The effect of the financial crisis of ’07 – ’09 on the relationship between option-implied volatility and realized volatility in the S&P 500 market

University of Amsterdam

Faculty of Economics and Business

BSc Economics and Business

Finance and Organization track

The effect of the financial crisis of ’07 – ’09 on the

relationship between option-implied volatility and

realized volatility in the S&P 500 market

“This paper examines the relationship between realized and implied volatility before the start of the financial crisis of 2007-2009 and after it. An OLS-regression together with an IV-regression and a Chow test is used. Data is obtained from Datastream and ranges from 2002 to 2016. Results are mixed and depend on the date considered to be the start date of the crisis.”

Date: 30.01.2017

Name: Victor Oprea

Supervisor: Liang Zou

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Statement of originality

This document is written by Student Victor Oprea who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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1. Introduction

Volatility, broadly defined as a measure of possible variation of an economic variable or a function of that variable and usually measured as a standard deviation of that variable (Aizenman & Pinto, 2005), is one of the most researched topics in the financial literature, and with good reason. Volatility is the most basic and commonly known measure of the risk of a given asset or company, and thus it is of great interest to all participants of the financial industry. Modeling volatility is needed for choosing trading strategies and can be used to assess whether financial options are over-valued or under-valued. Moreover, forecasting volatility can be used to predict large stock price movements and possibly take cautionary measures such as tightening credit requirements. This paper will focus on the standard deviation of continuously compounded returns - the realized volatility, and the volatility implied by financial options. The relationship between these two measures has been researched in numerous markets and settings, but the topic is still open for new approaches and datasets. Specifically, implied volatility is widely assumed to be a market forecast of future realized price volatility and therefore an important factor in modeling future volatility (Canina & Figlewski, 1993, p. 678). Judging by this fact, checking whether this is indeed true is of paramount importance to the academic literature since it will allow for greater predictive power of future risk.

Within the sub-field of implied and realized volatility, there could be many approaches. Most papers check whether the implied volatility is an unbiased and efficient predictor of future realized volatility, such as Latane and Rendleman (1976, pp. 369-381) or the paper upon which this study is based, written by Christensen & Prabhala (1998, pp. 125-150). The results are in favor of the opinion that implied volatility is indeed efficient and unbiased, although there are some contradictory results, such as those of Canina & Figlewski (1993, pp. 659-681). The current paper will follow suit and will also check whether implied volatility has any predictive power on future realized volatility and whether it is unbiased. Moreover, in this paper, the effect of the financial crisis of 2008 on the relationship between implied and future realized volatility will be investigated. Specifically, the paper will check if the predictive power of implied volatility, with regard to realized volatility, has changed after the financial crisis of

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’07-’09. This investigation has not been previously done before; while the link between implied and realized volatilities is well-documented, the relationship between the 2 measures of volatility in terms of crises is not that well-researched. The research question is of academic interest due to the fact that, in order to use implied volatility as a valid forecast for realized volatility, one would need to know whether the relationship still holds around extreme volatility events such as financial crises and how does it change. The answer could have implications for the usage of implied volatility as a forecast and for the efficiency of option markets. This paper will use a newer dataset, ranging from 2002 to 2016, and will use the methodology outlined by Christensen & Prabhala (1998, pp. 125-150). This paper will add a Chow test in order to check for a change in the regression line due to the crisis. The results of the paper vary depending on the date chosen as the start of the financial crisis. Choosing a certain date suggests, for the most part, implied volatility is an unbiased forecast of future realized volatility; however, choosing a later date suggests the opposite.

The rest of this paper will be structured as follows. In part 2, the relevant literature and results will be reviewed, and the seminal papers of the field will be discussed. Part 3 will provide a description of the data and sampling procedure, together with some descriptive statistics of the final dataset. Part 4 will outline the methodology used in this paper and the hypotheses that are tested. Part 5 presents the main results and a discussion based upon them. Part 6 consists of the conclusion and suggestions for further research, followed by the bibliography and appendix.

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2. Literature review

This section will describe the existing relevant literature on the topic and the most notable findings in this domain.

The relationship between option implied volatility and realized future volatility has been extensively covered by various authors. Most of the studies investigate whether the implied volatility is an efficient predictor of future realized volatility and whether this fact has any usefulness in further research. The earlier studies have provided varying results due to different sampling procedures and datasets. Newer studies that use newer sampling methods are more consistent in their findings, suggesting that implied volatility definitely has predictive power on future realized volatility and it most probably efficient.

The first paper that is worth mentioning is the famous study regarding the pricing of options and corporate liabilities done by Black & Scholes (1973, pp. 637-654). In the paper, an option-pricing model with simple inputs is derived; the model is still being used in the present times due to its flexibility and empirical robustness. The model takes into account the risk-free rate, the strike price and the stock price, all of which are observable. Moreover, the model also uses a certain value for the volatility of the stock, which has been proven to be of interest to the academic community.

One of the first studies on implied volatility is done by Latane & Rendleman (1976, pp. 369-381) that used the weekly closing prices of 24 stocks and options on those stocks traded on the Chicago Board Option Exchange, henceforth CBOE, with a span of 39 weeks, or around 10 months. The study investigated the predictive power of implied volatility and has found that it indeed does have predictive power and that it is a better predictor of future realized volatility when compared to past realized volatility (Latane & Rendleman, 1976, pp. 369-381). Subsequent studies have also found similar results. Chiras & Manaster (1977, pp. 213-234) and Beckers (1981, pp. 363-381), use a more general form of the Black & Scholes model, specifically Merton’s model (Merton, 1973, pp. 141-183), and also more observations; they find similar results to Latane & Rendleman (1976, pp. 369-381) – implied volatility generates better predictions than past realized volatility, which is the general conclusion of earlier studies on implied volatility.

