Do unusual events affect the predictive power of implied volatility? : the case of the "Brexit" referendum

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MSc. Quantitative Finance

Master Thesis

Do unusual events affect the predictive

power of implied volatility?

The case of the “Brexit” referendum.

Voda, Ada Student ID: 11925590

Supervisor: Dr. P.J.P.M. (Philippe) Versijp

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Abstract

This thesis investigates the effect of the Brexit referendum on the predictive power of implied volatility on the FTSE 100 index excess returns and variance. Using weekly observations, the volatility smile of FTSE 100 options is constructed. The return and variance analysis is defined by comparing the effect of several high volatility events determined by the Markov Switching volatility model. This thesis finds a predictive relationship between the implied volatility skew and realized volatility level during the Brexit referendum, however no significant results were observed for realized returns.

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Statement of Originality

This document is written by Student Ada Voda who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document are 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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Acknowledgements

I would like to express my sincere gratitude to my supervisor Dr. P.J.P.M. Versijp, who has guided me through the process of writing this Master’s thesis, and has pushed me to do my best research work so far. Additionally, I would like to thank my internship supervisor at ING Bank, Tijmen van Paasen, for giving me the opportunity to be part of the FX Derivatives team, and providing me with the highest quality data for this thesis. My sincerest thanks goes to my fellow intern Pieter van der Wal, for his valuable recommendations, and for making this internship experience unforgettable. Last but not least, I would like to dedicate this work to my family, who has supported me in all my years as a student, and who has always undoubtedly believed in me.

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List of Figures

4.1 FTSE 100 Index Level and ATM Implied Volatility . . . 19

4.2 FTSE 100 Index Level and Implied Volatility . . . 19

4.3 Aggregated FTSE 100 Volatility Smirk using the Sticky Strike Rule . . . 21

4.4 FTSE 100 Volatility Smirk Pre and Post Brexit using Sticky Strike . . . 23

5.1 Markov Switching Model Volatility Regimes . . . 26

5.2 Smoothed and Filtered Probability Distribution . . . 27

8.1 Aggregated FTSE 100 Volatility Smirk using the Sticky Delta Rule . . . 65

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List of Tables

4.1 Descriptive Statistics . . . 18

4.2 Fitting the Volatility Smile: Sticky Strike and Sticky Delta Rule . . . 20

4.3 Fitting the Volatility Smile Pre and Post Brexit using the Sticky Strike Rule . . 22

5.1 Summary statistics of Expectations Maximization (EM) Algorithm . . . 24

5.2 Excess return MSCI using Smoothed Probabilities . . . 33

5.3 Excess return MSCI using Filtered Probabilities . . . 34

5.4 Realized Volatility using Smoothed Probabilities . . . 37

5.5 Realized Volatility using Filtered Probabilities . . . 38

5.6 Excess return LIBOR using Smoothed Probabilities . . . 40

5.7 Excess return LIBOR using Filtered Probabilities . . . 41

5.8 Maximum and Minimum VIF measures for all regressors . . . 43

5.9 TSLS regressions using Smoothed Probabilities . . . 46

5.10 TSLS regressions using Filtered Probabilities . . . 47

6.1 Excess return MSCI using Smoothed Probabilities . . . 51

6.2 Excess return MSCI using Filtered Probabilities . . . 52

6.3 Realized Volatility using Smoothed Probabilities . . . 53

6.4 Realized Volatility using Filtered Probabilities . . . 54

6.5 Excess return LIBOR using Smoothed Probabilities . . . 55

6.6 Excess return LIBOR using Filtered Probabilities . . . 56

8.1 Fitting the Volatility Smile Pre and Post Brexit using the Sticky Delta Rule . . . 66

8.2 Robustness Check 2: Excess return MSCI using Smoothed Probabilities . . . 67

8.3 Robustness Check 2: Excess return MSCI using Filtered Probabilities . . . 68

8.4 Robustness Check 2: Realized Volatility using Smoothed Probabilities . . . 69

8.5 Robustness Check 2: Realized Volatility using Filtered Probabilities . . . 70

8.6 Robustness Check 2: Excess return LIBOR using Smoothed Probabilities . . . . 71

8.7 Robustness Check 2: Excess return LIBOR using Filtered Probabilities . . . 72

8.8 Robustness Check 3: No Decay Effect Smoothed Probabilities . . . 73

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Introduction

The Brexit referendum has received a lot of media attention and stirred even more emotion in financial markets. The reason being that the result of the Brexit referendum was unexpected and as such caused the volatility of both option and stock markets to increase. Evidence has shown that together with this increase in volatility, the market sentiment of risk changed as well.

Prior research such as Yan (2011) documents that there exists a lead-lag relationship between the option and stock market, and that implied volatility serves as a good proxy of future down-side risk. Additionally, it has been documented that the levels of implied volatility measured by the ‘VIX’ (volatility) index can be used as a risk factor that affects future stock returns and variance. Naturally, this raises the question, could a trading strategy be developed based upon implied volatility that could lead to excess returns during unusual political events?

This thesis attempts to assess the predictive power of implied volatility on realized returns and variance. More specifically, this research uses the Brexit referendum as the event study to gauge whether this theory holds in practice.

This particular topic is interesting for a multitude of reasons. Firstly, it tries to answer whether excess returns and variance can be predicted using implied volatility during the period of the Brexit referundum, which adds to the existing literature as this has not been studied in the past. Secondly, although the relationship between option markets and stock markets has been extensively covered in past research. This research looks at this relationship in both isolation as well as in conjunction with several other unique events. Thirdly, evidence has shown that the implied volatility levels during the Brexit referendum have increased to the highest level since the financial crisis in 2009, making this a truly unique event. More interestingly, the volatility smile of FTSE 100 options became steeper after this event. This paper incorporates these facts to shed light on what were the consequences of these structural changes on both a global and local level.

Existing literature has established several stylized facts for the behavior of implied volatility and excess returns, in a cross sectional (firm) level Xing et al. (2010) as well as in a portfolio level approach Giot (2005). However, little has been done in revealing the predictive power of implied volatility on index returns, which is where this thesis adds considerably value to the existing literature. Moreover, research that has been done based on implied volatility, use em-pirical evidence to construct the underlying volatility smile (Alexander (2000)). Conversely, this paper uses evidence from financial markets as defined by Derman (1999) to construct volatility smiles based on rules of thumbs used by option traders. Thirdly, the informational content of implied volatility on an event setting discussed by Lin and Lu (2015), provides evidence of a significant predictive power of implied volatility towards returns. This thesis replicates the return analysis of this paper, and contributes to the literature by defining events in a structural way. The Markov Switching volatility model (Hamilton (1989)) is used, to define high volatility events in the index returns that are comparable among each other. Finally, this thesis combines results of existing literature on relationship between the equity and currency option markets

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Busch et al. (2011), to assess the informational content of the currency options volatility smile towards equity index returns and variance.

To investigate the research question of this thesis, data from the FTSE 100 index is employed, from January 2010 to March 2018. Options data from this underlying index are retrieved, for a maturity of one month. After manipulating the retrieved time series the methodology of the research analysis is divided in three main components. First, the volatility smile before and after the Brexit referendum is constructed using the Sticky Strike and Sticky Delta rules described by Daglish et al. (2007). Second, high volatility events in the FTSE 100 index returns are de-fined using the Markov Switching volatility model implemented by Hamilton (1989). Finally, the identified events are used to determine the effect of unique events in the predictive power of implied volatility in realized returns and variance of the FTSE 100 index. The methodology used by Lin and Lu (2015) is adjusted, to capture the index dynamics. As an extension of this model, the EUR/GBP volatility skewness has been used to determine the effect the Brexit referendum had on the return and variance predictability. Final robustness checks on the return and variance analysis are constructed to unlock any additional predictive relationship that might not be captured by the original analysis.

