Arbitrage Opportunities with a Delta-Gamma Neutral Strategy in the Brazilian Options Market

FUNDAÇÃO GETÚLIO VARGAS ESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA

Lucas Duarte Processi

Rio de Janeiro 2017 Lucas Duarte Processi

Arbitrage Opportunities with a Delta-Gamma Neutral Strategy in the Brazilian Options Market

Dissertação para obtenção do grau de mestre apresentada à Escola de Pós- Graduação em Economia

Área de concentração: Finanças

Orientador: André de Castro Silva Co-orientador: João Marco Braga da Cu- nha

Rio de Janeiro 2017

Ficha catalográfica elaborada pela Biblioteca Mario Henrique Simonsen/FGV

Processi, Lucas Duarte Arbitrage opportunities with a delta-gamma neutral strategy in the Brazilian options market / Lucas Duarte Processi. – 2017. 45 f.

Dissertação (mestrado) - Fundação Getulio Vargas, Escola de Pós- Graduação em Economia. Orientador: André de Castro Silva. Coorientador: João Marco Braga da Cunha. . Inclui bibliografia.

1.Finanças. 2. Mercado de opções. 3. Arbitragem. I. Silva, André de Castro. II. Cunha, João Marco Braga da. III. Fundação Getulio Vargas. Escola de Pós-Graduação em Economia. IV. Título.

CDD – 332

Agradecimentos

Ao meu orientador e ao meu co-orientador pelas valiosas contribuições para este trabalho. À minha família pela compreensão e suporte. À minha amada esposa pelo imenso apoio e incentivo nas incontáveis horas de estudo e pesquisa. Abstract

We investigate arbitrage opportunities in the Brazilian options market. Our research consists in backtesting several delta-gamma neutral portfolios of options traded in B3 exchange to assess the possibility of obtaining systematic excess returns. Returns sum up to 400% of the daily interbank rate in Brazil (CDI), a rate viewed as risk-free. However, with bootstrap analysis, we find evidence consistent with the absence of arbitrage opportunities in the Brazilian options market.

Keywords: Options, Arbitrage, Brazilian Option Market, Delta Gamma Neutral Strategy List of Figures

1 Monthly Trade Volume in Brazilian Options Market...... 4 2 Cumulative Percentage of Volume Traded by Maturity...... 5 3 Risk-Free Yield Curves...... 6 4 Brazilian Stock Market Index – IBOVESPA...... 7 5 Physical Volatilies Estimated with Univariate GARCH(1,1) processes...... 9 6 Examples of Volatility Smiles/Skews...... 12

7 Delta-Gamma Neutral Strategy Performance...... 24 8 Histogram of Returns...... 26 9 Rolling Strategy Returns – 1-year Investment...... 26 10 Rolling Strategy Volatility – 1-year Investment...... 27 11 Rolling Sharpe Ratio – 1-year Investment...... 27 12 Rolling Accuracy Ratio – 1-year Investment...... 28 13 Actual Returns vs. Expected Returns...... 29 14 Bootstrap Confidence Intervals for Mean Return...... 29 15 Sensitivity Analysis on Brokerage Fees...... 30

vi List of Tables

1 B3 Stock Type Codes...... 3 2 Traded Options by Europeanness and Type in B3’s database...... 4 3 Data Sources for Calculating Implicit Volatilities...... 11

4 OLS Estimates of Volatility Convergence Models...... 16 5 Wald-Test Statistics – Volatility Convergence Model...... 17

6 Daily Returns Summary Statistics...... 25 7 Returns GARCH(1,1) Model...... 30 8 Arbitrage Determinants Regression...... 32

vii Contents

1 Introduction 1

2 Data and Methodology3 2.1 Data...... 3 2.2 Volatilities...... 6 2.2.1 Physical Volatilities...... 6 2.2.2 Implicit Volatilities...... 8

3 Volatility Convergence Models 13 3.1 Simple Convergence Model...... 13 3.1.1 Model Specification...... 13 3.1.2 Estimation and Results...... 15 3.2 Generalized Convergence Model...... 15 3.2.1 Model Specification...... 15 3.2.2 Estimation and Results...... 17

4 Delta-Gamma Neutral Strategy 18 4.1 Strategy Description...... 18 4.1.1 Delta-Gamma Neutrality...... 19 4.1.2 Optimization Problem...... 20 4.2 Results...... 23 4.3 Discussion...... 28 4.4 Arbitrage Determinants...... 31

5 Conclusion 34

Bibliography 35

viii Chapter 1

Introduction

We investigate arbitrage opportunities in the Brazilian options market. We first test the presence of short-term mean-reversion in implied volatilities of all option stocks traded in B3 - Brazilian Stocks, Options and Futures Exchange - from January/2009 to December/2016. We find that they mean-revert to a long-term level that is a quadratic function of moneyness and maturity. Our research consists in backtesting several delta-gamma neutral portfolios of options traded in B3 exchange to assess the possibility of obtaining systematic excess returns. These portfolios are constructed with the constraint of having the first and second derivatives with respect to the underlying asset equal to zero. This means that they could profit from volatility mean-reversion without being affected by underlying stock short-term movements.