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Later studies, however, find slightly different results than those mentioned above. Day & Lewis (1992, pp. 267-287), using a larger dataset of about 6.5 years, between March 1983 and December 1989, find mixed results. The study was based on the S&P100 and, respectively, OEX options; the results suggested that implied volatility might be efficient; however it contained no incremental information when compared to past realized volatility. The study also mentioned that no strong conclusions could be made regarding implied volatility (Day & Lewis, 1992, p. 286). Jorion (1995, pp. 507-528) examines implied volatility in the foreign exchange market and finds that although implied volatility does have predictive power and outperforms other types of volatility forecasts, it is biased and too variable.

The study done by Canina & Figlewski (1993, pp. 659-681) figures some more extreme results, suggesting that implied volatility has almost no predictive power and is outperformed by past realized volatility. The authors analyzed the S&P100 market and took into account over 17000 American option contracts. The data ranges from March 1983 to March 1987 and included options with up to four month maturities. Moreover, the authors eliminated certain options, such as those with seemingly negative implied volatilities. The result figured slope coefficients for implied volatility ranging from the vicinity of 0 to 0.2, while the slopes for past realized volatility were higher. The result were controlled for both nonsynchronous prices and forecasting errors. The study suggests that the methodology used to compute implied volatilities is faulty and there are other considerations that are worth taking into account, such as liquidity, investor preferences, interactions with other markets, etc. (Canina & Figlewski, 1993, p. 677).

However, as mentioned in the study done by Canina & Figlewski (1993, pp. 659-681) and in the study done by Jorion (1995, pp. 507-528), the academic literature focuses on daily observations and therefore on overlapping datasets. The option prices are sampled daily, but the maturity of the options is at least one month. This results in serial auto-correlation between observations which could distort results and further exacerbate problems related to errors in variable measurement. Further studies take this into account and adopt a different sampling procedure that was not possible before due to the unavailability of a sufficient amount of data.

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The first paper to introduce a new sampling procedure is written by Christensen & Prabhala (1998, pp. 125-150). The authors were first to introduce the method of a non-overlapping dataset, in which options are sampled while taking account the maturity and expiration date. Christensen & Prabhala (1998, p. 126) mention that previous studies suffered from the correlation problem described above and also the maturity mismatch problem – the predictive power is tested for a period that is shorter than option maturity. The authors use 139 months of option and stock index data on the S&P100 and use Ordinary Least Squares regressions, as well as an Instrumental Variables approach. Christensen and Prabhala (1998, pp. 125-150) find that implied volatility outperforms historical volatility and that it is an efficient and unbiased estimate of future realized volatility. The results are partly in line with those obtained by Jorion (1995, pp. 507-528), except that the coefficients are also found to be unbiased.

Following the above mentioned paper, most studies used the non-overlapping dataset method introduced by Christen & Prabhala (1998, pp. 125-150) and find similar results in different option markets. Li & Yang (2008, pp. 405-419) use a similar methodology and also include dividend payments for the Australian market S&P/ASX 200. Their results correspond to those of Christensen & Prabhala (1998, pp. 125-150). Christensen & Hansen (2002, pp. 187-205) use a more complex method to estimate the implied volatilities that aggregates different maturity call and put options into a single forecast. The authors report the same results as previous papers. These findings are further supported by Szakmary et al. (2003, pp. 2151-2175), who study 35 futures markets and present the same results.

The current paper will also base the methodology on the study done by Christensen & Prabhala (1998, pp. 125-150), but with a new dataset. Moreover, similar to the above mentioned study, this paper will investigate whether there is a change of the regression line due to a shock. In this paper, the shock is the start of the financial crisis.

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3. Data collection and sampling

In this section, the data that was used in this paper will be described. Moreover, motivation for specific choices regarding data collection will be provided and the data sampling procedure will be explained in detail.

3.1 Data collection

This section will explain what data has been collected and why this specific type of data has been chosen.

3.1.1 Description

All the data was obtained and compiled through Datastream. First, 2 time series of financial options with approximately 100% moneyness on the S&P500 were collected with daily frequency, one series for European put options and 1 for European call options. Each time series includes the price of the option, described by Datastream code OM, the strike price, with code OS, 1-month implied volatility, with code O1, and the interpolated implied volatility, with code VI. In order to test whether the implied volatilities of the call and put options are the same, a regression equation was fitted for the whole period and for 2 separate sub-periods:

𝑖𝑣

𝑐𝑎𝑙𝑙

= 𝛼 + 𝛽 × 𝑖𝑣

𝑝𝑢𝑡

+ 𝜃

(1)

where

𝑖𝑣

_{𝑐𝑎𝑙𝑙} represents the implied volatility of the call options,

𝑖𝑣

_𝑝𝑢𝑡 represents the implied volatility of the put options, and

𝜃

represents the error term. The results of the regression are presented in Table 7 of the Appendix and suggest that the two volatilities are the same – the coefficient for the put implied volatility is not significantly different from 1.

The daily observations range from 19.11.2004 to 16.12.2016, with a total of 3151 observations for each time series, or around 144 months. Besides that, another time series of the Total Return Index of the S&P500 was collected from Datastream with daily frequency for the same period as the options.