This thesis proceeds as follows; Chapter 2 discusses the existing literature concerning the implied volatility and stock returns, the shape of the volatility smile and the informational content of implied volatility in particular events. It is noticeable that previous literature has covered each topic individually but not considered them in interaction with each other. Chapter 3 presents the methodology used to answer the research question. The Markov Switching volatility model is used to define high volatility events, and Ordinary Least Squares regressions combined with decayed effects are used to unravel the relationship between implied volatility and realized returns and variance during these events. Moreover, Chapter 4 provides an overview of the data gathered to conduct the research analysis. Chapter 5 discusses the results of the proposed methodology, as well as discuss mis-specification issues such as endogeneity and multicollinearity. Chapter 6 extends this model to the informational content of FX implied volatility, and provides three robustness checks to validate the results. Finally, Chapter 7 provides the conclusion of this thesis.

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Literature Review

Option pricing for European type options has fundamentally changed since the introduction of the Black and Scholes (1973) model. By assuming that stock prices follow a Brownian Motion with no drift as a central piece of their methodology, the authors developed an equation to price any European type option on a trading underlying. This model has a number of assumptions, of which the following two are of a great importance: the lognormal distribution of the underlying’s price, and the assumption of constant volatility, where the return volatility is assumed to be constant for options of the same underlying. The many advantages of this model is its simplicity and the ability it offers the trader, to easily calculate the price of any European option, as long as the key inputs (stock level, strike price, time to maturity, risk free rate, dividend yield and volatility) are given. However, since this model was first developed, there has been copious changes in financial markets worldwide.

It is unreasonable to assume normality of stock returns because their risk neutral implied distribution experiences fat tails and is heavily skewed to the left. Most important for this the-sis, the volatility of returns is a non constant, due to the jump appearances in times of turmoil as pointed out by Kou (2002). An implication of this deviation from the normality assumption is that the price indicated by the Black and Scholes (1973) model is different from the price indicated in the market of the same option, with the same maturity and strike price. This gives rise to a level of volatility known as the implied volatility, which equalizes the empirical option price to the option price quoted in the market. If the Black and Scholes (1973) assumptions hold, and the volatility is constant throughout all options on the same underlying, then it would be logical to expect the implied volatility levels of these options to be the same.

Given the aforementioned, evidence from option markets, after ”Black Monday” in October 1987, shows that the implied volatility of call and put options of the S&P 500 index varied signif-icantly for different strike levels and maturities. This was the first instance when the volatility smile was observed in the market, which presented the relationship of implied volatility and strike levels. The volatility smile is a convex function which is generally centered around the at-the-money strike level, especially for currency options. For equities, this relationship is called the volatility smirk, due the skewed-to-the-left shaped convex function. As documented by Hull (2016), the price of deep out-of-the-money (OTM) call options - positioned in the right end of the smile - is lower when the implied distribution of the underlying stock price is used, compared to the one when the lognormal distribution is used. A lower implied price means a lower implied volatility for these options. The contrary holds for deep OTM put options - positioned in the left end of the smile, where higher implied prices indicate a higher implied volatility level compared to using a lognormal distribution. Hence, the shape of the volatility smile is for equities skewed to the left.

One possible explanation for this phenomenon is what is known in finance as the “crashopho-bia”. Rubinstein (1985) reports findings that indicate the traders’ concerns regarding the occur-rence of a possible crash, similar to Black Monday in 1987. As such, they price OTM put options higher than the call counter party, as an insurance towards a downturn. Another hypothesis

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for the existence of volatility smirks is the “the leverage effect”. It describes the phenomenon where negative shocks to the market value of equity, increase the amount of leverage a firm has, and subsequently this makes the remaining equity much riskier. Dennis et al. (2006) in their research in this area find that the this left skew shape, also known as the “Assymetric Volatility Phenomenon (AVP)”, is related to systematic volatility shocks such as interest rate movements or international economic downturns, more than to firm-specific volatility shocks.

2.1 Implied Volatility and Stock Returns

Since the origination of the discrepancies between the Black and Scholes (1973) price of an option and its market value, a variety of literature has considered implied volatility as a forward-looking market sentiment of risk. For instance, Bates (1991) sheds light into whether option prices could have predicted the “Black Monday” crash of 1987. The author argues that option prices gave a reliable market sentiment of risk because OTM put options in the year prior to the crash were remarkably more expensive than OTM call options. Since these OTM put options were viewed as crash insurance, Bates (1991) investigates whether the sudden jump in their prices, and hence implied volatility, could be an early indication of an imminent shock in the market. To test this hypothesis, Bates (1991) fits a jump-diffusion model of daily call and put S&P 500 options into S&P 500 futures for all days between 1985 and 1987, where among others, he estimates the probability of a jump using non-linear least squares. He finds significant negative jumps up to a year prior to the 1987 crash, but not 2 months prior and more importantly not the day prior to the crash. The work of Bondarenko (2014) concurs this result. He names the increase of put prices since ”Black Monday” and the 2009 financial crisis, as “the overpriced put puzzle”. In his work, he fails to find a non-parametric model that explains this phenomenon and argues that the lack of investor rationality may be a reason for such a result.

In efficient markets, a change in implied volatility would translate to a simultaneous change in the underlying’s returns, because in efficient market theory, prices convey all available public information present in the market. Consequently, there should be no predictive power beyond the normal risk-return relationship. However, evidence has shown that the risk-return relation-ship is not all there is to the dynamics of stock returns and implied volatility. The early study of Easley et al. (1998), investigating the informational content of implied volatility, finds that stock prices are not fully efficient and option prices deviate from the put-call parity without causing arbitrage. Since then, many authors have developed this idea and have documented important findings, as the following presents.

Giot (2005) looks into the simultaneous change in implied volatility levels and price change in the underlying stock indices. He finds that the S&P 100 and the NASDAQ 100 index returns, yield significant changes in the corresponding implied volatility index returns (VIX and VNX), whilst taking into account the index leverage effect. In addition, Giot (2005) examines the de-pendence of forward looking index returns on implied volatility index levels, by testing whether entering into a long position on the above stock indices triggered by high implied volatility levels yields positive returns. He finds a positive predictive relationship of VIX and VNX levels at time (t) on the n-day ahead returns of the SP100 and NASDAQ100 indices, however evidence was hardly conclusive.

Banerjee et al. (2007) extend on this theory by investigating the relationship between future returns, current implied volatility levels and innovations in portfolios rather than indices. They create 12 portfolios based on the constituents’ size, price to book ratio and beta. To align with

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(2007) use 30 and 60 day holding period returns as the dependent variables and regress them on both VIX levels and innovations. Their conclusion conveys strong evidence that VIX levels, as well as innovations, can predict future 60 day returns, especially for high beta portfolios. Addi-tional robust estimates were provided by testing whether the relationship between future returns and VIX levels and innovations depends on the market performance and volatility regime. For both scenarios, they find an insignificant difference between the states, which reaffirms their results that VIX levels and innovations can be used to predict future returns in any market condition.

