• No results found

The stability of implied volatility for Brexit and the US election in 2016 : a study based on call options

N/A
N/A
Protected

Academic year: 2021

Share "The stability of implied volatility for Brexit and the US election in 2016 : a study based on call options"

Copied!
37
0
0

Bezig met laden.... (Bekijk nu de volledige tekst)

Hele tekst

(1)

1

T

HE STABILITY OF IMPLIED VOLATILITY FOR

B

REXIT AND THE

US

ELECTION IN

2016:

A

STUDY BASED ON CALL

-

OPTIONS

.

Henk Gouws August 15 2018

A

BSTRACT

Given the options pricing framework provided by Black-Scholes 1973, this paper focuses on volatility, and implied volatility, as these are the only inputs not directly observable by market participants. Analysing the months in which Brexit and the US election occurred, this paper tests whether there are structural changes in the implied volatility regimes over the months in which these events occur, as well as identifying factors which affect implied volatility. The outcome of the study finds that there is indeed a structural change in the implied volatility regime in the month following these events, and intuitively a change in the shape of the volatility smile. The factors used to predict implied volatility (level of the S&P 500 and VIX indices, put-call ratio and whether a market event has occurred) prove inconclusive and requires further study.

(2)

2

1. I

NTRODUCTION

The world of option pricing has largely developed over the last 50 years. A brief look at the history of options, completed by Poitras (2008) finds that options contracts or similar contracts with embedded option features and gambles have been in use since Aristotlean and Biblical times. Tracing the exact history of options is rather indeterminate. Formalised options and forward trading becoming prevalent in 1531 on the newly opened Antwerp exchange, and an equivalent exchange was established in London in 1571 (de Roover 1949). During the 17th and

18th century forward contracts and options displayed many essential characteristics of modern

day derivative markets, with the put-call parity already being understood. The use of options were mainly for managing risk (Vallelado 1992), and it wasn’t until 1973, approximately 50 years ago, when two professors Black and Scholes developed a closed form formula to calculate option values. In the same year the Chicago Board of Options Exchange (CBOE) also started its activities. It was in 1973 that the first 16 formalised options contracts were exchanged. In recent years the value of options traded has increased from $12.13bln in 2013 to $14.84bln in 2017.1 Today, most exchanges and a collection of external data providers provide the market

with a wide array of historical options data, including all fields which could reasonably be expected to be required by analysts.

For the study we look at two recent events, Brexit and the 2016 US election, which, subject to certain subjective views, resulted in an outcome that was not expected by the market. The paper analyses the impact of the outcome of two market events on the implied volatility of options traded on the S&P 500: Brexit which occurred on 23 June 2016 , where the expected outcome was widely regarded that the UK will remain part of the European Union (EU). The second focus for data collection will look at the outcome of the US election which occurred on 8 December 2016 , where Donald Trump, representing the GOP, was seen as an unlikely victor of the US election and managed to clinch a victory. Both events occurred in 2016 and the short term market effects were easily noticeable on a global scale.

Given the availability and accessibility to a majority of option pricing data and the ability to computationally apply these data on the larger, accessible datasets we have seen the adoption of the Black-Scholes model (BS), introduced in 1973, as an industry standard with various other models aiming to solve the unknown volatility metric.

Initially the adoption of a uniform approach to options pricing by using the closed form BS formula allowed the market to understand the all the underlying components. This leads one to

1 Number of futures and options traded worldwide from 2013 to 2017 (in billions)

(3)

3

constructively analyse the importance and effects of index and option pricing volatility, and the implied volatility used in the calculation of option prices.

The adoption of this method and the subsequent newer methods have inspired this paper to investigate whether unexpected outcomes of market events affects the stability, or structure of implied volatility on S&P options. This paper applies the sticky strike rule, along with an f-test as prescribed by Chow (1960) to conduct a test of the stability of implied volatility given these uncertain market events.

To investigate the research question of this thesis, data from the S&P 500 index and VIX index for 2016 are used. The options data for the underlying S&P 500 index were collected from the CBOE. The methodology is discussed in section three and the regression models are specified. Section 4 describes the data and section 5 discusses the results. Section 6 discusses a robustness test applied to put options, and section 7 concludes.

(4)

4

2. L

ITERATURE REVIEW

The pricing of European style options has fundamentally changed after the introduction of the BS formula and the accompanying assumptions. The most marked assumptions of the BS model are the assumptions of constant volatility and a lognormal distribution. Even though volatility can be near-constant in the very short term, it is never constant in the longer term. It is due to this specific assumption that the implied volatility of an option can be determined: it is the value of volatility which solves the closed form equation. The second assumption assumes log normally distributed returns on the underlying. This is an acceptable assumption on real world stock prices, even though it does not always hold, and generally, stock returns tend to be distributed around a mean. Various studies have tested this: Fama (1992) concluded that returns are non-normal members of the stable class of distributions. Blume (2008) subsequently tested the monthly residuals gathered from the market model – with results consistent to those of Fama. Teichmüller (1971) also completed a similar study prior to Fama and Blume, and his findings were line, concluding that the distribution is a non-normal member of the stable class with fatter tails. Even though these assumptions are seemingly violated, they are used in a market where participants agree, and if all key inputs are known (strike price, stock price, risk-free rate, time-to-maturity, dividend yield and volatility) an option can be priced with relative ease.

The implied volatility is generally accepted as a measure which can be indicative of the future return volatility over the remaining life of the option. An additional assumption of the BS model is efficient markets, and if this holds, implied volatility should contain all the information in all other variables which would be reasonably expected to explain future volatility. Various studies have been concluded by amongst others Jorion (1995), Day and Lewis (1992) and Lamoureux and Lastrapes (1993). Christensen and Prabhala (1997) further tested this on a wider sample and found that implied volatility is a less biased forecast of future volatility than previous studies. Due to the larger dataset incorporated they also manage to show that volatility is more biased before a market crash than after due to “a poor signal-to-noise ratio prevalent prior to the crash, and perhaps also due to learning by market participants in the wake of the crash.” (Christensen and Prabhala (1997); p.127) This is an important finding for this paper as the stability of implied volatility will be tested in a later section. Christensen and Prabhala (1997) also find that past volatility has much less explanatory power than previously reported with almost no additional explanatory power over implied volatility.

(5)

5

Macbeth and Merville (1979) challenged the Black (1975)2 assumption where he states that

deep out the money (in the money) options have BS model prices which are less (greater) than the prevailing stock price, and find the opposite. They also found that different options’ market prices, given the same underlying have varying implied volatilities over time. This is to be expected as the change in volatility in the underlying should be reflected by a change in the price provided by the BS model: The BS model is a function that provides option prices based on volatility. If the option prices are the missing variable that is provided, it is possible to reverse engineer volatility, giving us the implied volatility.

Dupire (2004) discusses that if all maturities and all strikes that make up the price of a European call option are known, a risk neutral process for the spot can be attained. This results in a BS model that has the characteristics of a one-factor model that can be used to provide clarity on all European options. It also allows the observation for which volatility to use to price various options. Researchers have over time tried to enhance the BS model to reflect this property where volatility is used to price an option, and computed a theoretical smile. The researchers have (un)knowingly added additional non-visible sources of risk such as stochastic volatilities or transaction costs which makes the BS model lose its completeness. These techniques fall outside the scope of this paper, but the questions which in turn created the need for such a model helps the reader to better understand the dependency of implied volatility on the strike price. Is it possible to build a spot process which keeps the BS model complete and is compatible with these volatility smiles at the given maturities? And if so, do the implied volatilities change when there are structural changes in implied volatility regimes.

2.1 V

OLATILITY SMILES3

Modelling volatility smiles is the preferred method of assessing whether there is any form of stability visible in the implied volatility as the difference between two points on the curve can be differenced. In order to understand how this shape can provide us with additional insights on how the implied volatility is affected by external market events the paper discusses first how the volatility smile is shaped and thereafter, in section 5, if the structure of the shape is affected by market events.