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3.1.2 Measures of implied volatility

Datastream provides 2 measures of implied volatility – constant maturity, 1-month implied volatility, O1, and the interpolated implied volatility, VI. O1 is calculated as the implied volatility of the 1-month option with the strike price closest to the level of the underlying asset. The interpolated implied volatility, VI, is a more exact measure of implied volatility due to the fact that it takes into account the tick size and tries to compensate for the difference between the price of the underlying asset, namely the S&P500, and the closest strike price. Datastream calculates the interpolated implied volatility, VI, as a weighted average of the implied volatilities of the 2 at-the-money options closest to the level of the underlying asset, namely the S&P500. The weights are proportional to the distance between the strike price and the level of the underlying – if the S&P500 trades at 2148 and the closest strike prices are 2140 and 2150, then the implied volatility of the option with strike price 2140 has a weight equal to 2150−2148

2150−2140

=

0.2

and the implied volatility of the option with strike price 2150 has a weight equal to 2148−2140

2150−2140

= 0.8

. From now on, only the implied volatility VI will be used in the regressions. The measure O1 will be used for robustness checks presented in the Appendix.

3.1.3 Moneyness

One of the factors that affect the price of stock options is the option strike price and the ratio of the strike price over the price of the underlying asset, also known as moneyness. Moreover, the option-implied volatility also depends on the moneyness, i.e. options with different moneyness have different implied volatilities. In this paper, only options with 100% moneyness have been used, so that the strike price is as close as possible to the price of the underlying asset, the S&P500. This level of moneyness is considered optimal in the literature. Hull and White (1987, pp. 292-298) argue that when volatility is not constant and is related to the stock price, options with 100% moneyness produce the smallest bias. Options with 100% moneyness are also less affected by non-normally distributed returns and the least affected by the possibility of early exercise of American options (Agnolucci, 2009, p. 318).

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3.1.4 Dividends

Another factor that affects the price of stock options is dividend payments. Dividends reduce call values and could induce early exercise in American call options (Christensen & Prabhala, 1998, p. 131). In order to account for that, the Total Return Index of the S&P500 has been used, thus it is assumed that all dividends are reinvested.

3.2. Data sampling

In this section all the manipulations of the raw data will be explained and the final data set will be described.

3.2.1. Sampling

The sampling method used in this paper is similar to the method used by Christensen and Prabhala and subsequent papers, specifically a non-overlapping dataset (Christensen & Prabhala, 1998, pp. 125-150). By definition, options on the S&P500 expire on the 3rd _{Friday of each month. Thus, on the Monday following the 3}rd _{Friday of each} month, denoted by

𝑡

, the option price and implied volatility are sampled. The options expire the next month and have a time to expiry of one month. For the same period, the price of the S&P500 Total Return Index is recorded with daily frequency starting the Monday following the expiry date until the next 3rd Friday, denoted by

𝑡 + 1

. A whole sequence of prices and volatilities is recorded in this way for the whole period of 144 months.

3.2.2 Final dataset and descriptive statistics

The data will be split into 2 periods – before and after the financial crisis. In this paper, the start of the financial crisis is considered the collapse and bankruptcy filing of the Lehman Brothers company on the 15th _{of September, 2008. Thus, the first 46 data} points are taken as the period before the crisis and the other 98 data points as the period after the crisis. Descriptive statistics for the whole period, as well as the 2 periods before and after the crisis, are calculated.

Moreover, all of the literature on the topic uses natural logarithms for the implied and realized volatility. One reason for that is that the volatilities are skewed and taking

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the logarithm brings the distributions closer to normal (Christensen & Hansen, 2002, pp. 187-205). Table 1 shows descriptive statistics for the whole period for the price and implied volatility of call options, the natural logarithm of implied volatility, the realized volatility and its natural logarithm. The natural logarithm of the implied volatility measures has a slightly higher mean and a lower standard deviation than the logarithm of the realized volatility. These results are in line with most of the research done on this topic. Christensen & Prabhala (1998, p. 130) find similar qualitative results, however the results of this paper are somewhat higher. One reason for that could be the different time periods that are studied – the results suggest that both realized and implied volatility have increased in terms of standard deviation over time. Li & Yang (2008, p. 412) mention that implied volatility is less volatile than realized volatility and this result fits the informal definition of implied volatility as a smoothed out expectation of realized volatility.

Table 1.

Descriptive statistics for call options time series for the whole 144 months– price, 2 measures of implied volatility (O1 and VI) and their natural logarithms ( ln(O1) and ln(VI)) and the realized volatility and its natural logarithm (SDEV and ln(SDEV)).

Price O1 VI SDEV ln(O1) ln(VI) ln(SDEV)

Mean 24.64 0.171 0.169 0.159 -1.865 -1.882 -1.989

Std.Dev. 9.028 0.904 0.092 0.110 0.415 0.420 0.506

Skewness 1.552 2.303 2.472 3.031 1.036 1.094 0.910

Kurtosis 5.895 9.704 10.804 14.936 3.667 3.902 3.966

Table 2 shows the same descriptive statistics for the same period, but for put options. The mean of the market prices of put options over the time period is higher than that of call options, but the mean of the implied volatility is lower. This result does

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not correspond to the results of Li & Yang (2008, pp. 405-419), who find that the mean of the put implied volatility is higher than the mean of the call implied volatility. Since the paper of Li & Yang (2008, pp. 405-419) deals with the same type of volatility and options, it would be expected that the results would correspond. Possible reasons for the divergence would be that the market that is studied is different and that part of the methodology is not the same, since the authors take into account dividends in a different way.

Table 2.

Descriptive statistics for put options time series for the whole 144 months– price, 2 measures of implied volatility (O1 and VI) and their natural logarithms (ln(O1) and ln(VI)) and the realized volatility and its natural logarithm (SDEV and ln(SDEV)).

