Review of Quantitative Finance and Accounting

, Volume 42, Issue 4, pp 599–626

Volatilities implied by price changes in the S&P 500 options and futures contracts

Authors

    • Department of FinanceCollege of Business, Auburn University
  • Wei Li
    • Department of FinanceHenry B. Tippie College of Business, The University of Iowa
Original Research

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 hedging

JEL Classification

G13C61

1 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).

In this study we introduce another concept, that of price-change implied volatility, and compare its hedging effectiveness with traditional implied volatility. The Black1 model is typically viewed as a model of prices, or more specifically, of price levels. One way to estimate implied volatility (σt) is to equate the observed option price at time t (\(G_{t}^{\ast }\)) with the model-implied price (G). The model price is a function of the underlying (Ft), maturity (T), a constant interest rate (r), and volatility (σ)2:
$$ G_{t}^{\ast }=G(F_{t},T,r,\sigma). $$
(1)
However, the model can also be used to predict the changes in price that respond to both the passage of time and change in the underlying. Similarly, therefore, we may choose a volatility (σpciv) that equates the observed price changes with the model-implied price change:
$$ \Updelta G^{\ast }=G(F_{t+h},T-h,r_{t+h},\sigma _{pciv})-G(F_{t},T,r_{t},\sigma _{pciv}). $$
(2)

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

In this section, we examine the rationale for price change volatility more closely. By choosing implied parameters, as in Eq. (2), \(\Updelta G^{\ast }=\Updelta G,\) and for small but finite time intervals, \(\Updelta G\) can be approximated using a Taylor’ series approximation. For futures prices that follow a smooth diffusion, such as geometric Brownian motion (GBM), option price changes can be approximated by
$$ \Updelta G^{\ast }\approx \frac{\partial G}{\partial F}|_{\sigma _{pciv}}\Updelta F+\frac{\partial G}{\partial t}|\sigma _{pciv}h+\frac{ \partial ^{2}G}{\partial F^{2}}|_{\sigma _{pciv}}\Updelta F^{2}, $$
(3)
where h is the time interval, \(\frac{\partial G}{\partial F}=\hbox{delta},\;\frac{\partial G}{\partial t}=\hbox{theta},\) and \(\frac{\partial ^{2}G}{\partial F^{2}}=\hbox{gamma}\). We find that the direct computation in Eq. (2) differs little from the approximation in Eq. (3) in estimating implied volatilities from daily price changes. Thus, changes in the observed option price, \(\Updelta G^{\ast },\) are closely approximated by expanding the pricing model about the implied price change volatility, σpciv. To clarify the hedging implications, consider a portfolio V with a long position in an option and a position in \(-\frac{\partial G}{\partial F}|_{\sigma _{pciv}}\) units of the underlying futures contract, i.e.,
$$ V=G^{\ast }-\frac{\partial G}{\partial F}|_{\sigma _{pciv}}F. $$
(4)
By Eq. (3), the change in portfolio value over the interval h is
$$ dV\approx \frac{\partial G}{\partial t}|\sigma _{pciv}h+\frac{\partial ^{2}G }{\partial F^{2}}|_{\sigma _{pciv}}\Updelta F^{2}. $$
(5)
The result is a portfolio hedged to first order net of deterministic changes when implied price-change volatility is used to compute the position in the underlying. To implement an out-of-sample hedge, lagged σpciv may be chosen as an estimate of in-sample (contemporaneous) σpciv.4

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.

The resulting dataset contains 76,544 observations on call options and 101,010 observations on put options (Tables 1, 2) for this period. Put options therefore account for 56.9 % of trades in the S&P 500 futures options. This is consistent with Bollen and Whaley (2004), who document that put options in their study represented 55 % of trades in S&P 500 index options. The dataset contains both short- and long-term options. However, short-term options are markedly prevalent. The majority of options traded are out-of-the-money and at-the-money. Trading in the S&P 500 futures options and their underlying was the highest in 1998 but slowly declined during subsequent years because of the emergence of the GLOBEX electronic trading platform.
Table 1

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

Data are American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006. Each option is matched with an underlying future such that their trading occurred within 30 s. Only options with prices of at least $0.25 are included. The moneyness is expressed as F/K, where F is the price of the futures contract and K is the strike price. Calls are in the money for F/K > 1.

Table 2

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

Data used are American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006. Each option is matched with an underlying future such that their trading occurred within 30 s. Only options with prices of at least $0.25 are included. The moneyness is expressed as F/K, where F is the price of the futures contract and K is the strike price. Puts are in the money for F/K < 1

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.