2 Fischer Black. "Fact and Fantasy in the Use of Options." Financial Analysts Journal, (July- August 1975). 3 The volatility smile is a graph shape that can be viewed by plotting the implied volatility and strike prices of options. It generally results in a skewed, downward sloping “smile” and allows the reader to gain insight into how volatility differs for options with different maturities. This is further discussed in section 3.1.

(6)

6

A wide array of research has been completed on volatility smiles as volatility smiles allow traders to infer various option characteristics (price, greeks, moneyness etc. depending on the x-axis of the smile, discussed below), and academics have used the smile to assess anomalies in the theoretical pricing model presented by BS. The shape of the curve is easy to interpret as it provides the implied volatilities (y-axis) for options with the same time- to-expiration (x-axis). In this paper the x-axis is specified as moneyness, which also forms a smile shape. Intuitively the at-the-money options will have the lowest implied volatility as these options require less volatility to move into the money. The same holds for out-the-money options: the deeper they move out of the money the higher degree of deviations, or implied volatility they require to move back into the money. The lognormal assumption of the BS model allows the reader to understand why the shape of the smile is curved. If all options had constant volatility the shape of the smile should be flat, but due to the higher required volatility one should always anticipate a smile or a smirk. The relevance of this shape will allow the paper to discuss whether this shape supports a stable relationship between implied volatility and the known strike prices. Even though various research reports has been concluded on the shape of a smile, very few papers have measured the extent of the deviations of the volatility smile.

Dennis and Mayhew (2000) examine the violations of the constant volatility assumption in the BS model by building a regression model for the slope of the curve. They apply the model to equities options from 1986 – 1996. The period of the data was, at the time, one of the most comprehensive tests to investigate the importance of different factors which explain the volatility smile for options traded on the CBOE. Three possibilities are examined:

1. Stocks do follow the constant variance assumptions, but cannot follow a dynamic replicating strategy when trading costs are taken into account. Supply/demand imbalances may explain the digression from the BS model.

2. As leverage is added to firms’ balance sheets, constant variance which may be exhibited in asset values may not be reflected in the stock price.

3. A firm’s assets’ value does not follow an underlying process that has constant variance.

Using various works from Macbeth and Merville (1979) and Rubenstein (1985), which show that different options on the same underlying have different market prices, Dennis and Mayhew run their model for individual stocks, and compare the results to equivalent S&P500 index options. This is relevant to this paper as it adds more colour to the characteristics which are to be expected from index options. Their findings indicate that the slope of the volatility smile is somewhat negative, and inclines less gradually than that of the S&P500. They also find that

(7)

7

market risk could be an explanation for deviations of smiles from expected by the BS-model. Another important finding is that the put/call volume ratio is significant in explaining the inclination of the S&P500 smile.

2.2 I

MPLIED VOLATILITY AND MARKET EVENTS

Looking back at the original assumptions of the BS-model it could be implied that the option implied volatility should be an unbiased estimator of future return volatility in the underlying asset. From the discussion above it is clear that this does not always hold, and the point of entry for the study undertaken in this paper is to see whether stability within the implied volatility of call options with the S&P500 as underlying can provide the reader with insight into the behaviour of the implied volatilities, or smiles when market events occur.

Lin and Lu (2015) studied how the option traders- and analyst expected -outcomes transfer information between option- and stock prices. They test whether the analyst initiation of coverage, recommendation changes or forecast revisions (“tipping”), option traders’ provision of info to analysts (“reverse tipping”) and common information which is a combination of the two prior situations plus the private information available to the options trader could impact the expectation of stock price returns. Their findings indicate that the “tipping” hypothesis is the most consistent theory as options traders tend to take positions based on the market reports of analysts, and the impact the analyst recommendations have on the release date of the information. This is an example of an event with which the market can react. Generally, analyst recommendations can beat, meet or miss, with all outcomes having a different impact on the move of the share price, as well as the implied volatility and value of options on an underlying. For the purpose of this study, this indicates that expectations in the market, based on a consensus driven view by the market can impact the variability of excess returns.

Nikkinen and Sahlström (2001) examine the uncertainty in stock markets surrounding the release of scheduled macro-economic announcements. Macro-economic announcements create also creates the base for the study in this paper as the release of macro-economic data by countries across the world are well-scheduled and documented. Market consensus is also a key driver behind subsequent market moves, as the outcome of a GDP growth rate, as an example, can beat, meet or miss the consensus market view. As per the analyst expectations this measure of uncertainty will influence the levels of volatility and implied volatility in the market. From their study, Nikkinen and Sahlström (2001) examine the uncertainty in the U.S and Finnish markets around scheduled macro-economic news releases. Specifically focused on the producer price index (PPI) and the Consumer Price index (CPI), they measure whether news releases in the U.S have an impact on foreign exchanges, secondly they contribute to past literature by

(8)

8

Patell and Wolfson (1981) and Donders and Vorst (1996) that explains a significant rise in implied volatility before an earnings call, with a subsequent significant drop after announcement has occurred. Lastly they extend research on investigating the implied volatility on announcement days, and the days before and after the announcement. Donders and Vorst (1996) and Edrington and Lee (1996) were able to determine that there are indeed increases in implied volatility prior to announcements, but they could not determine an exact day the increase happens. On this Nikkinen and Sahlström (2001) explores the option that the implied volatility increase is not a gradual process. Using data from the Helsinki stock exchange, and the VIX as a proxy or market implied volatility, they find that implied volatility drops after economic releases. Using the announcement days as dummy variables in a regression formula they also conclude that implied volatility jumps a day prior to the release. Ultimately effects of these announcements are dissected and worked through in a day. This study allows this paper to see that it is not only earnings announcements but also macro-economic events which affect the implied volatility of options on underlying indices.

It is important for the reader to remember that the prior events that have been mentioned in this section, and all the prior research that has been done on earnings announcements, macro-economic releases and implied volatility presents a scenario where the impact is interpreted based on a three tier outcome: Beat, meet or missed. As part of this study, where the outcomes of (arguably) macro-economic events are interpreted, these events can be seen as binary in a sense. For example, the Brexit referendum provided a platform for British citizens to decide whether they would like to remain a member state of the European Union. The outcome of this vote was based on a majority of more than 50%. There is no beat or meet or miss, but rather result which can only have two predetermined outcomes, is widely documented and can occur with an equally likely outcome. The same is true for a presidential election. Based on this it is important to understand how news can influence implied volatility, as the change in expectation can influence implied volatility.

For the purpose of this paper, there is no additional research completed on the impact that news releases would have on the outcome of the events that are studied, and it does not form part of the model that is used to determine the stability in the implied volatility of S&P 500 options. The outcomes of events will be used as per Nikkinen and Sahlström (2001), that the occurance of such an event will be a dummy variable within a regression model. The focus remains on the impact that these binary outcomes have on the stability of implied volatility on the S&P 500 index.

(9)

9

2.3 I

MPLIED VOLATILITY AND THE

S&P

500

Up to this point the discussion has been around the general measures of implied volatility and how it is impacted by market events. This can be further applied to options specifically trading on the S&P 500. There is an abundance of research concluded on options trading on this index. Based on our discussions above it is clear that not only the events surrounding international markets, but also the theoretical inputs present by Black and Scholes can both have an indiscernible and a visibly impactful effect on the implied volatility of options.