Contradicting the results of Araújo and Ribeiro(2016), we do not find evidence of arbitrage opportunities in the Brazilian options market when transaction costs are considered. We impose brokerage costs of 0.045%, the highest fee charged on intraday option trades, according to B3’s standard brokerage fee tables. Returns sum up to 400% of the daily interbank rate in Brazil (CDI), a rate viewed as risk-free, and the strategy achieves an annualized Sharpe ratio of 0.20. However, with bootstrap analysis, we find evidence consistent with the absence of arbitrage opportunities in the Brazilian options market. This result is robust to both unconditional and GARCH-conditional variance tests. The t-statistics of these two tests are respectively 1.50 and 2.07.

This approach is different from other studies because the analysis is taken on several options on different underlying assets, which gives us the opportunity to investigate factors that influence the magnitude of arbitrage opportunities. After running LASSO OLS penalized linear regressions to estimate the relevance of several candidate factors, we conclude that Europeanness is the most important factor that drive returns in our delta-gamma neutral strategy.

These results are in line with with Fama(1970)’s efficient market hypothesis, which implies

1 that one cannot obtain systematic excess returns by executing an active trading strategy. Even if a trader can access information in a faster or more complete manner, additional returns should not compensate the costs of obtaining that information. Because there is no possibility of using information - even from other markets - more efficiently than the market, there can beno arbitrage opportunities in an efficient market. Since the 70’s, several studies tested the empirical validity of the efficient market hypothesis in many different markets. US options market was analyzed in the seminal worksof Fischer Black (1972) and Chiras and Manaster(1978). Other tests were more recently conducted by Noh et al. (1993) e Harvey and Whaley(1992) and suggested the existence of market inefficiencies in US options market. In particular, Kat(1996), Ibáñez(2009), Goltz and Ni Lai(2009) and Mastinsek (2012) use delta or delta-gamma neutral portfolios to assess empirical properties of returns in options market. In the Brazilian options market, Fuchs(2001) tested market efficiency using a delta-gamma neutral strategy with options on Telemar (a major telecommunications company in Brazil), concluding that the market is unable to value these instruments adequately, making possible to obtain systematic excess returns even when trade costs are considered. More recently, Araújo and Ribeiro(2016) obtained similar conclusions with delta-gamma neutral portfolios with Petrobras’ options using intraday data, with a backtest that performed 1600% of CDI. However, none of these studies conduct a multiple underlying analysis and the timespan of their data is too short to derive general conclusions about the Brazilian options market. Our research intends to derive a broader, yet simpler, view of implied volatilities in the Brazil- ian options market and assess the possibility of exploiting their mean-reversion characteristic to obtain systematic excess returns.

2 Chapter 2

Data and Methodology

2.1 Data

Our primary data source for implementing delta-gamma neutral strategies is B3 daily options database.1 It contains information on all options and futures traded in B3, in particular stock options. We read daily prices and option characteristics such as strike price, type of contract, underlying stock and maturity for stock calls (field CD_BDI = 78) and stock puts (CD_BDI = 82) from january/2009 to december/2016. More than 4 billion Brazilian Reals (BRL) worth of options (1.3 billion US dollars) is traded monthly on average in this market, as we can see in Figure1. Extracting underlying stock exchange codes from database involves combining information contained in two different fields: the first four letters of field NOMRES indicate thecompany that has issued the underlying stock, while the first three characters of field ESPECI indicate the kind of stock on which the option is written, according to the codes specified in Table1. For instance, if NOMRES begins with "PETR" and ESPECI with "ON ", then the underlying stock is Petrobras’ ordinary shares and its code in B3 exchange is "PETR4".