The resulting datasets for calls and puts are shown in Table 3. The dataset for calls with 1-day lags contains 9,562 observations. However, after applying the filters, the number of observations decreases to 2,274 (dataset Calls1). The maturity of calls in this dataset range from 14 to 106 days. The majority of options are out-of-the-money options. The data for calls with higher lags decreases sharply; there are 252 observations for 2-day lags, 201 for 3-day lags, and just 169 for 4-day lags.
Table 3

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 data are American-style options on the S&P 500 futures traded on the CME from January 1998 to September 2006. For each strike, only options traded closest to 10:00 a.m. are selected on each day and used to create datasets with 1-, 2-, 3-, and 4-day lags. The selection process and data requirement are described in Sect. 3. Datasets Calls1H and Puts1H are datasets used in Sect. 5 for the estimation of volatility surface of the Practitioner Black–Scholes model. The moneyness is expressed as F/K, where F is the price of the futures contract and K is the strike price

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

We compare the statistical properties of price-change implied volatility with both price-level implied volatility and with historical volatility based on a 60-day moving average. Historical volatility is dated at the midpoint of the period and computed as
$$ \hat{\sigma}_{t+30}=\sqrt{\sum_{i=0}^{59} \frac{(y_{t+i}-\bar{y})^{2}}{59},} $$
(6)
where \(y_{t}=ln\left( \frac{P_{t}}{P_{t-1}}\right) \) is the log of daily ending price ratios and \(\bar{y}\) is the sample mean of y over the 60-day period. Price-level and price-change implied volatilities are computed by equating model predictions to observed prices on days t − 1 and t, respectively. Both price-change and price-level implied volatilities use the same options trading closest to 10:00 a.m., as described above. Because this is a period of heaviest trading, we avoid possibility of stale prices and end-of-the day biases.

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.

Plots of historical volatility, price-level, and price-change implied volatilities are shown in Figs. 1, 2, 3, 4, and 5. Since several observations for price-level and price-change implied volatilities may exist on a given day, figures for these volatility measures are based on daily volatility averages across different strike prices. Figure 1 (Fig. 2) shows the daily average of price-change implied volatility for calls (puts), and Fig. 3 (Fig. 4) shows the daily average of price-level implied volatility for calls (puts), respectively. As can be seen from the plots, all volatility measures show similar patterns over time. However, the plots suggest important differences between price-level and price-change implied volatilities. First, for both puts and calls, the dispersion of price-change implied volatility is larger than that of price-level implied volatility. Second, the dispersion of put volatility is higher than that of call volatility. In fact, Tables 4 and 5 show that the sample standard deviation of price-level implied volatility for calls (puts) is 0.054 (0.0906). The sample standard deviation of daily price-change implied volatility for calls (puts) is 0.0830 (0.1341).
https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig1_HTML.gif
Fig. 1

Daily average of price-change implied volatility for calls. Data are American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006. Price-change implied volatility is calculated for calls with 1-day lags using prices of calls traded closest to 10:00 a.m. The figure shows daily averages of price-change implied volatility across different strike prices

https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig2_HTML.gif
Fig. 2

Daily average of price-change implied volatility for puts. Data are American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006. Price-change implied volatility is calculated for puts with 1-day lags using prices of puts traded closest to 10:00 a.m. The figure shows daily averages of price-change implied volatility across different strike prices

https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig3_HTML.gif
Fig. 3

Daily average of price-level implied volatility for calls. Data are American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006. Price-level implied volatility is calculated using prices of calls traded closest to 10:00 a.m. The figure shows daily averages of price-level implied volatility across different strike prices

https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig4_HTML.gif
Fig. 4

Daily average of price-level implied volatility for puts. Data are American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006. Price-level implied volatility is calculated using prices of puts traded closest to 10:00 a.m. The figure shows daily averages of price-level implied volatility across different strike prices

https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig5_HTML.gif
Fig. 5

Moving average of S&P 500 index volatility. This volatility was calculated as 60-days moving average of historical volatility of S&P 500 index

Table 4

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

Volatility statistics are shown for call options during different periods. These periods are based on the behavior of the S&P 500 as described in Sect. 3.1. Data are American-style options on S&P 500 futures contracts traded on the CME from January 1998 to September 2006