The first study viewed is by Shu and Zhang (2003) where they analyse the relationship between the realized and implied volatility of daily S&P 500 index options over a four year period (1995 – 1999). They test four different estimators of implied and realised volatility. This includes the standard deviation of return, the square root of intraday returns (Andersen, 2000), Yhang and Zhang (2000) estimator and the extreme volatility value estimator of Parkinson (1980). Secondly they test the error arising from model specification based on a comparison of the BS-model against the calibrated Heston (1993) stochastic option-pricing BS-model. They find that improved estimation of the realised volatility can significantly boost the forecast ability of implied volatility. The difference in error measurement between the calibrated Heston- and the BS-models does not significantly improve the forecasted realised volatility by using each model’s implied volatility measure. One key finding is that, when implied and historical volatilities are used to predict realised volatility, the implied volatility measure is a better measure than historical volatility. The model used to determine these results assists this paper in specifying its model.

Dumas et al. (1998) run empirical tests on S&P 500 options and implied volatility from June 1988 through December 1993. They make use of European style options and a deterministic volatility function option valuation model (DVF) which is based on the hypotheses by Derman and Kani (1994), Rubenstein (1994) and Dupire (1994). The latter three argued that return volatility is a deterministic function of price and time. This model has the potential of fitting observed cross sections of option prices exactly. The result of Dumas et al. finds that the DVF has no additional hedging or predictive performance to a procedure which that smooths the BS-Model implied volatilities. This is excellent guidance towards the stability of realised and implied volatility as the BS-model can, with basic adjustments of implied volatility prove useful at future guidance. In the methodology below we discuss the implications of this.

Lastly in the academic review, we discuss the volatility surface (volatility smile with time as a third dimension) of the S&P 500 options. Skiadopoulos (2000) discusses the dynamics of the S&P 500 implied volatility surface, whereas Gonçalves (2006) measure the predictive dynamics.

(10)

10

The former addresses shocks that can move the implied volatility smiles and surfaces using Principal Component Analysis whereas the latter models the surface along cross-sectional moneyness and maturity dimensions.

Skiadopoulos (2000) argues that current models developed by Hull and White 1987; Scott 1987; Wiggins 1987 that provide for stochastic volatility , or jump process as presented by Bates 1988; Merton 1976 do not fit the observed implied volatility patterns well. They explore how many variables are needed to explain the dynamics of a volatility surface (smile) and if these factors are correlated to the underlying price. They find that two factors explain approximately 78% of the variation in volatility smiles on the S&P 500. Building out from prior research by Dumas et al. (1998) they find that deterministic volatility models are not capable of explaining the dynamics of the volatility surface. It is for this reason that this paper keeps its focus on volatility smiles. Furthermore, their results contrast with Kamal and Derman (1997) who identified three principal components that explain 95% of the variance of a volatility surface for options traded on the S&P 500 and Nikkei 225. Even though the components differ slightly and Skiadopoulos only identifies two components, the study by Kamal and Derman also identify term structure of volatilities and the level of volatilities as two of the three key components. Expanding on the predictive dynamics of the volatility surface, Gonçalves (2006) observes that S&P 500 option prices describe non-constant surfaces of implied volatility, yet they find that a two stage model consisting of cross-sectional implied volatility variation and estimating parametric VAR-type models produces a high quality fit of the values over time.

2.4 A

DDING UPON EXISTING LITERATURE

From the discussion above it becomes clear that there is no certainty surrounding the stability or predictability of the implied volatility smile over time. Prior research has identified a myriad of factors which could possibly be used to model implied volatility or prove that it is stable over time. This paper aims to build on these models in 4 ways:

1. Adding shorter term observations as the event is approached, this paper will measure the stability of the implied volatility smile in the days leading up to the event and the subsequent cool-down period. By analysing the monthly values of options, with a specific focus on the two months in which the particular events occur.

2. Additionally this paper identifies interaction terms which could assist in explaining the levels of implied volatility. This is based on a normal ordinary least squares regression.

(11)

11

3. In order to do this the volatility smiles before and after events are based on the sticky strike model, and a chow test in order to see if there are structural changes in the implied volatility regime.

4. After sampling and testing the data, the months of the Brexit referendum and US elections are analysed and the findings are concluded. The key tests hope to find that there are structural changes which occur in the volatility smiles, which price options before and after the market events. If there are structural changes, it could be implied that options, which are purchased before market events, will have significant price changes after the events occur, and could turn out to be mispriced even if the outcome of the event is not known. This will be a great finding for traders and hedgers, as known structural changes in the implied volatility regimes can further help the decision of hedging or speculating on market events.

(12)

12

3. M

ETHODOLOGY

The goal of this research is to assess whether the political economic events which occurred in 2016, namely Brexit and the US presidential election, of which both events’ outcomes were against the expectation of the world consensus, changed the dynamics and stability of the implied volatility structure. Firstly the methods of getting the data into a usable format are discussed. Secondly, the modelling of returns and variance of implied volatility will be discussed, and lastly a discussion of interactive factors which could explain the changes in implied volatility are discussed, and a regression model is specified. The main study will only apply these changes to call-options, with robustness checks completed for put options.

It is important to note that the methods that will be used in this paper have not specifically been tested in any prior research that was addressed in the literature review, but there are various references to prior literature that will be made in the discussion below. An attempt is made however, to create a new methodology based on the factors or methods used in prior research.

3.1 I

MPLIED VOLATILITY STRUCTURE

Constructing volatility smiles which can be interpreted as being representative of the current level of implied volatility for different strike prices is the first step in creating models which can explain the movement of implied volatility to any variable.

In order to create consistent volatility smiles over any time horizon, given various strikes and implied volatilities. It is therefore important to see what is used in practice and what can be used as a rule of thumb4. Daglish et al. (2007) empirically test the “Sticky Strike and Sticky Delta

rules” which are used in practice. For the purpose of this paper the sticky strike rule is used to create our volatility smiles.

3.1.1 S

TICKY

S

TRIKE RULE

This rule states that implied volatility (σK, T) is a function of only K (strike) and T (time), and is a

plausible model in the exchange traded market. Generally in this market the exchange shows the options that trade, and traders will anchor their calculations on a volatility measure they first choose for any of these options. It brings more clarity to the argument by Macbeth (1979) that is discussed above, as it can be assumed that a fixed implied volatility can now be used for the calculation of daily implied volatility used for the volatility smile.

The model to determine σK, T is given by:

4 The “Rule of thumb” is exactly how Daglish et al. explain it in the paper entitled Volatility surfaces:

(13)

13

( ) ( ) ( ) ( ) (4.1) Seeing as the paper interprets the daily movement of implied volatility, it implies that (T-t) should equal 0 for all daily observations, and the correctly specified form for daily observations can be seen below in equation 4.2:

( ) (4.2)

Additionally the volatility smile will be presented as a function of implied volatility on the y-axis and moneyness (this will then include all strikes as a %; K/S) on the x-axis. In order to calculate the moneyness of an option using equation 4.2, K is simply divided by the current share price visible on the given observation.

( ) [ ] (4.3)

3.1.2 T

HE

C

HOW TEST

Chow (1960) tests the equality between sets of linear regression by comparing the coefficients. These comparisons can determine whether there are structural breaks in the data when there is a sudden change in the relationship being examined. This study follows a variation of the normal f-test with a restriction. The test is run in the following steps:

3.1.2.1 Run the regressions as specified in equation 4.3 for each month, January through December.

3.1.2.2 Then data for two months are appended (i.e January and February) and the same regression is run for this data. This provides us with a restricted model. 3.1.2.3 The SSE for two months individually, is added together and forms the

unrestricted SS (URSS; sum of pre- and post-structural-break months) where the appended (combined) data forms the restricted SS.