ESPECI Code Stock Type Exchange Code ON Ordinary 3 PN Preferred 4 PNA Preferred A 5 PNB Preferred B 6 UNT Unit 11 CI Index 11

Table 1: B3 Stock Type Codes

To identify whether the options are American – and can be exercised at any date prior to

1 This database is called COTAHIST and can be found at http://www.bmfbovespa.com.br/pt_br/servicos/ market-data/historico/mercado-a-vista/series-historicas/

3 Average 8 6 4 BRL (Billions) 2

2009 2010 2011 2012 2013 2014 2015 2016

Figure 1: Monthly Trade Volume in Brazilian Options Market expiration – or European – i.e., can only be exercised at expiration date – one should look at the fifth character of field NOMRES. Options with an "E" are European, all codes denotean American option. We read a total of more than one million lines from B3’s database, each line representing a different option and day of trade. From Table2 we see that the most frequently traded options in the Brazilian market are American calls (85% of total volume), followed by European puts (11%). Some European calls are traded (4%), but American puts were not traded in the period of analysis.

Europeanness Type Count (%) Volume (R$ million) (%) American Call 648,255 56.3 363,427.7 84.8 European Put 388,520 33.8 47,229.3 11.0 European Call 113,921 9.9 18,113.6 4.2 American Put 0 0.0 0.0 0.0 TOTAL 1,150,696 100.0 428,770.6 100.0

Note: Data range from jan/2009 to dec/2016. Count is the number of distinct options and dates of trade.

Table 2: Traded Options by Europeanness and Type in B3’s database

We calculate option maturities by counting the number of working days between the date of analysis – contained in field DT_PREGAO – and option expiration date – field DATVEN, using the holiday calendar supplied by ANBIMA – the Brazilian financial and capital markets

4 100 90 80 70 60 50 40 30 20 10 0 Cumulative percentage of volume (%)

0 10 20 30 40 50 60 70 80 90 100 (...)

Maturity (business days)

Figure 2: Cumulative Percentage of Volume Traded by Maturity association.2 To convert the measure to annual maturities, we divide the number by 252 working days/year. From Figure2 we can see that the options traded in Brazil are usually short-term options: about 70% of the volume traded in the period was originated from options that expire in less than one month (21 working days). In addition to that, volume-weighted mean maturity is less than 24 working days. Risk free yield curves are obtained by combining B3 futures database with CETIP daily rates on interbank deposits (CDI). We build our yield curves using CDI rates as risk-free rate for one-day maturity, and CDI futures as rates for longer maturities. We use flat forward method to interpolate inexistent maturities. If we want to calculate the yield rate 푟푡 for maturity 푡, which lies between the two adjacent vertex 푖 – with yield 푟푖 and maturity 푇푖 – and 푖 + 1, then:

(︁ )︁ (︁1 + 푟 )︁ 푡−푇푖 푡 푇푖 푖+1 푇푖+1−푇푖 (1 + 푟푡) = (1 + 푟푖) × (1) 1 + 푟푖 A sample of yield curves are shown in Figure3. We read underlying asset information and capital change events on 120 different stocks from Bloomberg database. These stocks constitute all underlying assets of the options contained in our database. Unadjusted opening and closing prices are used to build the delta-gamma neutral strategy, while quotes adjusted for dividends and capital changes are used to estimate physical

2http://www.anbima.com.br/feriados/feriados.asp

5 13 12 11 10 9 Risk-free rate (%)

dez/2010 8 dez/2012 dez/2014

7 dez/2016

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Maturity (years)

Figure 3: Risk-Free Yield Curves volatilities. We also get Brazilian Stock Market Index (IBOVESPA) daily quotes from the same source. Figure4 shows us that IBOVESPA oscillated around a 56k-points mean in the period of analysis with an average volatility of 29% per year.3

2.2 Volatilities

We call physical volatility (PV) – or current volatility – a measure of the underlying asset volatility obtained from its own dynamics in the real world. On the other hand, we call risk-neutral volatility – or implicit volatility (IV) – a measure of the stock volatility implicitly calculated from observed prices of one or more derivatives – such as options – with the help of risk-neutral pricing theories. In this section we explain different methodologies used to obtain both volatility measures, as they will be the pivotal elements for executing our delta-gamma neutral strategy.

2.2.1 Physical Volatilities

To estimate physical volatilities we use Bloomberg database to get daily closing stock prices adjusted by dividends and capital changes and calculate discrete returns. If 푃푡 is the price of the stock – or index – at time 푡, then its discrete return 푟푡 is defined as:

3Standard deviation of discrete returns, annualized with the square root of 252.