Table 5

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

Volatility statistics are shown for put options during different time periods. These periods are based on the behavior of the S&P 500 as described in Sect. 3.1. Data are American-style options on S&P 500 futures contracts traded on the CME from January 1998 to September 2006

Tables 4 and 5 further summarize basic statistics for price-change and price-level implied volatilities. In these tables, the period of investigation (from January 1, 1998, to December 31, 2006) is divided into three subperiods according to the behavior of S&P 500 index (Fig. 6). The first is a period of increasing value of the index (from January 1, 1998, to August 31, 2000). During the second (from September 1, 2000, to March 6, 2003), the value of the index was decreasing; and during the third (from March 7, 2003, to December 31, 2006), the value of the index was increasing again.
https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig6_HTML.gif
Fig. 6

S&P 500 index including distributions from January 1, 1998 to December 31, 2006

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.

The difference between implied volatility measures and historical volatility raises the question of volatility risk premiums. Carr and Wu (2009) use the new VIX to investigate the volatility risk premiums on a number of stocks and indexes, including the S&P 500. The new VIX is determined by the price of a zero NPV swap on realized variance with payoff defined by
$$ (RV-SW)L, $$
(7)
where RV is realized volatility, SW is the swap rate, and L is the notional principal. Thus, the volatility, VIX\(\sqrt{SW}, \) where \({SW=E^{\mathbb{Q} }[RV]}\) and \({\mathbb{Q} }\) is a risk-neutral probability measure. Consequently, the VIX is not an unbiased estimate of realized volatility unless volatility has no risk premium, i.e., unless the risk-neutral measure is the same as the physical measure. Instead, Carr and Wu find that realized volatility has a negative risk premium so that participants are willing to assume expected losses in order to hedge against increases in volatility. Thus, one expects that the VIX would be greater than average realized volatility. Carr and Wu indeed find that VIX > Avg(RV) for their dataset extending from January 1996 to February 2003.

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

To investigate whether price-change implied volatility shows moneyness and maturity behavior similar to that of price-level implied volatility, we estimate the θ vector using OLS and the linear model
$$ \sigma =\theta _{0}+\theta _{1}(T-t)+\theta _{2}(T-t)^{2}+\theta _{3}\left(\frac{F }{K}\right)+\theta _{4}\left( \frac{F}{K}\right) ^{2}+\theta _{5}\left(\frac{F}{K} \right)(T-t)+\epsilon , $$
(8)
using both price-change and price-level implied volatility data. The equation is estimated first for the period from January 1, 1998, to December 31, 2006, then separately for three subperiods according to the behavior of S&P 500 index (Fig. 6). The regression coefficients are reported in Table 6 (price-change implied volatility for calls), Table 7 (price-change implied volatility for puts), and Table 8 (price-level implied volatility for calls and puts). Standard errors are adjusted using the White estimator.
Table 6

Regression coefficients for price-change implied volatility for calls

Period

Dates

Number of records

a

b

c

d

e

f

Adj. r2

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

The equation \(\sigma= a + b(T-t) + c(T-t) ^{2}+ d((F/K)) + e((F/K))^{2} + f((F/K))(T-t) + \varepsilon\) is estimated using price-change implied volatility (σ) for calls. F/K is moneyness and T − t is time to maturity. For details, see Sect. 4

*** Refers to the significance at the 99 % level, ** to significance at the 95 % level, and * to significance at the 90 % level

Table 7

Regression coefficients for price-change implied volatility for puts

Period

Dates

Number of records

a

b

c

d

e

f

Adj. r2

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

The equation \(\sigma= a + b(T-t) + c(T-t)^{2}+ d((F/K)) + e((F/K))^{2}+ f((F/K))(T-t) + \varepsilon\) is estimated using price-change implied volatility (σ) for puts. F/K is moneyness and T − t is time to maturity. For details, see Sect. 4

*** Refers to the significance at 99 % level,** to significance at 95 % level, and * to significance at 90 % level

Table 8

Regression coefficients for price-level implied volatility

Period

Dates

Number of records

a

b

c

d

e

f

Adj. r2

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

The equation \(\sigma=a+ b(T-t) + c(T-t)^{2}+ d((F/K)) + e((F/K))^{2} + f((F/K))(T-t) + \varepsilon\) is estimated using price-level implied volatility (σ) for calls and puts with 1-day lags. F/K is moneyness and T − t is time to maturity. For details, see Sect. 4

*** Refers to the significance at 99 % level, ** to significance at 95 % level, and * to significance at 90 % level

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 R2s than price-level implied volatility in these regressions for both calls and puts.