Xing et al. (2010) argue that if stock markets are efficient, the information must be in-corporated quickly in stock prices and hence predictability of returns by volatility smirks will be temporary. They test this hypothesis by documenting the predictive power of volatility smiles/smirks in a cross-sectional (firm) level on daily option data for all listed options from January 1996 to December 2005. In the first part of their research, Xing et al. (2010), explain the predictability of future stock returns on firm level by the volatility skew, defined as the difference between the OTM put options’ implied volatility and the at-the-money (ATM) call options’ implied volatility. This choice of skew calculation is in line with the market perception that out of the money put options are in high demand when there is greater expected risk of negative jumps. Xing et al. (2010) regress firm stock returns on the skew variable, using the Fama and MacBeth (1973) regression model. From the analysis conducted, they find that there is economically large and statistically significant predictive power of the volatility skew towards firm stock returns. This relationship remains strong even while controlling for firm-specific vari-ables. Their conclusions suggest that stocks exhibiting the steepest smirks underperform stocks with least pronounced volatility smirks in their traded options, by approximately 11 percent per year. As for how long the predictability lasts, the authors find that the predictive power of the volatility skew lasts over a 28-week horizon, and after that it slowly dies out. In the second part of the research, Xing et al. (2010) investigate the volatility smirks and future earning surprises to assess what information is embedded in the volatility skew. By sorting portfolios based on the volatility skew and calculating the earnings surprise, they were able to identify a close con-nection between the slope of the volatility smirk and news regarding future firm fundamentals. In fact, they find that firms with the steepest volatility smirk experience the worst earnings shocks in subsequent months, suggesting that information is indeed embedded in the shape of the volatility smirk and it is related to firm fundamentals.

A contradicting view on the leading informational position of option markets compared to stock markets of Xing et al. (2010), is presented by Yan (2011). He assumes efficiency for both the options and stocks market. He argues that expected stock returns are a function of the volatility smile slope, which serves as a proxy of average price jump size. Yan (2011) advocates that, since option markets are forward looking, one can extract information about the jump distribution of the underlying from market option prices. By modelling the jumps as a Stochastic Discount Factor (SDF) and stock returns as a combination of a Brownian motion and a Poisson process, the author derives that in continuous time, the slope of the implied volatility smile can be approximated to a product of jump intensity and stock jump size. By ranking stocks on the last trading day of the month in ascending order of their volatility smile slope and grouping them into five quantiles, Yan (2011) finds that the realized returns of the quantile portfolios (where the first quantile portfolio includes the lowest level of slope and the fifth quantile portfolio the highest), are significantly decreasing when moving from the first to the last portfolio. The same holds when a risk-adjusted mean is compared between the five portfolios. Using this result, Yan (2011) shows that there exists a negative predictive relationship between the stock returns and the slope of the options volatility smile. After isolating the idiosyncratic and systematic components of the slope, the results show that the idiosyncratic component

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dominates in explaining the slope-return predictability.

2.2 The shape of the volatility smile

Modeling the volatility smile is the first step towards unlocking the predictive power of implied volatility on returns. Traders within the option trading industry, have historically identified stylized facts of the shape of volatility smiles, and various academics have used these insights to build well-defined models of smiles. Daglish et al. (2007) in their work on modeling the volatility surfaces derive a non-arbitrage condition for the relationship between the implied and instan-taneous variance of the option, by quantifying the risk-neutral drift of the implied variance of the option described as a Wiener process. Additionally, they test if the traders’ rules of thumb follow this non-arbitrage condition. The authors use monthly volatility surfaces from over-the-counter options for 47 months, with six different time to maturity considered, starting from 6 months to 5 years. The rules of thumb considered, based on the work of Derman (1999), belong to two distinct groups. In the first group, there are stylized facts regarding the manner in which volatility surfaces develop over time, which are important for calculating the Greek letters of the traded options. The rules include the ”Sticky Strike” and ”Sticky Delta”. The ”Sticky Strike” rule assumes that the volatility smiles can be modeled as a deterministic function of the strike price and time to maturity, whereas the ”Sticky Delta” assumes that the volatility smiles can be modelled using the moneyness and the time to maturity. In the second group, traders include rules on how volatility smiles for different maturities relate to each other at a given time, which are important for creating volatility surfaces. Such is the ”Square Root of time” rule, which describes the relationship between implied volatility of options with different strike prices and time to maturity.

Using the developed model on a non-arbitrage condition, they find that when volatility is in-dependent of the asset’s price, the volatility smile is U-shaped. Additionally, when implied volatility increases(decreases) with the asset’s price, there is positive(negative) correlation be-tween the volatility of the asset and the asset’s price. While testing whether the rules of thumb fit the proposed non-arbitrage condition, Daglish et al. (2007) find that the ”Sticky Delta” rule outperforms the ”Sticky Strike” rule for over-the-counter products. In turn, the ”Square root of time” rule performs much better than the ”Sticky Delta” rule for both in and out of the sample estimates. Another great representation of the models introduced by Derman (1999) applied in practice, is used by Alexander (2000), as she develops a Principal Component Analysis (PCA) on volatility skews and smiles to quantify the impact of change in the underlying’s price on any given fixed strike volatility level. Her work helps on understanding the dynamics of volatility surfaces and on the accurate calculation of deltas (hedge ratios).

There is consensus on modelling the volatility smile as a function of strike levels or money-ness1_{, but there is greater debate on which variable measures the slope of volatility smiles best.} As mentioned Xing et al. (2010) introduce a skew variable for the purpose of capturing the demand for out of the money put options when negative shifts in the underlying’s price are ex-pected, which is defined as the difference between the OTM put and ATM call implied volatility for a fixed delta. Another methodology for the slope is presented by Bollen and Whaley (2004), who while investigating whether buying pressure impacts the shape of volatility smiles, let the delta of the options vary when calculating the slope, as the difference between OTM and ATM implied volatility of put options. Conversely, in the attempts to model jump risk, Yan (2011) proposes that the ATM implied volatility converges to the diffusive volatility of stock returns as the option is closer to expiration. Since jump risk affects the local steepness of the volatility smile at levels closest to the money, he uses the difference between the implied volatility of a put

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and a call option around the ATM levels (delta of 0.5 and -0.5 respectively) as a measure of slope of the implied volatility. He finds that this measure of the volatility smile slope outperforms on its predictive power of stock returns compared to the measures presented above by the other authors, when used to model jump risk.

2.3 Informational content of implied volatility and events

Extensive research regarding the information transmission from option to stock markets during particular events is conducted by Lin and Lu (2015), who investigate whether option prices predict the stock market’s response to three types of analyst news: recommendation changes, forecast revisions and an initiation of coverage. They base their hypothesis on the results of Cremers and Weinbaum (2010), who argue that it takes several days for stock prices to pick up information already embedded in option prices. In their work on deviations from the put-call parity and stock predictability, Cremers and Weinbaum (2010) conclude that the stock return predictability is viewed as a proxy for informed trading. Lin and Lu (2015) consider three types of information flow between informed traders and analysts. Firstly, they consider tipping, a situation where analysts inform option traders about the three aforementioned analyst events. The second type of information flow considered is reverse tipping, where option traders tip analysts for the current trading state of options, and lastly, they introduce the common information type of flow where both channels mentioned above are combined. The authors use two types of proxies for informed trading, the IV spread2 and the IV skew, defined as the difference between the implied volatility of OTM put options and implied volatility of ATM call options. Lin and Lu (2015) regress the IV spread and the IV skew on stock returns, while adding interaction terms with dummy variables indicating days with the 3 types of analyst events. Their results show that option trading, specifically implied volatility, can predict excess stock returns because informed traders prefer the option market to trade their information towards an analyst related event.