3.1.2.4 The following formula is used to calculate the f-test, and the data is measured against the f-table:

(4.4)

RSS

break

post

RSS

RSS

break

pre

RSS

RSS

combined

RSS

k

n

RSS

RSS

k

RSS

RSS

RSS

F

c c

_

_

_

2

/

/

)

(

2 1 2 1 2 1

(14)

14

3.2 I

MPLIED

V

OLATILITY AND EVENTS

3.2.1 B

ASIC RETURN OVERVIEW

From the above literature review various factors come have been shown to potentially influence the implied volatility. For all return data, the natural logarithm of the index will be differenced to indicate the daily percentage move:

(4.5)

Furthermore the impact of events will be added into the regression equations as a dummy variable, taking a value of 1 or 0. The value of one will be included in the formula for one week leading up to the event and one week after and will be specified as:

( ) {

3.2.2 R

EGRESSION INPUTS AND MODEL

To determine which market factors influence the implied volatility over time and when events occur, three regression models are generated and tested for robustness.

Firstly it is important to test whether there is some form of stability between the level of the index and the implied volatility of the options traded on this index. According the research completed by Christensen and Prabhala (1997) they proposed the following two models to determine whether implied volatility has any effect on the realised volatility:

(4.6)

These models provided the key results in their study and were tested again Nillson (2008) where data on the S&P 500 and the OMSX30 provide results implied volatility contains some information about the realised volatility. These two methods focus on the realised volatility rather than the implied volatility that this paper focuses on.

As the first equation in the analysis of implied volatility the realised volatility is changed to implied volatility (IVt). The implied volatility is calculated by reverse engineering the BS model

and this is regressed against the presence of market events (ME):

(15)

15

This paper adjusts the above regression equation to regress implied volatility (IVt) against

various factors that were identified in the literature review. From the literature review the prescribed inputs the following factors are deemed as important for the purpose of this study:

1. The level of the S&P and the level of the VIX indices.

2. The current and historical implied volatility and realised volatility of the options and the indices.

3. The inclusion a specific market event (ME). 4. Options volume traded and put/call ratios

From these new factors the second regression equation is specified.

( ) (4.8)

Were Index is calculated as the natural logarithm of the level of the S&P500, Change is calculated as the weekly percentage change in the VIX index. Market events remain binary and assume a value of one or zero as specified above. The skew is included as specified in equation 4.5 and the

Put-Call ratio is the daily ratio of puts and calls traded over the duration of the study.

Using the selected variables which are identified in equation (4.8) it is possible to see whether the market events will have a significant effect on the implied volatility.

(16)

16

4. D

ATA AND DESCRIPTIVE STATISTICS

The data employed in this research study is daily data from the 1st of January 2016 to the 31st

December 2016. Both the S&P 500 and the VIX index data are retrieved from YAHOO finance. The options data are collected from the Chicago board of options exchange (CBOE). The CBOE provides large arrays of data and the data retrieved for this research are very granular, providing the daily price, bid-ask, implied volatility, implied underlying price, volume, greeks and some additional data, for all options traded on the S&P 500 underlying There are approximately seven- to ten thousand (monthly and weekly expiration)5 options traded daily on

the CBOE and pricing, greek and implied volatility is given daily. The key data analytics tool is Microsoft Excel, and the data is first broken up into daily data, with the applicable data sorted into monthly data, including all strikes, trading dates, expiry dates and implied volatilities and formulas required for regression.

For the purpose of sticky strike study, only call options within a moneyness range of 0.9 and 1,1 are included, but for all maturities. Giot (2005) has shown that the informational content on implied volatility is higher for shorter maturity options. For this reason the regression data as specified by equation 4.8 is run on short term options (28 – 32 days to expiration) and longer term options (58-62 days to expiration). The moneyness of these options also plays an important role, as closer to the money options are more liquid than and contain the bulk of the observations. For this reason, the paper uses options trading in the range of 10% OTM to 10% ITM. Due to the quality of CBOE data there is no need to calculate implied volatility.

The main focus of the paper tests if there is stability in the implied volatility smiles during the months of the Brexit referendum which occurred on 23 June 2016, and the US presidential election where Donald Trump unexpectedly won the US Electoral College on 8 November 2016. Below in Figure 1, the shaded areas in blue indicate the week before and after the events occurred, and both events are marked by a significant drop in the S&P 500, followed by a opposite upward movement. If any instability in the implied volatility of the options is noticeable, it would be possible for an investor to be in a better position to plan their hedging or trading positions in options. A selection of options based which are least affected by the structural change could result in a more stable position if unexpected outcomes occur.

5 New options included in the CBOE daily pricing files include S&P weekly options. These options are relatively new to market and are not included in the scope of the study. Only options with monthly expirations are included.

(17)

17

FIGURE 1: S&P 500 INDEX, 31-12-2015 TO 31-12-2016, WITH BREXIT (23 JUNE 2016) AND THE US ELECTION (8 NOVEMBER 2016) HIGHLIGHTED IN BLUE. BOTH EVENTS ARE HIGHLIGHTED FROM THE WEEK BEFORE TO THE WEEK AFTER THE EVENT OCCURRING.

When analysing the two events, Brexit occurs towards the end of the month of June, with one week left until month end, and three weeks left to the next option expiration date. It is also one week after the prior options expiration and listing date. This provides a good observation period for a shorter term implied volatility analyses, and analysis will be able to determine if there is a structural break in the implied volatility regime. Similarly the US election occurs one day after the first week of November. That provides approximately one week to option expiration for November, and three weeks after the prior expiration and listing date. As with Brexit, the observations would allow for analysis to determine if there is a structural break from one listing date to the next.

Given the above variables and the event dates, it is possible to view the descriptive statistics of the data to obtain some initial insight into the movement of the data. Table 1 below presents the descriptive statistics for the S&P 500 and the VIX based on daily data for 2016. Options data are discussed monthly, with the index provided both monthly (appendix) and annually, and additional care is taken to present the descriptive statistics two months, June and November, in which the events occur. Table 2 gives a summary of key descriptive statistics for all options traded in 2016. Table 3 and 4 discusses the descriptive statistics of options data related to the regression model specified in equation 4.8.

(18)

18

TABLE 1: DESCRIPTIVE STATISTICS FOR THE S&P 500 AND THE VIX INDICES, BASED ON DAILY OBSERVATIONS FOR 2016 TRADING DAYS. SOURCE: YAHOO FINANCE.

From the data above it is possible to see how the two indices acted throughout the year and over the two event months. From the data it is clear that a large range should be expected, as the shape of the graph in Figure 1 indicates a sharp drop in the S&P500 index6, and it climbs to a

new high just before the year end. When the months in which events occur, including the dip in March, the variability of the S&P 500 index becomes rather subdued. Excluding March, June and November, the range of the S&P 500 and VIX index are limited to 76.46 and 4.47 respectively. Comparing this range to the event months, it is possible to see that large dips were experienced as the S&P 500 had a range of 118.58 and 128.17 for June and November respectively. Additionally the standard deviations of the event periods, even though lower in June than November, still exceeds the average non-event months’ standard deviation over the period. This provides an indication of an increase in volatility in the event months and provides a base for the paper to see whether the implied volatility remains stable over the event months.

In table 2 below, the strike prices are divided and described according to levels of moneyness. For ATM options, all options within the range of moneyness -1% - +1% are included. OTM options are shown as all options which fall between 1% and 10% OTM and ITM options as 1% to 10% ITM. All implied volatilities over the period follow a normal volatility smile pattern where the mean of the implied volatility is decreasing as moneyness increases. This holds true across all months in the observation period. In total the monthly observations 6169 call options in July to a maximum of 9128 in August. This is equal to the amount of puts in the dataset. These

6

Descriptive statistics of the S&P 500 and VIX indices

2016 Observations Mean StdDev Min Max Range

S&P 500 252 2094.65 101.43 1829.08 2271.72 442.64

VIX 252 15.83 3.97 11.27 28.14 16.87

Event Months Observations Mean StdDev Min Max Range

June S&P 500 22 2083.89 29.19 2000.54 2119.12 118.58 VIX 22 17.77 3.46 13.47 25.76 12.29 November S&P 500 21 2164.99 40.61 2085.18 2213.35 128.17 VIX 21 15.24 3.28 12.34 22.51 10.17

Average non-event months Observations Mean StdDev Min Max Range

S&P 500 189 2106.65 23.62 2067.58 2144.04 76.46

(19)

19

data forms the basis for the Sticky Strike rule discussed in the methodology section as [ ]and [ ] can easily be extrapolated from the data.