Price O1 VI SDEV ln(O1) ln(VI) ln(SDEV)

Mean 25.449 0.171 0.169 0.159 -1.883 -1.895 -1.989

Std.Dev. 10.472 0.904 0.092 0.110 0.425 0.426 0.506

Skewness 1.795 2.303 2.472 3.031 1.108 1.113 0.910

Kurtosis 8.678 9.704 10.804 14.936 3.887 3.915 3.966

Tables 3 and 4 show descriptive statistics for both periods – before 15th _of September 2008 and after this date – for both call (Table 3) and put (Table 4) options. For call options, implied volatility has lower mean and is less volatile after the start of the crisis. Realized volatility has higher mean and is also less volatile. For put options, implied volatility has higher mean and is more volatile after the crisis, while realized volatility has higher mean and is less volatile.

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Table 3.

Descriptive statistics for call options time series for the period before and after the start of the crisis– price, 2 measures of implied volatility (O1 and VI) and their natural logarithms ( ln(O1) and ln(VI)) and the realized volatility and its natural logarithm (SDEV and ln(SDEV)).

Price ln(O1) ln(VI) ln(SDEV)

Before After Before After Before After Before After Mean 22.502 22.643 -1.933 -1.833 -1.974 -1.839 -2.066 -1.954

Std.Dev. 8.588 9.097 0.350 0.440 0.346 0.446 0.518 0.498

Table 4.

Descriptive statistics for put options time series for period before and after the start of the crisis– price, 2 measures of implied volatility (O1 and VI) and their natural logarithms ( ln(O1) and ln(VI)) and the realized volatility and its natural logarithm (SDEV and ln(SDEV)).

Price ln(O1) ln(VI) ln(SDEV)

Before After Before After Before After Before After Mean 19.701 28.147 -1.955 -1.850 -1.986 -1.853 -2.066 -1.954

Std.Dev. 8.691 9.097 0.354 0.452 0.351 0.452 0.518 0.498

3.2.3 Potential errors in variables

The results of this paper could be affected by measurement error, as mentioned by Christensen & Prabhala (1998, p. 131) and Li & Yang (2008, p. 412). Firstly, the measurements could be nonsynchronous i.e. the prices of puts, calls, the implied and realized volatilities could be sampled at different times in the day and that could create

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errors. Jorion (1995, p. 522) predicts that the errors in the implied volatilities could amount to 1.2%, which is a substantial result. However, this result is estimated for the S&P100, thus the error for the S&P500 could differ from the above-mentioned value. Moreover, there could be errors regarding the Black & Scholes model used in this paper i.e. the underlying assumptions, such as the assumption of a log-normal distribution of returns, are not met (Li & Yang, 2008, p. 412). Christensen & Prabhala (1998, p. 131) mention that even with errors in variables, the analysis of implied volatility is meaningful, but should not be seen as a test of option market efficiency.

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4. Methodology

In this section the methodology of this paper will be described, which includes providing definitions for the variables used in the hypothesis testing, the model and hypotheses used and the theoretical expected results.

4.1 Variable definitions

This section will address the definitions used in the research.

4.1.1 Implied volatility

The calculation of implied volatility is based on the Black-Scholes-Merton option pricing model (Black & Scholes, 1973, pp. 637-654). This model is based on the following assumptions (Beckers, 1981, pp. 363-381):

a) Stock prices follow a log-normal distribution.

b) Perfectly competitive capital markets.

c) The option price is a function of the stock price and time to maturity.

d) The short-term interest rates and the variance of stock return are constant until the expiration of the option.

Under these assumptions, the value of a call option is equal to:

𝐶

𝑡

= 𝑆

𝑡

𝑁(𝑑

1

) − 𝑋

𝑡

𝑒

−𝑟𝑡𝑇𝑡

𝑁(𝑑

2

)

(2)

The value of the put option is equal to:

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where

𝐶

_𝑡 is the call option price at time t

,

𝑁(. )

is the standard normal probability distribution function,

𝑋

_𝑡 is the strike price of the option at time

𝑡

,

𝑟

_𝑡is the risk-free interest rate,

𝑇

_𝑡 is the time to maturity, and

𝑑

₁,

𝑑

₂ are calculated in the following way (Black & Scholes, 1973, pp. 637-654):

𝑑

1

=

ln �𝑆

𝑡

𝑋

𝑡

� + �𝑟

𝑡

+

𝜎

_{𝑖𝑣,𝑡}2

2 � 𝑇

𝜎

𝑖𝑣,𝑡

√𝑇

(4)

𝑑

2

=

ln �𝑆

𝑡

𝑋

𝑡

� + �𝑟

𝑡

−

𝜎

_{𝑖𝑣,𝑡}2

2 � 𝑇

𝜎

𝑖𝑣,𝑡

√𝑇

(5)

where

𝜎

_{𝑖𝑣,𝑡}2 is the volatility of the underlying asset over the remaining life of the option. Most of the factors in the Black-Scholes model – the strike price, the underlying price, the risk-free rate (usually taken as the Treasury Bill rate) and the time to maturity – are directly observable. The only unobservable factor in the Black-Scholes modes is the volatility of the underlying over the remaining life of the option (Beckers, 1981, p. 364). This volatility will be henceforth called the implied volatility, corresponding to the notation

𝜎

_{𝑖𝑣,𝑡}2 in the Black-Scholes formula

,

and is considered in the literature as the market’s expectations of the volatility for the next period (Canina & Figlewski, 1993, pp. 659-681). Consequently, if the markets are efficient and assuming the Black-Scholes assumptions hold in the real world, the market’s expectations should correspond to the actual realized volatility over the period. The implied volatility is derived by applying the inverse of the Black-Scholes model to actual call and put option prices. There is no analytic solution for implied volatility, thus it is derived using numerical methods – the option price is a monotonically function of the implied volatility, thus the inverse function theorem states that there is at most one value of implied volatility corresponding to a single option price. This calculation is provided by Datastream.