2.4 Adding upon existing literature

This thesis will expand on these prior studies in four important ways. Firstly, it elaborates on the predictive power of volatility smiles towards index returns and variance, which builds upon the findings of Xing et al. (2010) on the predictive power of implied volatility skew on a cross-sectional level. Secondly, the volatility smiles before and after the Brexit referendum, are modeled by testing the Sticky Strike and Sticky Delta rules based on the methodology introduced by Daglish et al. (2007) for volatility surfaces. Thirdly, this thesis introduces an innovative way to define events similar to the Brexit referendum for comparison purposes, by using Markov Switching volatility regimes. The regression analysis viewed in an event study perspective resembles the methodology developed by Lin and Lu (2015) where the different events are established by the Markov Switching volatility regimes, however, it uses a smoothing operator to distinguish between the effect of the start of a particular event period and its end. Finally, an extension to the predictive power of currency volatility smiles on index returns is given, by constructing the slope of the EUR/GBP volatility smile.

2

Defined as the difference between the implied volatility of call and put options with the same strike price and maturities, which theoretically should be the same (as a result of the Put-Call parity).

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Methodology

The goal of this research is to assess whether the Brexit referendum had an impact on the predictive power of implied volatility towards realized returns and variance of the FTSE 100 index. In other words, the thesis investigates whether the downside risk of this high volatility event, was materialized in additional estimated effect of implied volatility on subsequent index returns and variance. To achieve this objective, the incremental effect of Brexit on the implied volatility-returns relationship should be compared with the effect of other events on the same rapport. This argument sets the ground for the variable framework described below.

Firstly, to capture the downside risk conveyed by implied volatility, this thesis uses a measure of slope of the volatility smile of the FTSE 100 options. The volatility smile is constructed based on the methodology of Daglish et al. (2007), whereas the slope of the smile is based on the skew measure proposed by Xing et al. (2010). Secondly, the slope of the constructed volatility smile is used to estimate its effect on realized returns and variance, similarly to Lin and Lu (2015), where the methodology used by these authors is adjusted to account for the different events that might affect the volatility smile slope. To build the regression equations, the events are constructed as high volatility periods based on the methodology of Hamilton (1989) on Markov Switching volatility regimes, where high volatility sections in the data are determined based on the probability states inferred by the model.

The sections below present the methodology of modeling the volatility smile, construct the Markov Switching high volatility events as well as use this information to build the regression analysis of return and variance.

3.1 Modeling the volatility smile

Correctly defining the shape of the volatility smile is key to understanding the predictive power of implied volatility towards any other variable. Hence, special attention is given to modeling the volatility smile before and after the Brexit referendum. To do so, this thesis implements the methodology recommended by Daglish et al. (2007) by testing the “Sticky Strike” and “Sticky Delta” rules of thumb on FTSE 100 option data, adjusted for volatility smiles.

3.1.1 Sticky Strike Rule

The first rule of thumb is called the ”Sticky Strike Rule” and is mainly used for retrieving the market delta and gamma of a traded option. This rule describes the scenario where the implied volatility of the option is independent of the underlying’s price 0S0. In this scenario, traders assume that when the underlying’s price changes, the new level of 0S0 will become the new at-the-money strike price and all other points within the smile can/will change. However the implied volatility of this strike level will remain the same. This implies that the sensitivity of the price of the option to0S0 is fully determined by its first derivative. Making this assumption

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option’s strike price 0K0, and its time to maturity, 0T0. According to Daglish et al. (2007), the “Sticky Strike Rule” is particularly visible in index options where the implied volatility of this options is less reactive to the changes in index levels, and much less visible in FX volatility smiles. This rule of thumb is tested using the regression equation below, using Ordinary Least Squares (OLS) estimation:

σT K = a0+ a1K + a2K2+ a3(T − t) + a4(T − t)2+ a5K(T − t) + (3.1)

Since this research focuses on the dynamics of the volatility smiles, equation (3.1) can be re-evaluated as follows:

σK= a0+ a1K + a2K2+ (3.2)

Equation (3.2) suggests that for each trading day, there is a volatility smile that captures the relationship between strike prices and implied volatility. To model an aggregate smile, for all dates in the dataset, the stike levels should be adjusted for the underlying’s price, which experi-ences daily changes. Taking into account this effect, the equation transforms into the equation below, by modeling the volatility smile as a function of the moneyness level:

σK t = a0+ a1 Kt St + a2 Kt St 2 + t (3.3)

3.1.2 Sticky Delta Rule

The second rule of thumb called the ”Sticky Delta Rule”, also used for retrieving the market delta and gamma of a traded option, describes the scenario where the implied volatility depends on0S0 and 0K0 through the moneyness measure0K/S0. In this case, the new at-the-money strike price will preserve the implied volatility level of the pre-change at-the-money strike price. In other words, the slope of the volatility smile, remains the same as a response to a change in 0S0. This rule is tested using the following regression equation, estimated using OLS:

σT K= b0+ b1ln K S + b2 ln K S 2 + b3(T − t) + b4(T − t)2+ b5ln K S (T − t) + (3.4)

Similarly to the previous rule testing, equation (3) is readjusted for volatility smiles:

σK t= b0+ b1ln Kt St + b2 ln Kt St 2 + t (3.5)

3.1.3 Square Root of Time Rule

The third rule of thumb introduced by Daglish et al. (2007) describes the relationship between the volatility of options with different strike prices and time to maturity. This rule has a similar construction as the ”Sticky Delta Rule”, by dividing each variable on the right hand side by the square root of the time to maturity, and including additional cubic and fourth power components. Since adding the time component is important in modelling the volatility surfaces, but not for volatility smiles, this rule of thumb is not tested in this research.

3.2 Markov Switching models

Given the fact that financial data is characterized by dramatic breaks, such as financial crisis or market corrections, it is often not enough to remove their non-stationarity by taking the first

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difference of the series. This idea has been extensively studied by Hamilton (1989), who pro-posed a new methodology to describe the consequences of a dramatic change in the behaviour of a single economic or financial variable. The idea behind this model is that there exists a probability law governing the analysis data - such as returns - which is unobservable, but can be distinguished through the available observed series, and can be described as a multi-step Markov Chain process. Hence, modeling high volatility regimes using index level data, can be done by using the Hamilton (1989) methodology as explained in the following.

Kole (2010) provides an example of a simple Markov Switching model with two regimes, that can be described by an equation of index returns, as a function of their mean µS, and an volatility term σS. Note that both mean and volatility depend on the volatility state S. Assuming that the index returns follow a given distribution R, with a known probability distribution function D that depends on the unobserved state/regime St, the returns can be modelled as below:

Rt∼ (

D(µ1, σ12), if St= 1 D(µ2, σ22) if St= 2

(3.6)

Given that the unobserved process St follows a first order Markov Chain, the probability of the index returns being in regime S at time t, only depends on the regime at time t − 1. These probabilities are named as the transition probabilities and can be expressed as:

P [St= 1|St−1= 1] = p11 (3.7)

P [St= 2|St−1= 2] = p22 (3.8)

P [St= 1|St−1= 2] = p12 (3.9)

where p11 + p21 = 1, p12 + p21 = 1 and p21 = p12. These probabilities can be computed by using an initial value ζ = p1 and 1 − ζ = p2. Having modeled the transition probabilities, allows one to make inferences on the current state St, based on current and past observations, combined with the probability distribution of returns. In other words, the probability of index returns being in state/regime 1 based on information gathered at time t = 1, represented by P [S1 = 1|R1 = return1]1, can be fully described by the transition probabilities and return distribution2_{. It is now possible to forecast the probability of being in any of the two regimes} in time t = 2, expressed as P [S2 = 1|R1 = return1], using the inference made about the state in time t = 1. As a recurrent step, inference about the state in t = 2, can be made by making use of the forecast and observations at time t = 2. As shown by Kole (2010), the inference probabilities are constructed as follows:

P [St|returnt, returnt−1, · · · , return1] = ξt|t= 1 ξt|t−1ft

ξt|t−1 ft (3.10)

where ft is the probability density functions of returnt conditional on the regimes/states, and illustrates the element by element multiplication of vectors. The forecast probabilities in this case are simply described as:

ξt|t+1 = P ξt|t (3.11)

1_{The probability of being in state/regime 2 based on information gathered at time t = 1, is expressed as} 1 − P [S1= 1|R1= return1].