Table 3 and Table 4 below provides an overview of the of the dates for which options traded that expire between 28 and 32 days for the short term observations and between 58 and 62 days for the longer term observations. Generally, both time frames are still considered short term for the purpose of this paper, and coincide with the view of Giot (2005) mentioned earlier. As the observation period is shortened to 5 days, it is possible to collect the data for the newly created monthly options which have expiry dates of 30 and 60 days. The data presents itself generally as described in this section, and the only notable move to note is the extreme change in the VIX index during the months in which events occur, which should imply changes in implied volatility. The longer term options (58-62) generally have less trading data and -days available for options within the moneyness requirement of this study. This is tested in regression model 4.8 and the results are interpreted in the results discussion later in the paper.

(20)

20

TABLE 2: TABLE TWO PROVIDES AN OVERVIEW OF THE DESCRIPTIVE STATISTICS OBTAINED FROM ANALYSING THE DATA PROVIDED BY THE CBOE. THIS DATA ARE USED IN THE STICKY STRIKE REGRESSION AND THE OUTPUT IS USED TO CONDUCT THE CHOW TEST FOR STRUCTURAL BREAKS IN THE DATA. THE STRIKE PRICES VIEWED AND DECRIBED FOR OPTIONS AND THE STOCK PRICES ARE DESCRIBED IN TABLE 1. SOURCE: CBOE, ANALYSIS.

Monthly descriptive statistics - Call options strike prices

MARCH ObservationsMean StdDev Min Max Range JUNE ObservationsMean StdDev Min Max Range SEPTEMBER ObservationsMean StdDev Min Max Range

OTM Options OTM Options OTM Options

5-10% OTM 2126 1643 38 1785 1960 175 5-10% OTM 2169 1690 39 1805 2010 205 5-10% OTM 2137 1758 36 1915 2075 160

1-5% OTM 1778 1964 34 1885 2040 155 1-5% OTM 1813 2022 36 1905 2095 190 1-5% OTM 1796 2092 31 2020 2160 140

Implied Vol 3904 0.20 0.05 0.00 0.71 0.71 Implied Vol 3982 0.21 0.05 0.00 0.44 0.44 Implied Vol 3933 0.19 0.05 0.00 0.61 0.61

ATM Options ATM Options ATM Options

0-1% OTM 425 2013 27 1960 2060 100 0-1% OTM 427 2074 29 1985 2115 130 0-1% OTM 475 2147 20 2105 2185 80

0-1% ITM 424 2035 27 1980 2080 100 0-1% ITM 458 2096 27 2005 2140 135 0-1% ITM 443 2169 20 2125 2205 80

Implied Vol 849 0.16 0.03 0.00 0.23 0.23 Implied Vol 885 0.16 0.03 0.00 0.23 0.23 Implied Vol 918 0.15 0.03 0.00 0.21 0.21

ITM options ITM options ITM options

1-5% ITM 1597 536 36 2000 2165 165 1-5% ITM 1791 494 38 2025 2225 200 1-5% ITM 1819 561 32 2150 2295 145

5-10% ITM 1480 583 40 2080 2250 170 5-10% ITM 1731 555 41 2105 2325 220 5-10% ITM 2095 459 37 2235 2400 165

Implied Vol 3077 0.14 0.05 0.00 0.67 0.67 Implied Vol 3522 0.14 0.05 0.00 0.67 0.67 Implied Vol 3914 0.12 0.05 0.00 0.69 0.69

APRIL ObservationsMean StdDev Min Max Range JULY ObservationsMean StdDev Min Max Range OCTOBER ObservationsMean StdDev Min Max Range

OTM Options OTM Options OTM Options

5-10% OTM 2204 1550 35 1840 1995 155 5-10% OTM 1898 1695 40 1880 2065 185 5-10% OTM 2164 1624 33 1915 2055 140

1-5% OTM 1699 2011 31 1940 2080 140 1-5% OTM 1542 2086 36 1985 2150 165 1-5% OTM 1690 2079 27 2020 2140 120

Implied Vol 3903 0.19 0.05 0.00 0.56 0.56 Implied Vol 3440 0.19 0.08 0.00 1.10 1.10 Implied Vol 3854 0.20 0.06 0.00 0.87 0.87

ATM Options ATM Options ATM Options

0-1% OTM 443 2065 20 2025 2100 75 0-1% OTM 368 2140 26 2070 2175 105 0-1% OTM 466 2133 12 2105 2160 55

0-1% ITM 401 2084 20 2045 2120 75 0-1% ITM 373 2162 27 2090 2195 105 0-1% ITM 417 2156 13 2130 2185 55

Implied Vol 844 0.15 0.03 0.00 0.22 0.22 Implied Vol 741 0.14 0.03 0.00 0.22 0.22 Implied Vol 883 0.15 0.03 0.00 0.20 0.20

ITM options ITM options ITM options

1-5% ITM 1580 579 31 2065 2205 140 1-5% ITM 1504 524 38 2110 2280 170 1-5% ITM 1702 584 27 2150 2270 120

5-10% ITM 1477 566 37 2145 2300 155 5-10% ITM 1484 543 43 2195 2390 195 5-10% ITM 2162 416 34 2235 2380 145

Implied Vol 3057 0.13 0.04 0.00 0.65 0.65 Implied Vol 2988 0.12 0.04 0.00 0.61 0.61 Implied Vol 3864 0.13 0.05 0.00 0.70 0.70

MAY ObservationsMean StdDev Min Max Range AUGUST ObservationsMean StdDev Min Max Range NOVEMBER ObservationsMean StdDev Min Max Range

OTM Options OTM Options OTM Options

5-10% OTM 2058 1638 34 1840 1990 150 5-10% OTM 2333 1754 32 1945 2080 135 5-10% OTM 2157 1737 47 1880 2100 220

1-5% OTM 1685 2001 30 1940 2075 135 1-5% OTM 1938 2112 27 2050 2165 115 1-5% OTM 1782 2102 45 1985 2190 205

Implied Vol 3743 0.20 0.05 0.00 0.67 0.67 Implied Vol 4271 0.18 0.05 0.00 0.69 0.69 Implied Vol 3939 0.20 0.07 0.00 1.06 1.06

ATM Options ATM Options ATM Options

0-1% OTM 375 2056 18 2020 2095 75 0-1% OTM 504 2168 11 2140 2190 50 0-1% OTM 426 2155 40 2065 2210 145

0-1% ITM 386 2076 20 2040 2120 80 0-1% ITM 453 2189 11 2160 2210 50 0-1% ITM 442 2177 40 2085 2235 150

Implied Vol 761 0.15 0.03 0.00 0.22 0.22 Implied Vol 957 0.13 0.03 0.00 0.21 0.21 Implied Vol 868 0.15 0.03 0.00 0.26 0.26

ITM options ITM options ITM options

1-5% ITM 1620 476 31 2065 2200 135 1-5% ITM 1877 582 26 2180 2295 115 1-5% ITM 1728 531 46 2110 2320 210

5-10% ITM 1688 475 33 2145 2300 155 5-10% ITM 2023 490 32 2265 2400 135 5-10% ITM 2022 476 51 2190 2425 235

(21)

21

TABLE 3: PROVIDES THE DESCRIPTIVE STATISTICS OF THE MONTHLY INDEX AND SHORT TERM (28-32 DAYS TO EXPIRATION) OPTIONS DATA USED IN THE