6 Volatility Returns Index (x1,000 pts)

0.2 0.3 0.4 0.5 0.6 -0.05 0.00 0.05 40 50 60 70 2010 iue4 rzla tc aktIdx–IBOVESPA – Index Market Stock Brazilian 4: Figure 2012 Index 2014 2016 Average 7 (푃푡 − 푃푡−1) 푟푡 = (2) 푃푡−1 Our estimate of current physical volatility is an estimate of the standard deviation of the process that generates 푟푡. If 푟¯ is the simple average of returns and 휀푡 is a gaussian unexpected shock at time with time-varying conditional variance 2 then: 푡 휎푡

N 2 (3) 푟푡 =푟 ¯ + 휀푡, 휀푡 ∼ (0, 휎푡 )

The Generalized Autoregressive Conditional Heteroscedasticity – GARCH conditional variance equation can be stated as:

(︀ 2 )︀ 2 (︀ 2 )︀ (4) 휎푡 |Ω푡−1 = 휔 + 훼휀푡−1 + 훽 휎푡−1|Ω푡−2

We estimate the value of 휔, 훼 and 훽 with the maximum-likelihood method, as proposed by Engle(1982), for all 120 stocks in our sample. The term Ω푡−1 means that we only use data available until 푡−1 to estimate the conditional variance at time 푡. Volatilities are then calculated √ with the square root of the variance annualized by a factor 252. Some of them are shown in Figure5.

√︁ √ 2 (5) 휎푡 = 휎푡 × 252

2.2.2 Implicit Volatilities

We obtain implicit volatilities with the help of Black and Scholes(1973) (BS) option pricing model. This model is obtained with the use of non-arbitrage arguments in an efficient market that allows continuous and infinitesimal trading, has no transaction costs and in which risk- free rates and volatilities are known and constant. The stock option traded in this market is European, and the underlying stock has normally distributed log-returns and pays no dividends during the life of the option. Suppose that the underlying stock follows a Geometric Brownian

Motion (GBM) with drift 휇 and variance 휎2:

푑푆 = 휇푆푑푡 + 휎푆푑푍 (6) where 푍 follows Wiener process. This means that 푑푍 is independent and identically distributed, following a normal distribution with mean zero and variance 푑푡. If 푉 (.) is a function that

8 PETR4 VALE5

1.2 ITUB4 BBAS3 1.0 0.8 Volatilities 0.6 0.4 0.2

2010 2012 2014 2016

Figure 5: Physical Volatilies Estimated with Univariate GARCH(1,1) processes

9 evaluates the price of a derivative of the stock that follows6 then, applying Itô’s lemma 4 we get:

(︁휕푉 휕푉 1 휕2푉 )︁ 휕푉 푑푉 = 휇푆 + + 휎2푆2 푑푡 + 휎푆푑푍 (7) 휕푆 휕푡 2 휕푡2 휕푆 If we assemble a portfolio that is short an option and long 휕푉 stocks: Π 휕푆

휕푉 푑Π = −푑푉 + 푑푆 (8) 휕푆

Joining Equations6,7 and8, we can reestate the dynamics of 푑Π:

(︁ 휕푉 1 휕2푉 )︁ 푑Π = − − 휎2푆2 푑푡 (9) 휕푡 2 휕푆2

As now we have eliminated the stochastic term 푑푍, we have – at least for a brief moment – a risk-free portfolio that therefore earns:

푑Π = 푟Π푑푡 (10) where 푟 is the instantaneous risk-free rate. Using9 and 10 we get the Black and Scholes(1973) partial differential equation (PDE):

휕푉 1 휕2푉 휕푉 + 휎2푆2 + 푟푆 − 푟푉 = 0 (11) 휕푡 2 휕푆2 휕푆 Black and Scholes(1973) solved this PDE for European options, developing a closed formula to price them:

[︀ 푟(푇 −푡) ]︀ 푉 (휔, 푆, 푡, 퐾, 푇, 푟, 휎) = 휔 푆푁(휔푑1) − 퐾푒 푁(휔푑2) (12) 푆 휎2 푙푛( 퐾 ) + (푟 + 2 )(푇 − 푡) 푑1 = √ (13) 휎 푇 − 푡 √ 푑2 = 푑1 − 휎 푇 − 푡 (14) where 푆 is the underlying stock price at time 푡, 푇 is the time at which the option expires, 푟 is the continuous risk-free rate and 휎 is the underlying asset volatility. 휔 is a parameter that equals 1 for call options and −1 for puts. 푁(.) is the standard normal cumulative distribution function. If we observe at time 푡 a certain stock option which is traded with a price 푃 *, we can find a volatility 휎* that is implicit in the price of this option, assuming that Black and Scholes(1973)