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4.1.2 Realized volatility

Realized volatility is defined as the square root of the variance of the historic daily log-returns over the remaining life of the option. Thus, if an option is sampled at time

𝑡

, the corresponding realized volatility is considered the standard deviation of the daily log-returns between

t

and

t + 1

. The daily log return is calculated as:

𝑅

𝑖

= ln (

_𝑆

𝑆

𝑖

𝑖−1

)

(6)

where

𝑖 = 1,2, . . . 𝑛

is day

𝑖

between period

𝑡

and

𝑡 + 1

. The derivation of (5) is provided in Explanation 1 of the Appendix. The annualized realized volatility is calculated in Excel based on the version used by Li & Yang (2008, pp. 405-419) and has the form:

𝜎

𝑟,𝑡

= �

_{𝑛 − 1 �(𝑅}

252

𝑖

− 𝑅�)

2 𝑛

𝑖=1

(7)

where

𝑅�

is the mean daily log-return between period

𝑡

and

𝑡 + 1

and

𝜎

_𝑟,𝑡is the annualized realized volatility between period

𝑡

and

𝑡 + 1

. In this case, 252

𝑛−1 was used as opposed to 252

𝑛−2 used by Li & Yang (2008, pp. 405-419) due to the particularity of Excel’s standard deviation formula. This formula is also used in the paper written by Christensen & Hansen (2002, p. 190), where they mention that this formula only generates a proxy for the annualized realized volatility. The most precise solution is to use intraday data, but Datastream does not provide that kind of information.

4.2 Model

The model used in this paper is similar to the model used by Christensen & Prabhala (1998, pp. 125 – 150). It consists of one Ordinary Least Squares regression and one Instrumental Variables regression. The regressions will be done for the whole period of 144 months, but also for 2 separate periods – the first 46 months before the start of the financial crisis and the 98 months following that event. After the regressions, an

F

-18

test, specifically the Chow test, will be performed on the 2 separate periods to check for structural breaks in the data. This feature of the model is not mentioned in any previous research, however Christensen & Prabhala (1998, pp. 125-150) perform similar regressions on sub periods for the 1987 market crash. The Chow test is used to check whether the regression coefficients have significantly changed after a certain moment. The regression equations used are the following:

𝑟𝑣

𝑡

= 𝛼

1,𝑡

+ 𝛽

1,𝑡

𝑖𝑣

𝑡

+ 𝜀

1,𝑡

(8)

𝑖𝑣

𝑡

= 𝑎

𝑡

+ 𝑏

𝑡

𝑖𝑣

𝑡−1 (9)

𝑟𝑣

𝑡

= 𝛼

2,𝑡

+ 𝛽

2,𝑡

𝑖𝑣

𝑡

+ 𝜀

2,𝑡 (10)

where

𝑟𝑣

_𝑡 is the natural logarithm of the realized volatility between

𝑡

and

𝑡 + 1

,

𝑖𝑣

_𝑡is the natural logarithm of implied volatility (VI) in period

𝑡

,

𝑖𝑣

_𝑡−1is the implied volatility in period

𝑡 − 1

, and

𝜀

is the error term. Natural logarithms are used in line with anterior research on implied volatility. The distributions of volatilities are highly skewed, thus taking logarithms brings them closer to log-normality (Christensen & Prabhala, 1998, p. 131). Equation (10) is an Instrumental Variables, also called a Two Stage Least Squares, regression: realized volatility at time

𝑡

is regressed against the fitted values of implied volatility at time

𝑡

, obtained from (9). The Instrumental Variables regression is used in order to correct for possible endogeneity of the implied volatility and for possible errors in variables (Christensen & Prabhala, 1998, p. 137). The results of the Instrumental Variables regressions will be presented in the Appendix.

The

F

-test has the form:

𝐹 =

_{(𝑆𝑆𝑅}

(𝑆𝑆𝑅

𝑇

− (𝑆𝑆𝑅

1

+ 𝑆𝑆𝑅

2

))/𝑘

1

+ 𝑆𝑆𝑅

2

)/(𝑁

1

+ 𝑁

2

− 2𝑘)

(11)

where

𝑆𝑆𝑅

_𝑇 represents the sum of squared residuals for the whole dataset,

𝑆𝑆𝑅

₁ represents the sum of squared residuals for the period before the crisis, and

𝑆𝑆𝑅

₂ represents the after crisis sum of squared residuals.

𝑘

represents the total number of parameters,

𝑁

₁ represents the number of data points before the crisis and

𝑁

₂the

19

number of data points after the crisis. The Chow test follows an

F

-statistic with k and

𝑁

1

+ 𝑁

1

− 2𝑘

degrees of freedom. In this case,

𝑘 = 2

and

𝑁

1

+ 𝑁

1

− 2𝑘 = 140.

4.3 Hypotheses

This section will deal with hypotheses and expected results.

4.3.1 Testable propositions

Using the model described above, several things will be tested. First of all, this paper will do a basic test whether the implied volatility has any predictive power on the realized volatility. Thus, this hypothesis is testing whether there is a relationship between the 2 measures of volatility. This hypothesis is crucial for testing structural breaks in the data. Moreover, the paper will check, for both periods, whether the implied volatility is an unbiased forecast of the realized volatility and thus will answer whether the relationship between implied and realized volatility (if there exists one) changes during crisis times. Finally, in this paper, the existence of any structural breaks will be checked and thus the research question will be answered.