REGRESSION EQUATION 4.8. WEEKLY CHANGE IN THE S&P AND VIX FOR OPTIONS IN EXPIRATION WEEK IS SHOWN. EVENT IS WHETHER THE EVENT HAS OCCURRED IN THE APPLICABLE MONTH, AND THE PUT CALL RATIO IS THE RATIO OF OPEN INTEREST EXPLAINED BY ‘CALLS’ AND ‘PUT’ COLUMNS. SOURCE: CBOE

TABLE 4: PROVIDES THE DESCRIPTIVE STATISTICS OF THE MONTHLY INDEX AND LONGER TERM (58 - 62 DAYS TO EXPIRATION) OPTIONS DATA USED IN THE REGRESSION EQUATION 4.8. WEEKLY CHANGE IN THE S&P AND VIX FOR OPTIONS IN EXPIRATION WEEK IS SHOWN. EVENT IS WHETHER THE EVENT HAS OCCURRED IN THE APPLICABLE MONTH, AND THE PUT CALL RATIO IS THE RATIO OF OPEN INTEREST EXPLAINED BY ‘CALLS’ AND ‘PUT’ COLUMNS. SOURCE: CBOE

Monthly descriptive statistics - Call options

MARCH S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS JUNE S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS SEPTEMBER S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS

18/03/2016 1.35% -16.29% 0 0.7421 501247 371957 15/06/2016 -2.27% 35.80% 1 1.1946 468505 559676 22/09/2016 1.38% -30.46% 0 1.1754 536246 630319 16/03/2016 1.89% -20.17% 0 0.8256 392062 323696 17/06/2016 -1.19% 13.08% 1 1.2026 522527 628414 23/09/2016 1.19% -22.36% 0 1.2384 538113 666402 15/03/2016 1.84% -10.32% 0 0.8216 377182 309893 14/06/2016 -1.76% 37.78% 1 1.1786 445392 524918 21/09/2016 1.74% -31.04% 0 1.2415 526594 653792 14/03/2016 0.89% -2.51% 0 0.8074 362817 292940 13/06/2016 -1.45% 42.94% 1 1.1679 428918 500947 20/09/2016 0.60% -11.44% 0 1.2954 513394 665044 17/03/2016 2.53% -22.31% 0 0.7720 440363 339960 16/06/2016 -1.79% 28.00% 1 1.1710 500490 586079 19/09/2016 -0.93% 2.41% 0 1.2689 514998 653461

APRIL S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS JULY S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS OCTOBER S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS

18/04/2016 2.53% -19.72% 0 0.7991 545348 435786 21/07/2016 0.07% -0.63% 0 0.8260 711011 587322 19/10/2016 0.24% -9.90% 0 1.5556 425593 662041 22/04/2016 0.52% -2.98% 0 0.7810 659006 514667 22/07/2016 0.61% -5.27% 0 0.7427 710885 527957 17/10/2016 -1.73% 19.19% 0 1.6895 403044 680938 21/04/2016 0.42% 1.66% 0 0.7751 651743 505182 19/07/2016 0.54% -12.40% 0 0.7637 676559 516681 20/10/2016 0.41% -19.38% 0 1.5536 448120 696216 20/04/2016 0.95% -4.13% 0 0.7521 600244 451414 20/07/2016 0.95% -10.25% 0 0.6786 681733 462623 21/10/2016 0.38% -18.93% 0 1.5506 462054 716459 19/04/2016 1.88% -11.48% 0 0.7897 565377 446505 18/07/2016 1.38% -8.47% 0 0.6589 671393 442349 18/10/2016 0.13% -0.52% 0 1.5507 421464 653560

MAY S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS AUGUST S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS NOVEMBER S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS

18/05/2016 -0.82% 8.23% 0 1.0164 1185192 1204632 17/08/2016 0.31% 1.16% 0 0.9082 1471052 1336011 16/11/2016 0.63% -4.70% 1 0.9055 1530522 1385859 17/05/2016 -1.80% 13.31% 0 1.0115 1175155 1188728 18/08/2016 0.06% -2.16% 0 0.9177 1494495 1371544 18/11/2016 0.80% -9.78% 1 0.8767 1649293 1446015 20/05/2016 0.28% 1.06% 0 1.0261 1232348 1264561 15/08/2016 0.42% 2.66% 0 0.8701 1432200 1246209 17/11/2016 0.90% -9.90% 1 0.8756 1610245 1409942 19/05/2016 -1.17% 12.51% 0 1.0521 1180234 1241729 19/08/2016 -0.01% -1.83% 0 0.9214 1520548 1401011 14/11/2016 1.52% -25.63% 1 0.9258 1524998 1411855 16/08/2016 -0.16% 8.07% 0 0.9045 1434261 1297345 15/11/2016 1.89% -33.76% 1 0.8921 1515765 1352271

Monthly descriptive statistics - Call options

MARCH S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS JUNE S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS SEPTEMBER S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS

22/03/2016 1.67% -17.26% 0 0.8718 275079 239815 20/06/2016 0.20% -13.24% 1 0.9486 336097 318834 21/09/2016 1.74% -31.04% 0 1.5408 266143 410076 23/03/2016 0.47% -0.33% 0 0.9364 276735 259135 22/06/2016 0.67% 4.99% 1 0.9291 343733 319371 20/09/2016 0.60% -11.44% 0 1.5252 265865 405507 21/03/2016 1.57% -20.46% 0 0.8445 265545 224257 21/06/2016 0.65% -10.37% 1 0.8935 337909 301907 19/09/2016 -0.93% 2.41% 0 1.5376 256427 394275

APRIL S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS JULY S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS OCTOBER S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS

18/04/2016 2.53% -19.72% 0 0.7619 924334 704295 19/07/2016 0.54% -12.40% 0 0.8157 1150493 938493 19/10/2016 0.24% -9.90% 0.9685 1087776 1053487 20/04/2016 0.95% -4.13% 0 0.7739 958971 742155 20/07/2016 0.95% -10.25% 0 0.7714 1166639 899970 17/10/2016 -1.73% 19.19% 0 1.0572 1004716 1062152 19/04/2016 1.88% -11.48% 0 0.7581 943763 715500 18/07/2016 1.38% -8.47% 0 0.7794 1102591 859381 18/10/2016 0.13% -0.52% 0 0.9736 1051554 1023833

MAY S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS AUGUST S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS NOVEMBER S&P Weekly ChangeVIX WeeklyEvent P/C ratio CALLS PUTS

18/05/2016 -0.82% 8.23% 0 1.2292 268146 329595 22/08/2016 -0.34% 3.82% 0 1.2987 254340 330308 23/11/2016 1.27% -9.87% 1 0.8459 482581 408235 17/05/2016 -1.80% 13.31% 0 1.2650 231763 293175 23/08/2016 0.40% -2.08% 0 1.2606 265851 335129 21/11/2016 1.56% -15.35% 1 0.8284 465542 385650 24/08/2016 -0.31% 9.84% 0 1.1914 285163 339742 22/11/2016 1.03% -7.45% 1 0.8284 475158 393644

(22)

22

5. R

ESULTS

5.1 I

MPLIED VOLATILITY STRUCTURE

S

HORT TERM

When calculating the linear approximation that is explained by the Sticky Strike rule, it is possible to view the implied shape of the volatility smiles over the research period. The Sticky Strike equation as described in equation 4.3 is run for each month in and an additional regression is run for two weeks over the Brexit event and two weeks of the Trump election (one week before and one week after), and the results are tabled below. The slope is negative across all months as witnessed by the negative K/S coefficient. These are highly significant as witnessed by the high t-values.