The regression coefficients from Eq. (8) are used to create surface plots of price-change implied volatility as a function of moneyness and maturity (Figs. 7, 8). As can be seen from Fig. 7, the price-change implied volatility for calls appears to show a smile, while this effect is not noticeable for puts (Fig. 8).
https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig7_HTML.gif
Fig. 7

Plot of price-change implied volatility versus maturity and moneyness for calls during the period from January 1998 to December 2006. Price-change implied volatility is calculated for calls with 1-day lags using American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006

https://static-content.springer.com/image/art%3A10.1007%2Fs11156-013-0354-z/MediaObjects/11156_2013_354_Fig8_HTML.gif
Fig. 8

Plot of price-change implied volatility versus maturity and moneyness for puts from January 1998 to December 2006. Price-change implied volatility is calculated for puts with 1 day lags using American-style options on S&P 500 futures traded on the CME from January 1998 to December 2006

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.

The delta hedge consists of taking a position in the underlying asset and, depending on the application, a risk-free bond.10 The hedging portfolio is formed on day-one using parameters estimated at that time (when the hedge is initiated). The hedge is liquidated on day-two. The hedging portfolio V is
$$ V=-G_{F}F+B, $$
(9)
where GF is the option delta, F is the underlying futures price, and B is the price of a default free bond. The bond has price B =  −C, the price of the option to be hedged. Since the futures contract is costless, the option plus the hedging portfolio is a zero cost portfolio. The price change in the hedging portfolio is computed as
$$ dV=-G_{F}dF+r_{t}Bdt, $$
(10)
where \(rdt=\frac{r}{365}\) corresponds to daily interest and r is the risk-free rate proxied by Libor. The hedging error is thus \(e=dG^{\ast }+dV,\) where \(dG^{\ast }\) is the observed change in the price of the hedged option. The estimated volatility \(\hat{\sigma}\) is the key input for the hedging strategy since \(G_{F}=G_{F}(\hat{\sigma}(\theta ))\) depends on the volatility estimate \(\hat{\sigma}\).
To compare the performance of price-change and price-level implied volatility in delta hedging, we implement the approach used by Christoffersen and Jacobs in their evaluation of the PBS model. Their key contribution is the insight that the estimation loss function and evaluation function should be functionally equivalent. Consider, in the abstract, an abbreviated form of their argument: Suppose that \(G^{\ast }=G(\sigma (\user2{\theta }))\) where \(G^{\ast }\) is observed price and G is a nonlinear pricing model. Using implied volatilities, the \(\user2{\theta }\) vector in Eq. (8) is estimated by OLS, written concisely as \(\sigma =h(\user2{\theta })+\epsilon\). The estimate \(G\left( h(\user2{\hat{\theta}})\right)\) of \(G^{\ast }\) will be biased even when \(E[\user2{\hat{\theta}}]=\user2{\theta }\) because
$$ E[G\left( h(\user2{\hat{\theta}})\right) ]\neq G^{\ast }=G(h( \user2{\theta })+\epsilon ). $$
(11)
Conversely, if \(\user2{\theta }^{+}\) is the nonlinear least squares estimate of \(\user2{\theta }, \) the implied model specification is
$$ G^{\ast }=G(h(\user2{\theta }^{+}))+\varepsilon , $$
(12)
and while \(E[\varepsilon ]\) may be nonzero, it is nonetheless likely to be small so that \(G(h(\user2{\theta }^{+}))\) should give better estimates of \(G^{\ast }\) than \(G\left(h(\user2{\hat{\theta}})\right)\). Furthermore, the nonlinear estimation and evaluation functions are aligned when the evaluation function is the mean square of \(\left(G^{\ast }-G(h(\user2{\theta }^{+})\right)\). As noted by Christoffersen and Jacobs, in the end the alignment of the estimation and loss function is an empirical issue. In support of this argument, they find a 50 % reduction in root mean square error in a study of S&P 500 options priced by the PBS model.

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.