4Itô’s lemma states that if 푑푆 = 푎(푆,푡)푑푡 + 푏(푆,푡)푑푍 and 퐹 (푆,푡) is a double differentiable scalar function, then (︀ 휕퐹 휕퐹 푏2 휕2퐹 )︀ 휕퐹 , where dZ is a Wiener process. 푑퐹 = 휕푡 + 푎 휕푆 + 2 휕푆2 푑푡 + 푏 휕푆 푑푍

10 Parameter Source Field 휔 COTAHIST CD_BDI 푆 Bloomberg PX_OPEN or PX_LAST1 푡 COTAHIST DT_PREGAO 퐾 COTAHIST PREEXE 푇 COTAHIST DATVEN2 푟 B3 and CETIP —3 푃 * COTAHIST PREABE or PREULT

1 Not adjusted by dividends and capital changes. 2 푇 −푡 in Brazilian working days over 252. 3 Spot and Future CDI with flat forward interpolation.

Table 3: Data Sources for Calculating Implicit Volatilities pricing model correctly prices the derivative. The problem of finding the implicit volatility is that of finding a 휎* such that:

푉 (휔, 푆, 푡, 퐾, 푇, 푟, 휎*) = 푃 * (15)

Being 푉 (.) a strictly increasing function of 휎, it is guaranteed that 15 has a unique solution for positive prices. Because there is no easy way to isolate 휎 in Equation 12, the problem of finding the implicit volatility must be solved numerically. To speed up calculations, aswehave to evaluate a great number of implicit volatilities, we use Newton’s method to find the root of function 푓(휎) shown in Equation 16, with a quadratic convergence rate. Evidently, comparing with Equation 15, the 휎 that satisfies 푓(휎) = 0 is the implicit volatility 휎*.

푓(휎) = 푉 (휔, 푆, 푡, 퐾, 푇, 푟, 휎) − 푃 * (16)

So, we begin the process by choosing an arbitrary guess for 휎푖 and then we iteratively update our guess with the formula shown in Equation 17, until we reach a value 휎푖 that yields a 푓(휎푖) sufficiently close to zero.5

푓(휎 ) 푉 (휔, 푆, 푡, 퐾, 푇, 푟, 휎) − 푃 * 휎 = 휎 + 푖 = 휎 + (17) 푖+1 푖 푓 ′(휎 ) 푖 휕푉 푖 휕휎 Using Equation 17 we calculated implicit volatilities for all opening and closing prices by joining B3 and Bloomberg databases. The sources are listed in Table3. Despite of Black and Scholes(1973) model assumption of constant volatility, when we calculate implicit volatilities using different options of a same underlying, we often find different values for each combination of strike price and maturity, with a characteristic shape that resembles a

5Note that 휕푉 is the derivative called vega, whose closed formula is √︀ , if is the standard 휕휎 휈 = 푆휑(푑1) (푇 − 푡) 휑(.) normal probability density function.

11 (a) PETR4: 2011-fev-14 (b) PETR4: 2012-mar-12

Maturity 5 bd

1.0 23 bd

43 bd 0.30 0.8 0.20 0.6 Volatility Volatility 0.4

0.10 Maturity

0.2 5 bd 24 bd 48 bd 0.0 0.00 20 22 24 26 28 30 32 20 22 24 26

Strike price Strike price

Maturity Maturity 4 bd 16 bd 1.5 24 bd 35 bd

47 bd 0.6 59 bd 1.0 0.4 Volatility Volatility 0.5 0.2 0.0 0.0 16 18 20 22 24 18 20 22 24 26 28

Strike price Strike price

Notes: Vertical and horizontal dotted lines represent, respectively, underlying spot prices and physical volatilities. Maturities in business days. Figure 6: Examples of Volatility Smiles/Skews smile. That is why the lines shown in Figure6 are often called volatility smiles or skews. Vertical and horizontal dotted lines are drawn at, respectively, underlying prices and physical volatilities. We can see that sometimes implicit volatilities are all above physical volatilities – subfigure (a) – and sometimes completely bellow them – subfigure (b). The intermediary cases may depend on maturities – subfigure (c) – or on strike prices – subfigure (d).

12 Chapter 3

Volatility Convergence Models

Our strategy’s primary assumption is that implicit volatilities are governed by short-term mean reverting processes that converge to a function of physical volatilities. It is a simple reasoning that comes from the intuition that whenever implicit volatility is too far from real physical volatility traders will think that the option is being valued too cheap or too expensive. If it is cheap – i.e., implicit volatility is too low – the agents will buy it and its price will rise. Because the option price is a monotonic increasing function of the underlying asset volatility, the price movement will collaborate to reduce the gap between implicit and physical volatilities. In theory, the gaps between implicit volatilities and their long-term level create arbitrage opportunities that quickly disappear as traders exploit them, and that is why we observe mean- reversion as a stylized fact in volatility time series. We specify the details of models that formally test if there is volatility convergence in the Brazilian options market and we present results. We find evidence of mean-reversion in the process that drives implicit volatilities.