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where P stands for the transition probability matrix. This type of recursion where forecasts and inferences about the state, are made incrementally using the information gathered up until the time of inference, is called a filtered recursion.

If all information history is known upfront, another type of recursion can be used, namely the smoother recursion. This type of recursion provides the smoothed inference probabilities, which characterize the regimes/states throughout the entire period:

P [St|returnT, returnT −1, · · · , returnt, · · · , return1] = ξt|T = ξt|t P0(ξt+1|T÷ξt+1|t)

(3.12)

The estimation of the model parameters (µ1, σ1, µ2, σ2, p11, p22, ζ) can be done using the Maxi-mum Likelihood estimation method, specifically by implementing the Expectation Maximization Algorithm3. Both filtered and smoothed inferences are used to construct distinct events of high volatility, and possible deviations from their effects are interesting to understand investor’s an-ticipation of volatility.

3.3 Modeling returns and variance

In this section, the methodology of Lin and Lu (2015) is discussed and is adjusted for the pur-pose of analyzing the effect of the Brexit referendum event, on the predictive power of implied volatility on index returns. Lin and Lu (2015) compare the effect of three similar analyst related events on the the predictive power of implied volatility on average excess returns. The similarity of the events selected in this thesis, is established by the Markov Switching volatility model, which selects time periods characterized by high return volatility. The next step, is to model the independent variables that will be used to measure the relationship between implied volatility and index returns, in an event setting.

The measure of the slope of the volatility smile, can be adjusted to the skew variable used by Xing et al. (2010), which is also implemented in the work of Lin and Lu (2015). This is possible as the volatility smileof the FTSE 100 index is historically skewed to the left. The implied volatility skew is defined as the difference between the implied volatility of out-of-the-money (OTM) put options and the implied volatility of at-the-money (ATM) call options:

skewt= IVtOT M P − IVtAT M C (3.13)

In this thesis, since the options considered correspond to FTSE 100 options, the index call and put implied volatility is equivalent. This is due to the Put Call Parity condition, where for the same underlying at time to maturity, the implied volatility of put and call options must be equivalent. Since all market players assume market efficiency, the Put Call Parity holds. Consequently, equation (3.13) is rewritten as the difference between the 90% to the money and 95% to the money implied volatility. This is to account for the result of Lin and Lu (2015) that the skew calculated using the 100% at the money implied volatility performs worse than by using a moneyness level close enough to the money but not at it. The revised equation of skew is described as follows:

skewF T S E ,t= IVtOT M − IVtAT M (3.14)

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3.3.1 Excess Returns Regression Analysis Smoothed Probabilities

The first set of return analysis is constructed by looking into the relationship between index returns and implied volatility. This relation is captured by the portion of index returns in excess to a benchmark, that is explained by the skew of the index volatility smile. The benchmarks with which the returns are compared to, are the MSCI World (Morgan Stanley Capital International) index returns and the risk free rate based on the GBP (Pound Sterling) LIBOR overnight rate. The MSCI World index returns are constructed by deducting the portion4 _{of the FTSE 100} constituents that are simultaneously part of the MSCI index, and adding the GBP/USD spot rate returns to convert the MSCI returns in the returns that a British investor would obtain. To construct the GBP LIBOR overnight rate, the US LIBOR is retrieved. By using the available GBP Forward points as well as the GBP/USD spot rate, the GBP LIBOR is calculated by re-verse engineering the Forward value equation5. The regression equation is constructed as follows:

exrett+1 = β0+ β1 skewF T S E ,t+ t (3.15)

where exrett+1 is calculated as the log returns of the FTSE 100 index at time t + 1, in excess of the MSCI index returns as well as in excess of the GBP LIBOR overnight rate. The variable skewF T S E ,t is the slope of the index volatility smile/smirk at time t calculated as mentioned in equation (3.14). Visibly the required relationship cannot be captured by one regressor, due to the large possible concentration of explanatory power in the regression residual. Hence a set of control variables, based on Lin and Lu (2015), is added and will serve as a starting point for building a regression equation that will capture the predictive power of implied volatility on index returns for Brexit and other identified events. Equation (3.16) illustrates this idea:

exrett+1= β0+ β1 skewF T S E ,t+ β2 V olumet+ β3 Intraweek V olt+ β4 rett−1 + β5 Hskewm−1+ β6 Reweightt+ t

(3.16)

Equation (3.16) includes five control variables, based on the work of Lin and Lu (2015), as well as from equity option traders. V olumet is a measure of the logarithm of trading volume of FTSE 100 index futures at time t, which is used to capture the market liquidity and is adjusted for seasonality. In absence of a turnover measure for index options (due to their cash settlement when they expire), the trading volume of index futures is a representation of investor’s uncer-tainty about the stock market direction. The second control variable used is a measure of index return volatility, similar to the one introduced by Lin and Lu (2015). IntraweekV olt is calcu-lated as a function6 of the FTSE 100 futures High, Low, Open and Close prices based on the work of Rogers and Satchell (1991) for estimating variance from High Low and Closing prices. Additionally, the rett−1 variable captures the FTSE 100 index returns at t − 1 that describes the index conditions the week prior to the analysis date. The variable Hskewm−1 measures the historical index return skewness up to a month before the analysis date. This measure, introduced by Lin and Lu (2015), is important to isolate the effect of the skewness in the FTSE 100 volatility smile on index returns, without being affected by the skewness of returns. Finally, several traders within the ING Equity Derivatives team, suggested that the “reweighting” days of indices have a significant effect on index returns on the day of the weighting scheme change

4_{Based on https://www.msci.com/world, the weight of UK listed constituents is 6%.} 5

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announcement. Hence, the Reweightt variable, is computed as a binary variable that takes the value 1 every second week of March, June, September and December which are weeks identified as the FTSE 100 reweighting periods7, and 0 otherwise.

To understand the influence of the Brexit referendum as a stand-alone event (without com-paring it to similar events) on the FTSE 100 index returns, the exrett+1variable is regressed on a binary variable that takes the value of 1 for three consecutive weeks, the one before the Brexit referendum, the one of the Brexit referendum, and the one after. In addition, to capture the effect of the Brexit referendum on the predictive power of the skewF T S E ,tvariable on the FTSE 100 excess returns, this equation is extended using an interaction term skewF T S E ,t∗ Brexitt. Both equations described above are illustrated as follows:

exrett+1= β0+ β1 Brexitt+ δ controlst+ t (3.17) exrett+1 = β0+ (β1+ β2 Brexitt) ∗ skewF T S E ,t+ β3 Brexitt+ δ controlst+ t (3.18)

Equation (3.16) captures the informational content of the volatility skew for all sample data. To explain the changes in these dynamics when high volatility events emerge in the market, the Markov Switching model is used to identify several high volatility regions based on index returns. There is an important distinction between the use of the smoothed or filtered proba-bilities to define regions of high volatility in the Markov Switching model. Using the smoothed probabilities, allows one to make inferences on the volatility regimes using all available past in-formation. Hence high volatility regions are on average longer periods than the ones determined using filtered probabilities, because all event occurrences are known in advance. On the other hand, the use of filtered probabilities regimes is more interesting, since the information used to make inferences is the one only known at the point of inference and the past. For this reason, using filtered probabilities better describes the investor uncertainty about the market state.