4.3.2 Statistical hypotheses

The main hypotheses are:

𝐻

1

: 𝛽

1,𝑡

= 0

This hypothesis will test whether the implied volatility has any predictive power on the realized volatility. Under the null hypothesis, implied volatility has no predictive power on the realized volatility.

𝐻

2

: 𝛼

1,𝑡

= 0, 𝛽

1,𝑡

= 1

This hypothesis will check whether the implied volatility is an unbiased forecast of the realized volatility. Under the null hypothesis, implied volatility is unbiased.

𝐻

3

: 𝛽

1,𝑡,𝑏𝑒𝑓𝑜𝑟𝑒

= 𝛽

1,𝑡,𝑎𝑓𝑡𝑒𝑟

20

where

𝛽

_{1,𝑡,𝑏𝑒𝑓𝑜𝑟𝑒} is the slope coefficient for the period before the crisis and

𝛽

_{1,𝑡,𝑎𝑓𝑡𝑒𝑟} is the slope coefficient for the period after the crisis. This hypothesis will check if the relationship between implied and realized volatility changed after the start of the crisis.

21

5. Results and analysis

In this section, the results of the paper will be presented, analyzed and compared with previous research. The results of the OLS regression (8) on call option implied volatility is presented below.

For the whole period, the intercept value is not significantly different from 0 and the slope value is not significantly different from 1. Therefore, the implied volatility of call options is an unbiased predictor of future realized volatility –

_𝐻

₁ is rejected and

_𝐻

₂ is not rejected. It is worth noticing that these results differ from Christensen & Prabhala (1998, p. 135), Li & Yang (2008, p. 414) and most other research in this area. Previous papers have found significant, albeit biased, coefficients while in this paper the coefficients are unbiased. Nevertheless, the above mentioned papers do find unbiased coefficients after controlling for endogeneity in an Instrumental Variables setting. The results of this paper suggest that the implied volatility perfectly predicts the realized volatility. This fact could occur due to market choice – in this paper, the market under question is the S&P500, while previous research focuses on S&P100, ASX, KFX and other option markets. Additionally, previous research cited in this paper is performed on earlier time periods and that could influence the results. Moreover, the adjusted R2 is significantly higher in this paper compared to previous papers, meaning more of the variance of

𝑟𝑣

_𝑡 is explained by the model. Specifically, 63.44% of the variance in the dependent variable is explained by the variance of the independent variable. When looking at separate periods, we find unbiased coefficients for implied volatility before the crisis – intercept not significantly different from 0 and slope not significantly different from 1. After the crisis, the slope coefficient is not significantly different from 1, but the intercept is significantly different from 0 at a level of 5%. At 1% significance level, the intercept coefficient is not significantly different from 0. That means that at 5% significance level, implied volatility is a biased predictor of future realized volatility i.e. it overestimates the future realized volatility (intercept is negative and significant). The adjusted R2 _{for the separate periods are also different from the whole period}

adjusted R2. This paper finds a higher adjusted R2 before the crisis and a lower value after the crisis, which means the explanatory power of the model decreased after the

22

start of the crisis. Before the crisis, the model explains 66.18% of the variance of the future realized volatility, while after the crisis this value decreases to 63.08%. The value of the

F

-statistic done for the Chow test is equal to 2.616. The

F

-statistic follows an

F

distribution with 2 and 140 degrees of freedom. The value obtained is not significant at a significance level of 5% thus the relationship between realized and implied volatility did not change after the start of the crisis.

Table 5.

Results of OLS regression (8) on call option implied volatility. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 Chow test

𝑖𝑣

𝑡 Intercept Slope Whole period -0.182 (0.1173) 0.96** (0.0608) 0.6344 Before crisis 0.349 (0.2597) 1.224** (0.1297) 0.6618 2.616 After crisis -0.317* (0.1304) 0.89** (0.0689) 0.6308 ** : significant at 1% * : significant at 5%

The results of the same regression, but for put option implied volatility, are presented in Table 6. For the whole period, a negative intercept value is found, however it is not significantly different from 0 at a 1% significance level. The slope is also significant at 1% and not significantly different from 1. As in the previous case, the coefficients we get are unbiased, thus

𝐻

₁ is rejected and

𝐻

₂ is not rejected. The adjusted R2 _{is equal to 0.6354, which is slightly higher than the value obtained for call}

implied volatility. This could suggest that put implied volatility explains more of the variance of the realized volatility, but the difference is too small to be significant. The regressions for separate sub periods also reveal interesting results. When looking at the period before the start of the crisis, we get an intercept value equal to 0.401; however it

23

is not significantly different from 0. The reason for this finding is that the intercept has a large standard error, more than twice of that for the whole period and that for the period following the start of the crisis. The slope value is equal to 1.24, and it is significantly different from 1 at 5% significance level, which means that the coefficients are biased – both

𝐻

₁ and

𝐻

₂ are rejected. Similar to the intercept coefficient standard error, the standard error of the slope coefficient is also large when compared to the other two regressions. The significantly different from 1 coefficient suggests that implied volatility underestimates future realized volatility for the period before the crisis. The adjusted R2 is higher for this period when compared to the other two periods and is equal to 0.7025. The Chow test has a value of 3.252191 and is significant at 5%, which means that the relationship between realized and implied volatility has changed following the start of the crisis. One explanation for that could be that due to the fact that put options are sometimes used for portfolio insurance and hedging against extreme case events such as crisis (Li & Yang, 2008, p. 411). Li & Yang (2008, p. 411) mention that demand on put options is usually higher than on call options. Following the start of the crisis, the demand for put options has increased which in turn has increased the price and consequently the implied volatility, which can be noticed from the table. The coefficient has changed from 1.24 (underestimating the future realized volatility) to 0.886 (overestimating the future realized volatility).