3.1 Simple Convergence Model

3.1.1 Model Specification

The simplest mean-reversion model states that implicit volatilities follow a mean reverting autoregressive AR(1) process, such that:

푃 N 2 (18) 휎푡 = 훽0 + 훽1휎푡−1 + 훽2휎 + 휀푡, 휀푡 ∼ (0, 휎푡 )

푃 where 휎푡 is the implicit volatility observed in time 푡, 휎 is the physical volatility and 휀푡 is white 1 noise. If we substract 휎푡−1 from both sides then: 1For more on mean-reversion property of autoregressive models, see Tsay(2010).

13 푃 Δ휎푡 = 휎푡 − 휎푡−1 = 훽0 + (훽1 − 1)휎푡−1 + 훽2휎 + 휀푡 (19) 푃 = 훽0 + 훾휎푡−1 + 훽2휎 + 휀푡

푃 where 훾 = 훽1 − 1. This process mean reverts exactly to physical volatility (휎 ) in the long-run if the following conditions are satisfied:

훽0 = 0 (20)

−1 < 훾 < 0 (21)

훽2 = −훾 (22)

Using these conditions, Equation 19 can be restated as:

푃 Δ휎푡 = 훾(휎푡−1 − 휎 ) + 휀푡 (23)

and we have that 휎푡 mean reverts if 훾 < 0. If implicit volatility 휎푡−1 is higher than physical 푃 푃 volatility 휎 then 훾(휎푡−1 −휎 ) < 0, which tends to bring implicit volatility back to its mean. On 푃 푃 the other hand, if implicit volatility is lower than physical volatility, 휎 then 훾(휎푡−1 − 휎 ) > 0, which creates an upward trend proportional to the difference between current volatility and its long-term level 휎푃 .

Coefficient 훾 drives the rate of convergence: the higher its absolute value, the faster our volatility mean-reverts. Suppose that the difference in parenthesis in Equation 23 is currently of

푃 푑0. As 퐸[휀푡] = 0, we expect that 휎푡 takes a step of size |훾|푑0 in the direction of 휎 , reducing the gap to 푑0 −|훾|푑0 = 푑0(1−|훾|). In the next period, the step size expectation will be |훾|[푑0(1−|훾|)], 2 and the new distance will be 푑0(1 − |훾|) − |훾|푑0(1 − |훾|) = 푑0(1 − |훾|) . If we keep on calculating expected distances, we will notice that the distance 푑ℎ after ℎ steps is governed by:

ℎ 푑ℎ = 푑0(1 − |훾|) (24)

Using 1 we can calculate the half-life for our mean-revert process by isolating : 푑ℎ = 2 푑0 ℎ

1 푙표푔(2) 푑 = 푑 (1 − |훾|)ℎ → ℎ = − (25) 2 0 0 푙표푔(1 − |훾|)

14 3.1.2 Estimation and Results

We estimate the coefficients of Equation 19 with opening and closing implicit volatilities calculated with the steps described in Section 2.2.2, and used underlying physical volatilities obtained as stated in Section 2.2.1, for all options contained in our database, with data ranging from January/2009 to December/2016. The OLS estimates of Equation 19 can be seen on column 2 in Table4. The estimated coefficient for opening implicit volatility, 훾 = −0.476, is negative and significant at a 1% level, meaning that Condition 21 is satisfied and we observe mean-reversion in implicit volatilities. The magnitude of the coefficient found implies a half-life of 1.1 days. In comparison with astandard AR(1) stated in column 1, we note that including physical volatilities adds about 18 percentage points to the explanatory power of the model. However, Conditions 20 and 22 are not separately or jointly satisfied. Wald-test statistics are supplied in Table5. This means that, although we observe mean reversion in implied volatilities, their long-run values are not exactly physical volatilities.