As done by Lin and Lu (2015), a binary variable All EventsS ,t where S stands for the use of smoothed probabilities, as well as an interaction term with the volatility skew is added to the regression, to study the same relationship when several events in the market are studied. This variable includes all events, hence all periods identified as high volatility regions by the Markov Switching regimes, as explained above. The regression is expanded in this fashion:

exrett+1= β0+(β1+β2All EventsS ,t) ∗skewF T S E ,t+β3All EventsS ,t+δcontrolst∗+t (3.19)

An important observation in (3.19) is the set of control variables controlst is extended to controlst∗. Specifically, the interaction term between the All EventsS ,t variable and Reweightt is introduced, to capture the effect of an event occurring at the same time as a reweighting in the index portfolio. By adding the Reweightt∗ All EventsS ,t control variable, the impact of an event All EventsS ,t week on index excess returns, is then not attributed to any news related to the reweighting of the FTSE 100 index.

Finally, to compare the effect of the Brexit referendum on the predictability of excess returns with other similar events identified by the Markov Switching volatility model, the All EventsS ,t

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variable is segregated, by dividing it into two distinct groups represented by two binary vari-ables: BrexitM S and Other EventsM S where M stands for the period identified by the Markov Switching model and S stands of the period identified using smoothed probabilities. An impor-tant observation here, is that the BrexitM S variable might not correspond to the same number of observations as Brexitt, because BrexitM S is defined as a high volatility period according to the Markov Switching model, which may start or end, before or after the three consecutive weeks used to construct the Brexitt variable. For this reason, no comparison should be drawn between the two.

As an innovation to previous literature, a smoothing effect has been added to the binary event variables, to exponentially distribute weights to the observations closer to the events. The regression equation capturing this effect is constructed as follows:

exrett+1= β0+ (β1+ β2 e−λ|(t−t0)|BrexitM S ,t+ β3 e−λ(t ∗_−t 0∗)_{∗ OtherEvents} M S ,t) ∗ skewF T SE,t + β4e−λ|(t−t0)|BrexitM S ,t+ β5e−λ(t ∗_−t 0∗)_{∗ OtherEvents} M S ,t+ δcontrolst∗+ t (3.20)

where λ consists of the smoothing parameter which is chosen in such a way, that its half-life, which describes the time taken to exactly half the variable effect8, corresponds to half of the average length of the high volatility periods identified by the Markov Switching model. Note that after constructing the Markov Switching high volatility periods, the smoothing parameter λ chosen is 0.9, since its halflife matches the half of the average length of the high volatility period, namely 6.57 weeks.

Additionally, t0 marks the 23rd of June 2016 and t describes all other weeks that are part of the BrexitM S high volatility region. The absolute difference between t and t0 multiplied by the decay factor, describes the exponentially increasing importance of observations before t0, as well as the exponentially decreasing importance of observations after t0. As for the smoothing factor of the variable Other EventsM S, the t0∗ variable marks the start date of every high volatility region identified by the Markov Switching model, with the exception of the BrexitM S region. The variable t∗ captures all other high volatility region weeks, after the t0∗ starting dates. The reason why there is no particular date used as t0∗, is that such events were unexpected (financial crisis, Greek sovereign debt crisis, VIX corrections) unlike the Brexit referendum, whose date was determined several weeks prior to the event. For the same reason, days prior to the high volatility OtherEventsM S are given 0 weight, hence no absolute difference is required for this decay factor.

Filtered Probabilities

For completion purposes, the same regression equations are ran for the Markov Switching regimes identified using filtered probabilities, and hence a subscript of F is assigned to the binary vari-ables describing the Brexit or all other events. Equations (3.19) and (3.20) can be reformulated as described below:

exrett+1= β0+(β1+β2All EventsF ,t) ∗skewF T S E ,t+β3All EventsF ,t+δcontrolst∗+t (3.21)

8_{Defined as t}

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exrett+1= β0+ (β1+ β2 e−λ|(t−t0)|BrexitM F ,t+ β3 e−λ(t ∗_−t 0∗)_{∗ Other Events} M F ,t) ∗ skewF T S E ,t + β4e−λ|(t−t0)|BrexitM F ,t+ β5e−λ(t ∗_−t 0∗)_{∗ Other Events} M F ,t+ δcontrolst∗+ t (3.22)

Within equation (3.20) and (3.22), the coefficients of interest are β2 and β3, since they capture the interactive predictive power of the slope of the index volatility smile on index re-turns, for the Brexit referendum event compared to the same relationship in other similar events.

3.3.2 Realized Volatility Regression Analysis

Having modeled the index returns, the same methodology can be applied to the index volatility RV, which captures the degree of realization of implied volatility in the market in subsequent periods. This variable is calculated as the realized volatility of log returns one week after the week of the analysis and its relationship with the index volatility smile slope is described as follows:

RVt+1 = β0+ β1 skewF T S E ,t+ t (3.23) RVt+1= β0+β1skewF T S E ,t+β2V olumet+β3rett−1+β4Hskewm−1+β5Reweightt+t (3.24)

which captures the informational flow of implied volatility in realized volatility in the entire sample. It is important to note that the Intraweek V oltvariable introduced in the return anal-ysis is excluded from the set of control variables used for the realized volatility regressions. This choice is made due to its high correlation of with the dependent variable RVt+1, which both measure the volatility of index returns, one through futures trading levels and the other directly.

The Brexit referendum’s impact in realized volatility as well as its impact in the predictive power of skewF T S E ,t on realized volatility after the week of the analysis, is captured by the following equations:

RVt+1= β0+ β1 Brexitt+ δ controlst+ t (3.25)

RVt+1= β0+ (β1+ β2 Brexitt) ∗ skewF T S E ,t+ δ controlst+ t (3.26)

Similarly to the index excess return analysis, the Brexitt variable is defined as a binary variable that is 1 for three consecutive weeks before, at, and after the Brexit referedum on the 23rd of june 2016, while the rest of the weeks obtain the value 0.