For price-level implied volatility, we define \(\user2{\theta }^{+}\equiv \user2{\theta }_{iv}, \) and solve the nonlinear equations
$$ \user2{\theta}_{iv}(MSE) =Arg\min_{\user2{\theta }_{iv}}\frac{1}{n} \sum_{j=1}^{n}(G_{j}^{\ast }-G_{j}(\sigma _{iv,j}(\user2{\theta } _{iv})))^{2}\quad\hbox{and} $$
(13)
$$ \user2{\theta}_{iv}(AE) =Arg\min_{\user2{\theta }_{iv}}\frac{1}{ n}\sum_{j=1}^{n}\left\vert G_{j}^{\ast}-G_{j}(\sigma _{iv,j}(\user2{\theta }_{iv}))\right\vert , $$
(14)
where \(G_{j}^{\ast }\) is the observed price of an option and Gj is the model price. The vector \(\user2{\theta }_{iv}(MSE)\) denotes coefficients estimated by minimizing mean square errors (Eq. 13), and \(\user2{\theta }_{iv}(AE)\) denotes coefficients estimated obtained by minimizing absolute errors (Eq. 14).11
For price-change implied volatility, we estimate the vector \(\user2{\theta }^{+}\equiv \user2{\theta }_{pciv}\) as
$$ \user2{\theta}_{pciv}(MSE) =Arg\min_{\user2{\theta }_{pciv}}\frac{1 }{n}\sum_{j=1}^{n}(dG_{j}^{\ast }-dG_{j}(\sigma _{pciv,j}(\user2{\theta } _{pciv})))^{2}\quad\hbox{and} $$
(15)
$$ \user2{\theta }_{pciv}(AE) =Arg\min_{\user2{\theta }_{pciv}}\frac{1}{ n}\sum_{j=1}^{n}\left\vert dG_{j}^{\ast }-dG_{j}(\sigma _{pciv,j}(\user2{ \theta }_{pciv}))\right\vert, $$
(16)
where \(dG_{j}^{\ast }\) is the observed price change of the option and dGj is the model price change of the option. The vector \(\user2{\theta }_{pciv}(MSE)\) denotes coefficients obtained by minimizing the mean square errors (Eq. 15) and \(\user2{\theta }_{pciv}(AE)\) denotes estimates obtained by minimizing the absolute errors (Eq. 16).
Since our focus is on hedging, we also estimate the \(\user2{\theta}\) -vector as implied by the hedging setup. We therefore solve:
$$ \user2{\theta }_{ivh}(MSE) =Arg\min_{\user2{\theta }_{ivh}}\frac{1}{n} \sum_{j=1}^{n}(dG_{j}^{\ast }-dV_{j}(\sigma _{ivh,j}(\user2{\theta } _{ivh})))^{2}\quad\hbox{and} $$
(17)
$$ \user2{\theta }_{ivh}(AE) =Arg\min_{\user2{\theta }_{ivh}}\frac{1}{n} \sum_{j=1}^{n}\left\vert dG_{j}^{\ast }-dV_{j}(\sigma _{ivh,j}(\user2{ \theta }_{ivh}))\right\vert, $$
(18)
where \(dG_{j}^{\ast }\) is the observed change in the price of the hedged option, dVj is the price change in the hedging portfolio, and σivh is hedging implied volatility.
Volatility surfaces are updated daily, using current \(\user2{\theta }^{\prime }\)s for all volatility measures, and subsequently used in 1-day, out-of-sample delta hedges. To compare the effectiveness of different approaches of estimating the fitted volatility, we subject the resulting hedging strategy from different estimation methods to the same performance measures, namely the mean square error (MSE) and the absolute error (AE):
$$ MSE =\frac{1}{n}\sum_{j=1}^{n}(dG_{j}^{\ast }-dV_{j}(\sigma _{j}(\user2{ \theta })))^{2}\quad\hbox{and} $$
(19)
$$ AE =\frac{1}{n}\sum_{j=1}^{n}\left\vert dG_{j}^{\ast }-dV_{j}(\sigma _{j}( \user2{\theta }))\right\vert, $$
(20)
where \(dG_{j}^{\ast }\) is the observed change in the price of hedged option and dVj is the change in the price of hedging portfolio. Note that the evaluation loss function in (19) is identical to the estimation loss function in (17), and the evaluation loss function in (20) to the estimation loss function in (18). Following the lead of Christoffersen and Jacobs, we expect that the best performing delta hedge would be the hedge based on the implied hedging volatility (σivh), since the loss function used in the estimation period exactly matches the loss function used in the evaluation period.

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 sections12 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.