3.2 Generalized Convergence Model

3.2.1 Model Specification

We have empirical and theoretical reasons to believe that factors such as moneyness and maturity also influence implied volatility levels.2 Implied volatilities are a function of option prices, and prices are influenced by supply and demand movements. It is reasonable to assume that supply and demand for a deeply in-the-money option are somehow different from supply and demand for deeply out-of-the-money options. These differences end up resulting in different equilibrium volatilities, which is probably one of the reasons why we observe the volatility smiles shown in Figure6. We can also notice that maturity seems to influence implied volatility levels. To account for these differences, we could generalize the model represented in Equation 23 to allow for convergence to a function of physical volatility, moneyness (푀) and maturity (휏):

(︁ 푃 )︁ Δ휎푡 = 훾 휎푡−1 − 푓(휎 , 푀, 휏) + 휀푡 (26)

The function 푓(.) could take any form of relationship, but we try to capture the possible non-linearities between maturity/moneyness and long-term levels by OLS regressing on multiple Forsythe orthogonal polynomials of these variables.3

2We use the concept of simple moneyness: 푀 = 푆/퐾. 3As suggested by Hayes(1974).

15 Table 4: OLS Estimates of Volatility Convergence Models

Delta Implicit Volatility (IV) is the difference between Black-Scholes implicit volatilities calculated with daily closing (Closing IV) and opening (Opening IV) prices. Physical Volatility (PV) is a GARCH(1,1) volatility calibrated with returns of the underlying stock. Moneyness and maturity terms are orthogonalized. We run these OLS regressions with daily data on all options traded in B3 between january/2006 and december/2016. Column 2 shows that IV’s exhibit mean-reversion, but their long-term levels are not given exactly by PV’s. The half-life implied by this model is 1.03 days. Columns 3-5 shows that adding moneyness and maturity polynomials helps explaining IV long-term levels. Using AIC and BIC criteria we opt to use models with second-order polynomials in moneyness and maturity, like the one in column 4, to predict option closing prices.

Dependent Variable: Delta IV = Closing IV - Opening IV

Δ휎푡 Explanatory Variables (1) (2) (3) (4) (5)

*** *** *** *** *** Opening IV (휎푡−1) −0.187 −0.476 −0.487 −0.490 −0.490 (0.001) (0.001) (0.001) (0.001) (0.001)

PV (휎푃 ) 0.385*** 0.395*** 0.396*** 0.396*** (0.001) (0.001) (0.001) (0.001)

Moneyness 3.864*** 30.181*** −19.394* (0.261) (1.287) (10.320)

Squared Moneyness 38.066*** −74.403*** (1.808) (23.299)

Cubic Moneyness −74.706*** (15.429)

Maturity −3.570*** −4.652*** −4.668*** (0.091) (0.102) (0.102)

Squared Maturity 1.455*** 1.423*** (0.093) (0.094)

Cubic Maturity 0.004 (0.092)

Constant 0.072*** 0.030*** 0.031*** 0.043*** 0.036*** (0.0004) (0.0004) (0.0004) (0.001) (0.002)

Observations 337,818 337,818 337,818 337,818 337,818 R2 0.119 0.298 0.302 0.303 0.303 Adjusted R2 0.119 0.298 0.302 0.303 0.303 Residual Std. Error 0.092 0.082 0.082 0.082 0.082 F Statistic 45,764.530*** 71,698.500*** 36,552.680*** 24,521.960*** 18,395.570***

Note: *p<0.1; **p<0.05; ***p<0.01 The numbers in the middle part of the table are the OLS coefficient estimates; Standard deviations are shown in parenthesis under OLS coefficients; IV: Black-Scholes implicit volatilities using stock options’ opening and closing prices; PV: Physical volatilities estimated with univariate GARCH(1,1) processes on stock returns; Moneyness and Maturity terms are orthogonalized.

16 Hypothesis F-stat Pr(>F)

훽0 = 0 5993.45 0*** 훾 = 0 143370.62 0*** 훾 = 0 and 훾 + 훽2 = 0 71698.50 0*** 훾 + 훽2 = 0 11494.71 0*** 훽0 = 0 and 훾 = 0 and 훾 + 훽2 = 0 48450.08 0*** 푃 Note: OLS Model: Δ휎푡 = 훽0 + 훾휎푡−1 + 훽2휎 + 휀푡 Table 5: Wald-Test Statistics – Volatility Convergence Model

If we have empirical confirmation that this process describes a part of volatility smile intraday movements, we could create a strategy that buys cheap options (whose implicit volatility is sufficiently bellow its long term-level) and sell expensive options, therefore obtaining profits with their convergence.

3.2.2 Estimation and Results

Columns 3 to 5 in Table4 test other forms of reversion to a function of physical volatilities, with long-term levels determined by orthogonal polynomials of first, second and third degrees in moneyness and maturity. Column 3 shows us that including moneyness and maturity adds about 0.4 percentage points to the explanatory power of the model. All coefficients are statistically significant at a 1%level.