When the model is applied to the market events surrounding the Brexit and US election events, some material changes come to light in Table 5 below. During the Brexit event the slope coefficient reaches a low of -8.9 which is lower than all months excluding January and February. This implies a much flatter volatility smile during the Brexit event which could indicate a cautious, low risk taking environment. The opposite occurs during the US election where the slope coefficient actually reaches a high for the year of -25.46, indicating the steepest smile of the year. This indicates high changes in implied volatility as one moves across the moneyness axis of the volatility smile. This could be indicative of a much higher risk tolerance and an increased expectation of future volatility implied in the options’ prices.

The Chow test that is run on the options data provide an indication of whether there is a structural change in between monthly data and two months put together. The output of the Chow test is presented in Table 6 and includes the unrestricted (individual months) and restricted (combined months) squared errors. The test was run on the whole data set and the results interpreted on a 5% and 1% confidence level. An interesting observation comes out of the Chow test, as structural changes in the volatility smile do not seem to occur in the month leading up to the event and the month in which the event occurs, but rather in the month the event occurs and the month following the event.. As viewed by the data and the result of the F-test, there is no significant, structural change in June, when Brexit occurs, and only at the 5% in November, when the US election occurs. It is indeed noticeable that the month after the event indicates a structural change in the implied volatility regime of the options. This is true on all confidence levels for the US election and at the 5% level for Brexit. 7

7 Starting January and February 2016 there was an unexpected crash in the Chinese equities markets that was a continuation of the Chinese stock market bubble which popped in 2016. Even though this event is not addressed in this study, the reverberations experienced by this unexpected crash is also shown by the Chow test to cause a structural change in the implied volatility regime that existed prior to March 2016.

(23)

23

For an options trader this would be an important finding, as the implied volatility regimes used to price options one month before an event would not be structurally different from those in the month of the event occurring, but if the trader buys options in the month of the event, the implied volatility regimes could be different the following month, influencing future price expectations.

TABLE 5: OVERVIEW OF THE MONTHLY REGRESSIONS RUN FOR THE STICKY STRIKE TEST – RESULTS FOR CALL OPTIONS WITH 60 DAYS OR LESS TO EXPIRATION. (ALL T-STATISTICS SIGNIFICANT AT 1% LEVEL, THUS NOT INDICATED INDIVIDUALLY)

TABLE 6: OVERVIEW OF THE CHOW TEST FOR CALL OPTION DATA, INDICATING STRUCTURAL BREAKS IN THE IMPLED VOLATILITY REGIMES IN THE MONTHS FOLLOWING BIG MARKET EVENTS. COLUMN “JAN-FEB” INDICATES AN URSS SUM JANUARY AND FEBRUARY – AND AN RSS OF JANUARY AND FEBRUART TOGETHER AS ONE DATA SET, AND SO FORTH FOR ALL OTHER MONTHS.

January Coefficients Standard Error t Stat July Coefficients Standard Error t Stat

Intercept 3.6122 0.4000 9.0298 Intercept 9.6982 0.5365 18.0767

K/S -6.1943 0.8021 -7.7223 K/S -18.5176 1.0823 -17.1096

K/S^2 2.7788 0.4010 6.9291 K/S^2 8.9355 0.5446 16.4076

February Coefficients Standard Error t Stat August Coefficients Standard Error t Stat

Intercept 4.2548 0.3275 12.9910 Intercept 11.0224 0.3418 32.2479

K/S -7.4480 0.6566 -11.3431 K/S -21.2568 0.6886 -30.8712

K/S^2 3.3941 0.3282 10.3411 K/S^2 10.3412 0.3460 29.8915

March Coefficients Standard Error t Stat September Coefficients Standard Error t Stat

Intercept 8.3621 0.4048 20.6577 Intercept 7.9869 0.3624 22.0401

K/S -15.9894 0.8147 -19.6256 K/S -15.1965 0.7271 -20.8989

K/S^2 7.7653 0.4090 18.9879 K/S^2 7.3324 0.3638 20.1540

April Coefficients Standard Error t Stat October Coefficients Standard Error t Stat

Intercept 7.9334 0.3823 20.7529 Intercept 9.1895 0.3537 25.9800

K/S -15.2185 0.7733 -19.6792 K/S -17.4841 0.7092 -24.6548

K/S^2 7.4074 0.3903 18.9798 K/S^2 8.4232 0.3545 23.7607

May Coefficients Standard Error t Stat November Coefficients Standard Error t Stat

Intercept 8.0955 0.3620 22.3620 Intercept 11.4936 0.4724 24.3314

K/S -15.3398 0.7289 -21.0459 K/S -22.0321 0.9484 -23.2319

K/S^2 7.3763 0.3660 20.1541 K/S^2 10.6780 0.4748 22.4914

June Coefficients Standard Error t Stat December Coefficients Standard Error t Stat

Intercept 7.6272 0.3915 19.4836 Intercept 7.9166 0.4150 19.0774

K/S -14.3980 0.7895 -18.2380 K/S -15.2228 0.8350 -18.2302

K/S^2 6.9125 0.3971 17.4077 K/S^2 7.4198 0.4190 17.7066

Coefficients Standard Error t Stat Coefficients Standard Error t Stat

Intercept 4.9157 0.5739 8.5661 Intercept 13.2876 0.6382 20.8195

K/S -8.9477 1.1548 -7.7480 K/S -25.4617 1.2799 -19.8931

K/S^2 4.1876 0.5796 7.2254 K/S^2 12.3353 0.6400 19.2735

Regression output: Sticky Strike

F-Test Jan-Feb Feb-Mar Mar-Apr Apr-May May-Jun Jun-Jul Jul-Aug Aug-Sep Sep-Oct Oct-Nov Nov-Dec

RSS (unrestricted) 24.58 26.11 21.78 21.28 25.05 31.96 31.41 27.14 32.65 42.52 42.17

RSS (restricted) 24.66 30.71 22.25 21.64 25.17 32.62 31.60 27.47 32.91 43.26 45.33

r=k (coefficients + constant) 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00

Degrees of Freedom (UR) 6554.00 6821.00 6048.00 6427.00 6763.00 6299.00 6686.00 7318.00 7596.00 7591.00 7159.00

f-statistic 6.37 400.76 *** 43.2 ** 35.72 ** 10.87 43.6 ** 13.42 29.57 ** 20.01 ** 43.9 ** 178.97 ***

Critical 0.05 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50

Critical 0.01 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50

*** indicates significance at the 1% level ** indicates significance at the 5% level

(24)

24

5.2 I

MPLIED VOLATILITY STRUCTURE

L

ONG TERM

When analysing the same results, but for options that have more than 60 days left to expiration, there are clear differences to the shorter term counterparts. The slope coefficient remains negative over the sample but has a much flatter slope than any of the shorter term observations. This makes sense because the implied volatility for the implied volatility for longer term options, and in this case includes 3 year options, should be an average of the volatility experienced over the time to expiration. Loosely speaking this would imply a more gradual and toned down range of volatility which would explain the less steep slope when compared to the shorter term counter parts.

Be this as it may, there are still large spikes in the slope coefficients over the event periods, with the month of June exhibiting the largest coefficient of the data set. Seemingly it would imply that this impact was caused due to Brexit, but on closer inspection the slope coefficient during the two-week Brexit period indicates that the slope of the volatility smile actually declined. What is noticeable from figure 1, and the timing of Brexit happening on the 23d of June, is that volatility only spiked in the final week of June, implying that traders were already pricing in increased volatility and using a steeper implied volatility smile to price longer term options.