The hedging results are summarized in Tables 9 and 10. Panels A and B of Table 9 include results of hedges based on volatility surfaces. Hedges based on price-level implied volatility correspond to the loss function in (13) and (14), hedges based on the price-change implied volatility correspond to the loss function in (15) and (16), and hedges based on hedging implied volatility to the loss function in (17) and (18). The appropriate evaluation functions for hedges in panel A are RMSEs, for hedges in panel B, AEs. For completeness, we also report other measures, such as average error and minimum and maximum errors.
Table 9

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

The table shows the results of 1-day, out-of-sample delta hedges for different volatility measures for calls. The delta hedge is created on day 1, using parameters known on day 1. The hedge is liquidated on day 2. Panels A and B show results of delta hedges based on volatility surfaces relating volatility measures to moneyness and maturity (Sect. 4, Eq. 8). Surfaces are estimated using nonlinear least squares (Sect. 5, Eqs. 1318). Panel C shows results for fitted price-change volatility. The relation between fitted price-change volatility and price-level implied volatility (Sect. 5, Eqs. 21 and 22) is estimated daily using nonlinear least squares. For comparison, we also report hedging results for contract volatilities (panel D) and the unhedged positions (panel E). For detailed description of parameters and the hedge setup, please refer to Sect. 5

Table 10

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

The table shows the results of 1-day, out-of-sample delta hedges for different volatility measures for puts. The delta hedge is created on day 1 using parameters known on day 1. The hedge is liquidated on day 2. Panels A and B show results of delta hedges based on volatility surfaces relating volatility measures to moneyness and maturity (Sect. 4, Eq. 8). Surfaces are estimated using nonlinear least squares (Sect. 5, Eqs. 1318). Panel C shows results for fitted price-change volatility. The relation between fitted price-change volatility and price price-level implied volatility (Sect. 5, Eqs. 21 and 22) is estimated daily using nonlinear least squares. For comparison, we also report hedging results for contract volatilities (panel D) and the unhedged positions (panel E). For detailed description of parameters and the hedge setup, please refer to Sect. 5

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 σpciv compared to 0.9453 for σiv. The corresponding numbers for puts are 0.6798 for σpciv compared to 0.8661 for σiv. 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 σpciv , compared to 1.4285 (1.4460) for σiv. Surprisingly, we find no advantage in estimating the implied hedging volatility (σivh) 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.

A significant limitation in using price-change implied volatility is the requirement for large datasets. To address this problem, we examine the relation between price-change and price-level implied volatility (σiv) estimated by a \(\user2{\beta }\) vector defined as
$$ \sigma _{fpciv}=\beta _{0}+\beta _{1}\sigma _{iv}. $$
(21)
We refer to this volatility (σfpciv) as fitted price-change implied volatility. The fitted price-change implied volatility can be used to create a price-change volatility surface, as in the standard PBS surface, and used to create hedge ratios. The \(\user2{\beta }\) vector is estimated by nonlinear least squares:
$$ \user2{\beta }(MSE)=Arg\min_{\beta }\frac{1}{n}\sum_{j=1}^{n}(dG_{j}^{ \ast }-dG_{j}(\sigma _{j}(\user2{\beta })))^{2}. $$
(22)

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.

Footnotes
1

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.

 
2
The Black formula for a European call option on a futures contract, F, is given by
$$ G(F,T,r,\sigma )=e^{-rT}\left( FN(d_{1})-KN(d_{2})\right) $$
using standard notation and where N is the cumulative standard normal and
$$ \begin{aligned} d_{1} &=\frac{\ln (F/K)+\frac{\sigma ^{2}}{2}T}{\sigma \sqrt{T}}\\ d_{2} &=d_{1}-\sigma \sqrt{T.} \end{aligned} $$
 
3

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.

 
4

For constant σpciv = 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.

 
5

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).

 
6
The t statistic is calculated as follows:
$$ t=\frac{0.2632-0.1837}{\sqrt{\frac{0.0906^{2}}{3061}+\frac{0.0543^{2}}{2274}} } $$
Since t statistic assumptions are not met, the measure should be considered ad hoc.
 
7

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).

 
8

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.

 
9

See also Blenman and Wang (2012) for a comparison of implied and realized volatilities.

 
10

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.

 
11

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).

 
12

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.).

 
13

The only exception is the MSE-based loss function using percentages for calls (Table 9, panel A2), where price-level implied volatility has slightly lower RMSE than price-change and implied hedging volatilities.

 
14

Comparison of the impact of price-change implied volatility vis-à-vis the impact of other volatility measures on Greeks in a regression context can be found in Hilliard (2012).

 

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© Springer Science+Business Media New York 2013