Reciprocating the analysis on index returns on the subsequent realized volatility of the FTSE 100 index, the Markov Switching identified high volatility regions, are used to evaluate the im-pact of high volatility events on the predictive power of volatility smirk skew, on the following weeks index realized volatility. Equation (3.27) displays this idea, again by using the controlst∗ set of control variables:

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RVt+1= β0+ (β1+ β2All EventsS ,t) ∗ skewF T S E ,t+ β3All EventsS ,t+ δcontrolst∗+ t (3.27)

Most importantly, equation (3.28) illustrates the exponentially smoothed impact of the Brexit referendum, as well as other events identified by the Markov Switching volatility model. Simi-larly to equation (3.20), coefficients β2 and β3, depict the predictive power of implied volatility on index returns after the Brexit referendum compared to other events:

RVt+1 = β0+ (β1+ β2 e−λ|(t−t0)|BrexitM S ,t+ β3 e−λ(t ∗_−t 0∗)_{∗ Other Events} M S ,t) ∗ skewF T SE ,t + β4e−λ|(t−t0)|BrexitM S ,t+ β5e−λ(t ∗_−t 0∗)_{∗ Other Events} M S ,t+ δcontrolst∗+ t (3.28)

Finally, equations (3.27) and (3.28), can be reevaluated using the filtered probabilities, as represented below:

RVt+1= β0+ (β1+ β2All EventsF ,t) ∗ skewF T S E ,t+ β3All EventsF ,t+ δcontrolst∗+ t (3.29)

RVt+1 = β0+ (β1+ β2 e−λ|(t−t0)|BrexitM F ,t+ β3 e−λ(t ∗_−t 0∗)_{∗ Other Events} M F ,t) ∗ skewF T S E ,t + β4e−λ|(t−t0)|BrexitM F ,t+ β5e−λ(t ∗_−t 0∗)_{∗ Other Events} M F ,t + δcontrolst∗+ t (3.30)

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Data and descriptive statistics

The data employed in this empirical research are weekly data from 1st of January 2010 through 29th of March 2018. Both FTSE 100 index weekly returns and implied volatility levels are retrieved from Bloomberg, in cooperation with ING Bank N.V. All data manipulation such as constructing the volatility smile, estimating the Markov Switching model, estimation of returns and variance are conducted in Rstudio.

First, the data to construct the volatility smile models is retrieved. The most liquid traded options, the FTSE 100 options closest to the money below and above by 10%, are used because they maximize the number of observations. The implied volatility of these options is given in the UKX BVOL index, where the implied volatility measure is equivalent for put and call options of the same maturity and same exercise price. The maturity selected for the index options in this study is 1 month, due to the fact that the informational content of implied volatility is found stronger in short maturity options1. Additionally, the price of the index (coded as UKX in Bloomberg) is retrieved for the same dates.

Second, attention is given to the strike (exercise) price retrieval. The strike price is implied by the index price, implied volatility and a given delta. More specifically, the strike price 0K0 is retrieved by reverse engineering the Black and Scholes (1973) specification for the Greek letter delta2, defined as:

K = S0

exp(Φ−1_{(delta ∗ exp(T )) ∗ IV ∗}√_{T − (r} f+ IV

2

2 ) ∗ T )

(4.1)

where S0is the FTSE 100 index level, Φ−1represents the inverse normal probability distribu-tion funcdistribu-tion, T is the time to maturity (in this case 1 month), rf is the GBP LIBOR overnight rate as a proxy for the risk free rate, IV is the implied volatility for a given moneyness, and delta is a fixed value of the option’s delta, that depends on the moneyness level of the given implied volatility3. This means that the 90% ATM implied volatility is used to solve equation (4.1) for a delta value of 0.9, the 95% ATM implied volatility is used with a delta value of 0.75, the 100 % ATM implied volatility is used with a delta of 0.5, 105% ATM implied volatility is used with a delta of 0.25, and finally the 110% ATM implied volatility is used to solve equation (4.1) with a delta of 0.1. For simplicity purposes, it is assumed that the FTSE 100 does not distribute dividends (dividend yield = q) in this analysis period.

Table 4.1 below provides a summary of the data. There are overall 427 weekly observations and 11 important variables in the dataset. It is immediately visible that out of the money options, the IV 90% ATM and IV 95% ATM variables, have a significantly higher mean compared to the rest of the implied volatility levels in the data. This is an indication that the implied volatility

1 Giot (2005) 2_{∆ = exp(−q ∗ T )Φ(d} 1) where d1= ln(S/K)+(r−q+σ2 2 )∗T σ √ T 3

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is skewed to the left, and suggests that a volatility smirk can be constructed with the gathered data. Moreover, there is large variation between the minimum and maximum of the FTSE 100 related variables. These statistics suggest that different cycles exist or that regimes in the market, cause the index level as well the implied volatility to vary significantly over time.

Table 4.1: Descriptive Statistics

This table presents descriptive statistics of the variables retrieved for the construction of the FTSE 100 volatility smile. The IV variables are measured in percentages, and the Strike Level variables are constructed as indicated in equation (4.1). The measures displayed are N the number of observations, the mean, standard deviation, minimum, 25th Percentile, 75th Percentile and maximum.

Statistic N Mean St. Dev. Min Pctl(25) Pctl(75) Max Index Level 427 6,345.348 663.192 4,805.750 5,822.075 6,820.465 7,778.640 IV 90% ATM 427 20.838 4.628 12.080 17.390 23.235 37.930 IV 95 % ATM 427 18.934 5.246 11.690 15.210 21.060 40.180 IV ATM 427 14.601 5.195 7.190 10.865 17.010 35.440 IV 105% ATM 427 13.043 4.072 7.550 10.100 14.660 30.970 IV 110% ATM 427 14.049 3.216 7.550 11.920 15.275 28.280 Strike 90% ATM 427 5,603.526 710.229 3,923.595 5,069.682 6,125.223 7,135.696 Strike 95% ATM 427 6,048.742 689.717 4,440.746 5,525.705 6,543.136 7,542.534 Strike ATM 427 6,322.534 664.634 4,778.123 5,802.648 6,796.754 7,763.003 Strike 105% ATM 427 6,497.702 642.071 5,022.625 5,976.088 6,953.549 7,902.383 Strike 110% ATM 427 6,680.075 645.608 5,235.884 6,143.225 7,115.824 8,162.046

4.1 FTSE 100 Index and Implied Volatility

The dynamics of implied volatility with respect to the option’s underlying is fundamental to this thesis. The long studied relationship between the stock and option markets becomes visible in Figure 4.1 where the FTSE 100 index level is plotted against the at-the-money (ATM) implied volatility of FTSE 100 options. Since the ATM options are commonly the most liquid compared to the other moneyness level options, according to Derman (1999), their implied volatility is a great measure for the forthcoming volatility in the market. It is evident in 4.1, that there is a negative correlation between the FTSE 100 index levels and the ATM implied volatility. Additionally, a short lag exists between the decrease in the index level and the increase in the ATM implied volatility (and vice versa), clearly visible in periods such as mid-2010 as well as mid-2017. This confirms the notion that a lead-lag relationship exists between the two variables.

Furthermore, in figure 4.1, it is visible that there are different regimes in the movements of implied volatility. Apart from the negative skew that is present in the time series, there are periods with higher or lower implied volatility than others, for a corresponding drop or increase in the FTSE 100 levels. A similar analysis is conducted for in-the-money as well as out-of-the-money options, shown in Figure 4.2, where the negative skew between the implied volatility levels and the FTSE 100 levels is present, and the out-of-the-money implied volatility levels are the highest.

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Figure 4.1: FTSE 100 Index Level and ATM Implied Volatility

This figure represents the FTSE 100 index level (the upper line), and the at-the-money FTSE 100 index options’ implied volatility levels (the lower dashed line) from January 2010 to March 2018, with 1 month maturities, for both call and put options. The negative correlation between the two variables is a distinctive feature of this figure.

Figure 4.2: FTSE 100 Index Level and Implied Volatility

This figure represents the FTSE 100 index level (the upper line), and the at-the-money(dashed), out-of-the-money(bold) and in-the-money(solid) FTSE 100 index options’ implied volatility levels from January 2010 to March 2018, with 1 month maturities, for both call and put options. The negative correlation between the two variables is a distinctive feature of this figure.