The US election exhibits a similar outcome when analysing the month leading up to- and the month after the US election, indicating a similar risk profile for options traded, before, during and after the US election. The reason for this could be that the expectation of a Trump victory was low in terms of market expectation, and traders could have accepted additional risk in order to mitigate the risk of an unexpected outcome. Following a Trump victory, the market would have continued into a risk-on trade due to the capitalist tendencies of Trump and the expectation of higher short term returns. When addressing the question of structural changes during the year for longer term options it becomes clear that the longer term options are more prone to exhibit a structural change in the implied volatility regime. This makes sense, as a longer time to expiration would increase the probability of volatility swings, market level adjustments and a consequential change in options pricing structure. When viewing the results of the Chow test the first noticeable observation is that there are structural changes in the implied volatility regime for the months following the June Brexit and the November election8. The significance of July and December indicates that the structural

change is significant on the 1% level. Additionally the two event months, June and November again show that there is no structural changes in implied volatility regimes leading up to the event, but rather in the month after the event has occurred.

(25)

25

For a trader who trades longer term options this could yet be an indication of a new risk factor to take into account when trading options prior to an event occurring. The structure of the implied volatility regimes significantly change in the months following large market events, holding true even for the event in February not included in this study.

TABLE 7: OVERVIEW OF THE MONTHLY REGRESSIONS RUN FOR THE STICKY STRIKE TEST – RESULTS FOR CALL OPTIONS WITH > 60 DAYS TO EXPIRATION. (ALL T-STATISTICS SIGNIFICANT AT 1% LEVEL, THUS NOT INDICATED INDIVIDUALLY)

January Coefficients Standard Error t Stat July Coefficients Standard Error t Stat

Intercept 0.90904 0.09509 9.55934 Intercept 2.40669 0.12974 18.55079

K/S -0.87622 0.19070 -4.59488 K/S -3.96915 0.26032 -15.24731

K/S^2 0.17678 0.09535 1.85398 K/S^2 1.71760 0.13025 13.18716

February Coefficients Standard Error t Stat August Coefficients Standard Error t Stat

Intercept 1.1707 0.0839 13.9527 Intercept 2.6766 0.1163 23.0223

K/S -1.3950 0.1683 -8.2894 K/S -4.4994 0.2332 -19.2948

K/S^2 0.4396 0.0842 5.2224 K/S^2 1.9702 0.1166 16.8931

March Coefficients Standard Error t Stat September Coefficients Standard Error t Stat

Intercept 1.8544 0.1090 17.0094 Intercept 2.0972 0.1162 18.0463

K/S -2.8374 0.2194 -12.9332 K/S -3.2771 0.2328 -14.0739

K/S^2 1.1559 0.1101 10.4976 K/S^2 1.3385 0.1163 11.5041

April Coefficients Standard Error t Stat October Coefficients Standard Error t Stat

Intercept 1.8256 0.1104 16.5294 Intercept 1.7579 0.1112 15.8070

K/S -2.7850 0.2219 -12.5508 K/S -2.6313 0.2229 -11.8056

K/S^2 1.1233 0.1112 10.1047 K/S^2 1.0349 0.1114 9.2919

May Coefficients Standard Error t Stat November Coefficients Standard Error t Stat

Intercept 1.5966 0.1127 14.1686 Intercept 2.3237 0.1098 21.1679

K/S -2.3054 0.2261 -10.1952 K/S -3.7922 0.2201 -17.2266

K/S^2 0.8762 0.1131 7.7435 K/S^2 1.6256 0.1101 14.7671

June Coefficients Standard Error t Stat December Coefficients Standard Error t Stat

Intercept 3.7957 0.1266 29.9741 Intercept 2.1411 0.0787 27.2032

K/S -6.6564 0.2544 -26.1640 K/S -3.6345 0.1582 -22.9775

K/S^2 3.0209 0.1275 23.6980 K/S^2 1.6313 0.0793 20.5778

Brexit Coefficients Standard Error t Stat Trump Coefficients Standard Error t Stat

Intercept 1.6311 0.1408 11.5861 Intercept 2.0745 0.1335 15.5447

K/S -2.2968 0.2821 -8.1414 K/S -3.2610 0.2677 -12.1834

K/S^2 0.8445 0.1410 5.9899 K/S^2 1.3470 0.1339 10.0633

(26)

26

TABLE 8: OVERVIEW OF THE CHOW TEST FOR CALL OPTION DATA, INDICATING STRUCTURAL BREAKS IN THE IMPLED VOLATILITY REGIMES IN THE MONTHS FOLLOWING BIG MARKET EVENTS. (> 60 DAYS TO EXPIRATION)

5.3 R

EGRESSION EQUATION

S

HORT TERM

(28-32

DAYS

)

The regression equations posed certain problems due to the nature of implied volatility. Given that the implied volatility calculation is reverse engineered from the BS model, providing a closed form formula to solve implied volatility with, the selection of variables used in equation 4.8 could always run a risk of offering no explanatory value to the calculation of implied volatility.

For the regression, the options used for the shortest term observation of 28-32 days, provides the results as shown below in table 9. A discussion will be held on each individual regression coefficient and the subsequent significance of the coefficient with the intercept being discussed last. A t-statistic of 1.96 would allow us to conclude significance at the 5% level.

 S&P 500 weekly change: Generally the size of this coefficient is small when the market has experienced large changes over the past week, and large when there has been extremely small changes. Intuitively this makes sense as larger regression coefficients will make a small change more accentuated in the implied volatility result. The coefficient generally has the largest coefficient due to the size of the moves seen in Table 9. Overall the weekly change in the S&P 500 does not significantly explain implied volatility with the largest (insignificant) t-stat of 1.89 observed in January.

 VIX index change: The VIX index is generally expected to be a proxy to the volatility of the S&P 500 and is indicative of current market volatility. There is also an expectation that this variable would do well to explain implied volatility. Based on the regression result for the shortest term options in the study, the VIX index has only one month with significant explanatory power in the regression output with the largest t-stat observed in January at 1.98. This is a disappointing result as there are no notable events that occur in January. During the event months the VIX coefficient provides no significant result.

 The event: Seeing as the only two months which have events are June and November these months are discussed first. The events unfortunately have coefficients of 0.00 in both months and provide no significance in explaining the Implied volatility experienced during F-Test February March April May June July August September October November December

RSS (unrestricted) 2.74 3.88 4.59 4.39 7.15 7.75 6.07 6.52 5.42 4.88 4.10

RSS (restricted) 2.84 7.85 4.79 4.41 7.29 8.07 6.18 6.76 5.44 4.90 5.34

r=k (coefficients + constant) 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00

Degrees of Freedom (UR) 8880.00 9948.00 9571.00 9174.00 10592.00 10413.00 9593.00 10556.00 9753.00 9551.00 10367.00

f-statistic 104.63 *** 3396.35 *** 135.58 *** 8.83 66.87 ** 142.08 *** 60.79 ** 131.3 *** 13.69 16.45 1050.61 ***

Critical 0.05 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50 19.50

Critical 0.01 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50 99.50

*** indicates significance at the 1% level ** indicates significance at the 5% level

Referenties

GERELATEERDE DOCUMENTEN

We analyze the content of 283 known delisted links, devise data-driven attacks to uncover previously-unknown delisted links, and use Twitter and Google Trends data to

An opportunity exists, and will be shown in this study, to increase the average AFT of the coal fed to the Sasol-Lurgi FBDB gasifiers by adding AFT increasing minerals

The package is primarily intended for use with the aeb mobile package, for format- ting document for the smartphone, but I’ve since developed other applications of a package that

Most similarities between the RiHG and the three foreign tools can be found in the first and second moment of decision about the perpetrator and the violent incident

The proposed simulation algorithm schedules the heat pump (i.e., determines when the heat pump is on or off) whilst taking the uncertain future demand for heat and supply of

To answer the question whether an active portfolio strategy based on the implied volatility spread earns abnormal returns, it is necessary to have an expected return and

30 dependent variable intention to enroll and the mediator variable attitude as well as the extent of knowledge about Brexit, favorite country, visited the UK, and studied in the

This would provide flexibility for the H-gas market, as well as for L-gas if enough quality conversion capacity exists.. o Export/import transportation capacity tends to be held