0000 |
Point estimates of the best hedges are those based on price-change implied volatility surfaces for both calls and puts.^{13} For example, the AE for calls is 0.8114 for σ_{pciv} compared to 0.9453 for σ_{iv}. The corresponding numbers for puts are 0.6798 for σ_{pciv} compared to 0.8661 for σ_{iv}. Furthermore, the relative ranking of point estimates is invariant using the MSE and AE loss functions. The RMSE for calls (puts) is 1.2301 (0.9133) for σ_{pciv} , compared to 1.4285 (1.4460) for σ_{iv}. Surprisingly, we find no advantage in estimating the implied hedging volatility (σ_{ivh}) by minimizing the hedging MSE or the hedging AE. Although these hedges perform better than hedges based on price-level implied volatility surfaces, they underperform the hedges based on price-change volatility surfaces for both calls and puts. For reference, the unhedged portfolio has AE of 2.7449 for calls and 2.2139 for puts (panels E of Tables 9, 10). Our results suggest that the choice of volatility surface matters and that price-change implied volatility surfaces are more useful than traditional price-level implied volatility surfaces in hedging applications.
The results for the performance of hedges based on fitted price-change implied volatility are shown in panels C of Tables 9 and 10. For call options, the fitted price-change implied volatility produces the best hedging results, slightly outperforming even hedges based on the price-change implied volatility surface. For puts, the fitted price-change implied volatility results in hedges with similar MSE and AE as the price-change implied volatility surface.
We also evaluate the performance of delta hedges based on contract price-level and contract price-change implied volatilities (panel D). Contract volatility is the volatility calculated directly from a single observation on a specific contract (i.e., an observation on a contract with specific maturity and the strike price). For example, a price-change implied volatility might be estimated from the price changes from Monday to Tuesday. The delta hedge is then set up on Tuesday and evaluated on Wednesday. For both puts and calls, contract price-level implied volatilities produce hedges with better performance than price-change implied volatility. Contract price-level implied volatility, however, does not dominate fitted price-change implied volatility in hedges for calls.
Superior performance of price-level implied volatility in hedges for puts inevitably raises a question of whether the concept of price-change implied volatility is justifiable. To answer this question, consider the following scenario: A financial institution wishes to hedge a position using a put option with certain strike price and maturity. There is a high probability that this particular contract was not traded on previous day. Therefore, contract price-level implied volatility (as well as contract price-change implied volatility) cannot be estimated. In this case, the option will be priced based on the volatility surface, and it is the price-change implied volatility surface that gives the best hedging results.
6 Conclusion
In this study we introduce the concept of price-change implied volatility as an alternative volatility measure to traditional implied volatility (here, price-level implied volatility). The price-change implied volatility is estimated using price changes in an option contract and its underlying, rather than price levels, as in price-level implied volatility estimates. We compare the time series behavior, moneyness, and maturity biases of price-change implied volatility with both price-level implied volatility and historical volatility. Price-change implied volatility shows time series behavior similar to that of price-level implied volatility and the moving average of historical volatility. However, it is more disperse than either of these measures. We also find that price-change implied volatilities are smaller (larger) than price-level implied volatilities for calls (puts) for all subperiods. Money and maturity biases are also consistent with those found in price-level implied volatilities, although the maturity bias is not so pronounced. Calls exhibit more bias than puts.
The importance of a price-change implied volatility concept depends primarily on its hedging performance relative to price-level implied volatility. We find that delta hedges based on a price-change implied volatility surface produce better results than hedges based on a traditional price-level implied volatility surface for both calls and puts. More broadly, this suggests that for hedging purposes, the Practitioner Black–Scholes model formed on grids developed using price-change implied volatilities might well be superior to grids developed using traditional price-level implied volatilities. The troublesome issue of sample size can be addressed using fitted price-change implied volatility. This solution produces hedges with effectiveness better than price-change or price-level implied volatility surfaces. Fitted price-change implied volatility uses multiple data points to estimate price-change implied volatility and incorporates volatility information that is useful in hedging next-day price changes.^{14}
Hedging performance tends to be better for puts than for calls. The difference in performance is puzzling but suggests a type of market separation for puts and calls. As has been noted by Gârleanu et al. (2009), prices of S&P 500 options may be driven up by buying pressure from institutional investors that trade mainly for insurance purposes. Therefore, it appears that volatility measures for put options contain hedging information that is different from that found in call options.
The Black–Scholes model and Black model are the same model, but applied to different underlying. The underlying in our study are the S&P 500 futures options. Therefore, we use the term Black model instead of Black–Scholes model.
Although we use tick data, we do not investigate micro-structure issues since we use 1-, 2-, and 3-day price changes taken from trading hours with the greatest liquidity.
For constant σ_{pciv} = σ, geometric Brownian motion and \(h\rightarrow 0, \)\(\Updelta F^{2}\rightarrow \sigma^{2}h\). In this case, Eq. (5) is exact and the returns on the portfolio are deterministic. Under standard assumptions, equating the right-hand side of the equation to the returns on a risk-free portfolio gives the Black partial differential equation for option price.
In our lattice, we use 1,000 time steps, which slightly exceeds 960 steps used by Chang et al. (2012) and the number of steps used by Hilliard and Schwartz (1996, 2005).
This can be the result of market imperfections, such as transaction costs, the inability of market makers to fully hedge their positions at all times (Gârleanu et al. 2006), capital requirements, and sensitivity to risk (Shleifer and Vishny 1997).
VXO is defined as "the average over 8 near-the-money Black-Scholes implied volatilities at the two nearest maturities on S&P 100 index." As such, it is essentially an estimate of the 1-month, at-the-money implied volatility.
Consider a scenario in which a financial institution wishes to lock in the profits from selling an overvalued (vis-à-vis Black’s value) option to a client. It does this by creating an equivalent synthetic long position in the option. Absent the synthetic position, the firm’s profit is at risk. For further discussion, see Hull (2008). Our application is slightly different in that we only investigate the 1-day portion of a hedged portfolio that would otherwise be maintained until option expiration.
In addition to estimates based on dollar values, we obtained estimates using percentage values, i.e., we used \(\frac{G_{j}^{\ast }-G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))}{G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))}\) instead of \(G_{j}^{\ast }-G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))\) in the Eqs. (13 and 14).
These datasets are based on careful selection of data, preventing overlapping of observations and using trades in time of the highest liquidity (10:00 a.m.).