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4.2 Volatility Smile

The Sticky Strike and Sticky Delta models introduced in Section 3.1 are fitted to the available data and the best performing rule of thumb is used as the leading model. To construct a re-gression using the methodology developed by Daglish et al. (2007), manipulation of the data is required. The weekly observations are stacked for the five different moneyness/delta levels (producing 427*5 = 2135 observations), to provide an aggregated table of four variables: date, index level, strike price and implied volatility. Additionally, the strike level is scaled to the one-thousandth point to match the implied volatility measure. The Sticky Strike and Sticky Delta rules in equations (3.2) and (3.5) are then tested on this data, and the regression results are presented in Table 4.2 below.

Table 4.2: Fitting the Volatility Smile: Sticky Strike and Sticky Delta Rule

This table outputs the regression results of the Sticky Strike and Sticky Delta Rule. The dependent variable is the implied volatility of at-the-money, in- and out-of-the-money FTSE 100 options. The regression in column 1 corresponds to the Sticky Strike Rule. The regressors for this regression are the strike and strike squared levels. The strike level is divided by 1000 to match the implied volatility scale. The second column corresponds to the regression output of the Sticky Delta Rule. The regressors in this column are the log moneyness and log moneyness squared levels. As indicated in the notes, three stars indicate the significance level with 99% probability, two stars with 95 % and one star with 90%.

Dependent variable: Implied Volatility (1) (2) Strike Level −25.742∗∗∗ (1.082) Strike Squared 1.621∗∗∗ (0.087)

Log Moneyness Level −17.657∗∗∗ (2.252) Log Moneyness Squared 355.437∗∗∗

(20.051) Constant 112.805∗∗∗ 14.293∗∗∗ (3.319) (0.109) Observations 2,135 2,135 R2 0.701 0.424 Adjusted R2 _0.701 _0.423 Residual Std. Error (df = 2132) 2.985 4.143 F Statistic (df = 2; 2132) 2,497.708∗∗∗ 783.340∗∗∗ Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

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The output in the first column of table 4.2 shows that the volatility smile constructed with the Sticky Strike specification is depicted by a downwards sloping curve, due to the negative coefficient of Strike Level and positive coefficient of Strike Squared. All coefficients character-izing the smile are significant, and there is a 70% goodness of fit of the model to the available data. The results in column 2 show that when the Sticky Delta rule is fitted on the data, the Log Moneyness Level is also negative and is accompanied by a positive Log Moneyness Squared coefficient. Hence, the fitted Sticky Delta rule also points toward a downward sloping volatility smile, however a lower goodness of fit (42%) is observed. From the regressions in table 4.2, it can be concluded that the equation that governs the volatility smile best, follows a Sticky Strike rule. To illustrate the shape of the volatility smile, the fitted equations in Table 4.2 are plotted, by showing the implied volatility and the (log)moneyness levels. It is important to note that instead of using the strike prices on the horizontal axis, the moneyness level is used for the Sticky Strike rule, in order to construct an aggregated volatility smile independent of the changes to the index level.

Figure 4.3 illustrates the volatility smile of the FTSE 100 index options by fitting the Sticky Strike rule. It is clear that the leverage effect is present in the data due to the higher implied volatility levels for out-of-the-money levels represented by 90 % and 95 % of at the moneyness. This effect is confirmed by the data since the confidence bounds are wider for these moneyness levels, compared to the 105 and 110 % of the moneyness presented further right on the money-ness axis. The wider bounds indicate that the data points for these moneymoney-ness level are more dispersed. As a consequence, one can refer to this relationship as the volatility smirk of FTSE 100 options. In the appendix, figure 8.1 fits the Sticky Delta rule from the output of column 2 in table 4.2. A similar shape of the FTSE 100 volatility smirk can be observed when this rule is used.

Figure 4.3: Aggregated FTSE 100 Volatility Smirk using the Sticky Strike Rule

This figure represents the aggregated volatility smile using the Sticky Strike Rule of FTSE 100 options from January 2010 to March 2018, with 1 month maturity, for both call and put options. The grey line overlaying the smile, represents the 90 % confidence interval of the fitted line.

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The volatility smile constructed in figure 4.3 does not allow one to make inferences on the effect of a particular event in the data, such as the Brexit referendum. Therefore, the sample is divided in two parts, for days prior to and after the Brexit referendum. The Sticky Strike relationship between the implied volatility and strike levels indicated by Daglish et al. (2007) is tested for both subsamples and the regression outputs are presented in Table 4.3. A similar regression output can be found in table 8.1 in the appendix for the Sticky Delta rule.

Table 4.3: Fitting the Volatility Smile Pre and Post Brexit using the Sticky Strike Rule

The table shows the regression output of the Sticky Strike rule fitted in the two subsets before and after the Brexit referendum. The first subsample includes dates after the 1st of January 2014 until the 15th of June 2016. The second subsample includes all dates after and including the 16th of June 2016.

Dependent variable: Implied Volatility (1) (2) Strike Level −26.691∗∗∗ −43.434∗∗∗ (3.495) (3.825) Strike Squared 1.472∗∗∗ 2.680∗∗∗ (0.279) (0.276) Intercept 125.728∗∗∗ 185.757∗∗∗ (10.911) (13.201) Observations 635 465 R2 0.768 0.711 Adjusted R2 0.767 0.710 Residual Std. Error 2.308 (df = 632) 2.128 (df = 462) F Statistic 1,045.294∗∗∗ (df = 2; 632) 568.578∗∗∗ (df = 2; 462) Note: ∗p<0.1;∗∗p<0.05;∗∗∗p<0.01

As indicated in Table 4.3, the number of observations clearly diminishes in the post-Brexit subsample (635 observations in the pre-Brexit subsample, compared to 465 for post-Brexit), since this is a very recent event. However, the Sticky Strike model is present in both periods, since all three regressed variables are significant with 99 % probability and there is a goodness of fit measure of 70% for both regressions. The biggest difference observed is that the post-Brexit subsample seems to indicate a steeper smirk, due to a more negative Strike Level coefficient and a deeper curvature due to the higher Strike Squared coefficient. Moreover, the constant (intercept) increases in the post-Brexit period, indicating that there is an additional shift in the volatility smirk compared to the period prior to the Brexit referendum.

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Similarly to aggregated FTSE 100 volatility smirk, the visualization of the smirk is also presented for subsamples, to create a better picture of how the shape of the volatility smirk has changed in the post-Brexit data sample. Figure 4.4 plots the two smirks. It is visible that in the post-Brexit period, for the most out of the money strike levels, the same strike price leads to a higher implied volatility. This result illustrates that the skewF T S E ,t defined as the difference between 90% and 95 % at the money implied volatility, is lower in the pre-period compared to the after Brexit period. Additionally, the pre-Brexit smirk is flatter than the post-Brexit smirk, a feature that becomes detectable when analyzing closer to the money strike levels. This outcome presents evidence that the volatility smirk of the FTSE 100 index was flatter prior the Brexit referendum event. The change in the volatility smile slope (measure by the skew variable) is in line with the expectation that the downside risk of this high volatility event was incorporated in the volatility smile of the index, and it serves as a foundation of the analysis in this thesis.

Figure 4.4: FTSE 100 Volatility Smirk Pre and Post Brexit using Sticky Strike

This figure represents the aggregated volatility smile of FTSE 100 options for the pre-Brexit period (1st of January 2014 until 15th of June 2016), and for the post-Brexit period ( 15th of June 2016 onwards), with 1 month maturities, for both call and put options. The grey line overlaying the smile, represents the 90 % confidence interval of the fitted line.

Do unusual events affect the predictive power of implied volatility? : the case of the "Brexit" referendum

MSc. Quantitative Finance

Master Thesis