The regressions described above have also been done on the other measure of implied volatility, O1. Results are presented in Tables 12 and 13 of the Appendix. Moreover, in order to control for heteroskedasticity, robustness checks have been made. Results are presented in Tables 14 and 15 of the Appendix, along with the results of the Instrumental Variables regression (10). Both the robustness check and the IV regression have been performed on the whole period. The results do not differ from those presented above and confirm the previously drawn conclusions.

24

Table 6.

Results of OLS regression (8) on put option implied volatility. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 _{Chow test}

𝑖𝑣

𝑡 Intercept Slope Whole period -0.167 (0.1135) 0.96** (0.0584) 0.6354 3.252191* Before crisis 0.401 (0.242) 1.24**a (0.12) 0.7025 After crisis -0.312* (0.1275) 0.886** (0.0668) 0.6429 ** : significant at 1% * : significant at 5%

a : significantly different from 1

However, the results are not in line with the test performed in Table 7. The fact that the coefficient of the put implied volatility before the crisis is significantly different from 1 and the coefficient of the call implied volatility before the crisis is not contradicts the fact that the implied volatilities are the same. Moreover, under the put-call parity, the coefficients should be the same. In order to check the results, some additional tests were performed.

First of all, a new measure of implied volatility was created, equal to the arithmetic average of call and put implied volatility, under the assumption that it will be more precise. Regression (8) was repeated on the new measure and the results are presented in Table 10. The results do not differ from those performed on the call implied volatility – slopes are not significantly different from 1 and intercepts are not significantly different from 0, except the intercept after the start of the crisis, which is significantly different from 0.

Additionally, it was checked whether the date chosen as the start date of the crisis is optimal or the results would be more meaningful with another date. Regression (8) was performed again on the 2 separate sub-periods for call implied volatility, put

25

implied volatility and the average of those 2. However, the 2 sub-periods were changed – the start date of the crisis was now taken next month, so the 20th _{of October 2008. Thus,} the pre-crisis sub-period has now 47 observations and the after-crisis sub-period has 97 observations. The results are presented in Tables 8, 9, and 11, for the call, put, and average implied volatility, respectively. The results significantly differ from the previous ones. For all 3 measures of implied volatility, the slopes are significantly different from 1 and the intercepts are negative and significantly different from zero. Before the new date of the start of the crisis, slopes are significantly larger than one, while after the new date they are significantly smaller than one. These results would suggest that implied volatility is a biased predictor of realized volatility. Moreover, the standard deviations of the independent variables are smaller when the regression is done with the new date. Thus, it might be possible that the put implied volatility responded faster to the crisis, and therefore the coefficients were biased even with the old date, while the call options responded later and therefore the coefficients were biased only when including another observation in the pre-crisis sub-period. The new results are in line with the put-call parity – slope coefficients of call and put implied volatility are similar.

26

6. Conclusion, limitations and further research

In this paper, an analysis on the relationship between implied and realized volatility has been done. The main findings are in line with the literature: implied volatility is an unbiased predictor of future realized volatility, both for put and call options. Moreover, when checking for separate periods, implied volatility has remained unbiased in the 2 separate periods and the regression line has not changed for call options. For put options, the coefficients for the period before the crisis are biased and the regression line has changed following the start of the crisis. Moreover, the same regression was done on different sub-periods, and the results suggest that the coefficients are significantly biased for all measures of implied volatility, and that the relationship between realized and implied volatility significantly changed. Nevertheless, implied volatility should be extensively used in forecasting future realized volatility and should be seen as an important factor in volatility modeling.

Limitations of this research are firstly, the problem of errors in the measurement of variables discussed before. This could significantly affect the results. Additionally, the analysis has focused separately on put and call options. The implied volatility calculation could also be improved to include jumps. The volatility has been assumed continuous in this paper. However, there are models in which implied volatility is assumed to follow a stochastic distribution, such as the one by Jiang & Tian (2005, pp. 1305-1342). Finally, one of the limitations of this paper is using the Black & Scholes model. Although robust, the assumptions of this model may not hold in reality and thus calculating the implied volatility based on the Black & Scholes model may not be accurate.

27

7.Bibliography

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Aizenman, J., & Pinto, B., (Eds.). (2005). Managing economic volatility and crises: A

practitioner's guide. Cambridge University Press.

Beckers, S., (1981). Standard deviations implied in option prices as predictors of future stock price variability. Journal of Banking and Finance, 5(3), 363-381.

Black, F., & Scholes, M., (1973). The pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.

Canina, L., & Figlewski, S., (1993). The informational content of implied volatility. Review of Financial studies, 6(3), 659-681.

Chiras, D.P., & Manaster, S., (1978). The Information Content of Options Prices and a Test of Market Efficiency. Journal of Financial Economics, 6(2-3), 231-234.

Christensen, B. J., & Hansen, C. S., (2002). New evidence on the implied-realized volatility relation. The European Journal of Finance, 8(2), 187-205.

Christensen, B.J., & Prabhala, N.R., (1998). The Relationship between Implied and Realized Volatility. Journal of Financial Economics, 50(2), 125-150.

Day, T. E., & Lewis, C. M., (1992). Stock market volatility and the information content of stock index options. Journal of Econometrics, 52(1-2), 267-287.

Hull, J., & White, A., (1987). The pricing of options on assets with stochastic volatilities. The journal of finance, 42(2), 281-300.

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Jiang, G. J., & Tian, Y. S., (2005). The model-free implied volatility and its information content. Review of Financial Studies, 18(4), 1305-1342.

Jorion, P., (1995). Predicting volatility in the foreign exchange market. The Journal of

Finance, 50(2), 507-528.

Latane, H.A., & Rendleman, R.J., (1976). Standard Deviations and Stock Price Ratios Implied in Option Prices. The Journal of Finance, 31(2), 369-381.

Li, S., & Yang, Q., (2008). The relationship between implied and realized volatility: evidence from the Australian stock index option market. Review of

Quantitative Finance and Accounting, 32(4), 405-419.

Merton, R. C., (1973). Theory of rational option pricing. The Bell Journal of economics

and management science, 141-183.

Szakmary, A., Ors, E., Kim, J. K., & Davidson, W. N., (2003). The predictive power of implied volatility: Evidence from 35 futures markets. Journal of Banking &

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8. Appendix

Explanation 1.

Continuously compounded return:

𝑆

𝑡

= 𝑆

𝑡−1

𝑒

𝑅𝑡�𝑡−(𝑡−1)�

ln(𝑆

𝑡

) = 𝑙𝑛(𝑆

𝑡−1

) + 𝑅

𝑡

𝑅

𝑡

= ln (

_𝑆

𝑆

𝑡

𝑡−1

)

Table 7.

Results of OLS regression (1). Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑖𝑣

_{𝑐𝑎𝑙𝑙}

Independent variables Adj. R2

𝑖𝑣

𝑝𝑢𝑡 Intercept Slope Whole period -0.038 (0.027) 0.973** (0.014) 0.9708 Before crisis -0.08 (0.0597) 0.955** (0.0294) 0.9591 After crisis -0.033 (0.032) 0.974** (0.017) 0.9719 ** : significant at 1% * : significant at 5%

30

Table 8.

Results of OLS regression (8) on call option measure of implied volatility VI with the different start date of the crisis (one more observation in the pre-crisis period included). Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 _{Chow test}

VIcall Intercept Slope Before crisis 0.465* (0.23) 1.28**a (0.116) 0.7236 5.433** After crisis -0.4* (0.131) 0.849**a (0.069) 0.6103 ** : significant at 1% * : significant at 5%

a : significantly different from 1

Table 9.

Results of OLS regression (8) on put option measure of implied volatility VI with the different start date of the crisis (one more observation in the pre-crisis period included). Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 Chow test VIput Intercept Slope Before crisis 0.44* (0.207) 1.26**a (0.104) 0.7614 5.546** After crisis -0.389* (0.131) 0.848**a (0.068) 0.6150 ** : significant at 1% * : significant at 5%

31

Table 10.

Results of OLS regression (8) on the average of the call and put implied volatilities VIavg. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 _{Chow test}

VIavg Intercept Slope Whole period -0.161 (0.115) 0.968** (0.059) 0.6487 - Before crisis 0.397 (0.249) 1.24** (0.124) 0.6884 After crisis -0.303* (0.129) 0.894** (0.068) 0.6415 ** : significant at 1% * : significant at 5%

a : significantly different from 1

Table 11.

Results of OLS regression (8) on the average of the call and put implied volatilities VIavg with the different start date of the crisis (one more observation in the pre-crisis period included). Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 Chow test VIavg Intercept Slope Before crisis 0.472* (0.217) 1.28**a (0.109) 0.7485 5.636** After crisis -0.384** (0.130) 0.854**a (0.068) 0.6279 ** : significant at 1% * : significant at 5%

32

Table 12.

Results of OLS regression (8) on call option measure of implied volatility O1. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 _{Chow test}

O1 Intercept Slope Whole period -0.199 (0.12) 0.96** (0.063) 0.6185 - Before crisis 0.17 (0.273) 1.16** (0.139) 0.6032 After crisis -0.305* (0.132) 0.899** (0.07) 0.6279 ** : significant at 1% * : significant at 5%

a : significantly different from 1

Table 13.

Results of OLS regression (8) on put option measure of implied volatility O1. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

_𝑡

Independent variables Adj. R2 Chow test O1 Intercept Slope Whole period -0.195 (0.115) 0.953** (0.0597) 0.6393 - Before crisis 0.256 (0.255) 1.19** (0.128) 0.6533 After crisis -0.319* (0.128) 0.884** (0.067) 0.6397 ** : significant at 1% * : significant at 5%

33

Table 14.

Results of OLS regression (8) with robustness check and regression (10) on call option implied volatility. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

𝑡

Independent variables Adj. R2

𝑖𝑣

𝑡 Intercept Slope Robust regression -0.182 (0.12) 0.96** (0.062) 0.6369 Instrumental Variables -0.155 (0.147) 0.975** (0.077) 0.6342 ** : significant at 1% * : significant at 5%

a : significantly different from 1

Table 15.

Results of OLS regression (8) with robustness check and regression (10) on put option implied volatility. Standard deviations of the estimates are given in parentheses. Dependent variable:

𝑟𝑣

𝑡

Independent variables Adj. R2

𝑖𝑣

𝑡 Intercept Slope Robust regression -0.167 (0.11) 0.96** (0.056) 0.6558 Instrumental Variables -0.156 (0.142) 0.967** (0.074) 0.6534 ** : significant at 1% * : significant at 5%