However, increasing polynomial degrees fail to increase significantly OLS 푅2’s. We calculate Akaike and Bayesian Information Criterion – AIC and BIC – for various polynomial degrees of moneyness and maturity. Minimum BIC happens at second order polynomial for maturities, suggesting that the fit improvement that comes from adding cubic or higher terms does not compensate additional complexity. In addition to that, cubic maturity is not statistically significant according to column 5 in Table4. On the other hand, the use of cubic terms of moneyness was not rejected by AIC, BIC or t-statistics criteria. However, we decide for simplicity in this case and keep only two degrees in moneyness’ polynomial, in line with the treatment given to maturity. As a result of this analysis, we use models like the one in column 4 – i.e., quadratic in moneyness and maturity – as the regressions that drive our delta-gamma neutral strategy. Finally, convergence rates seem to be consistent with requirements for implementing a daily strategy. The 휎푡−1 coefficient in column 4 is compatible with a half-life of 1.03 days. Thismeans that every day we expect IVs to be almost halfway closer to their long-term levels. However, we do not know yet if that is enough to build a profitable strategy. In order to investigate this problem, we define and backtest such a strategy in the following chapter.

17 Chapter 4

Delta-Gamma Neutral Strategy

Given that we observe mean reversion in volatility smiles, one could create a strategy that gains from volatility convergence. Suppose that the implied volatility of a particular option is below its long-term level, estimated with Equation 26 using second order polynomials of maturity and moneyness. According to the model, given that a lower volatility is associated to a lower price, its current price is lower than its long-term price (cheap option). If an investor buys this option and its volatility increases toward its long-term mean, the price of the derivative also increases and the buyer achieves a profit. On the other hand, if an option’s current implied volatility is above its long-term level (expensive option), one should sell it and profit from the falling prices that follow its downward mean reversion.

The strategy we use is a delta-gamma neutral strategy. This means that the portfolios are created with the constraint of having its first and second derivatives with respect to the underlying stock price – i.e. delta and gamma – equal to zero. Because of this characteristic, we can expect movements in stock prices to have a limited impact on portfolio returns.

In this chapter, we backtest delta-gamma neutral strategies that profit from volatility convergence, achieving returns of 400% of Brazilian interbank rate CDI. Despite of this fact, we find that it is not possible to reject the null hypothesis that excess mean return is greater than zero. Analyzing the factors that drive returns on this strategy, we conclude that Europeanness is the most relevant measures to determine excess returns.

4.1 Strategy Description

We know from Black and Scholes(1973)’s pricing formula (Equation 12) that the risk factors that directly influence options prices are underlying asset spot price, underlying volatility and

18 risk-free rate. Strike prices are contractually defined, and maturity follows a deterministic1 path. Risk-free rates are assumed to be constant throughout the day, so the only unwanted risk factor is the underlying stock spot price. In our quest for gaining from volatility convergence, we cannot be exposed to underlying price movements that could ruin our profits and add unnecessary volatility to our portfolios. We also do not want to cloud our results with profits or losses that come from forces other than volatility mean reversion. Hence, we implement strategies by creating portfolios in which returns are protected against underlying spot price movements.

4.1.1 Delta-Gamma Neutrality

Suppose a portfolio 푃 of 푛 financial contracts that can be priced adequately with 푛 pricing 2 functions 푉푖(푆), 푖 = 1, . . . , 푛, where 푆 is the spot price of a stock. If 푞푖 is the quantity of asset 푖 held, then the value of 푃 is:

푛 ∑︁ 푉 (푆) = 푞푖푉푖(푆) (27) 푖=1 Using Equation 27 and observing that derivatives are linear operators, we can calculate the first and second derivatives of the portfolio value with respect to the stockprice 푆:

푛 푛 ′ ∑︁ ′ ∑︁ (28) Δ푃 = 푉 (푆) = 푞푖푉푖 (푆) = 푞푖Δ푖 푖=1 푖=1 푛 푛 ′′ ∑︁ ′′ ∑︁ (29) Γ푃 = 푉 (푆) = 푞푖푉푖 (푆) = 푞푖Γ푖 푖=1 푖=1 where 푉 ′(푆) is the first derivative of 푉 (푆) with respect to stock price 푆, usually referred to as delta (Δ); and 푉 ′′(푆) is the second derivative of 푉 (푆) with respect to 푆, usually called gamma (Γ). Suppose that stock price moves to 푆 + Δ푆. The new value of 푃 can be approximated by a Taylor expansion of second degree:

1 푉 (푆 + Δ푆) = 푉 (푆) + 푉 ′(푆)(Δ푆) + 푉 ′′(푆)(Δ푆)2 + 푂(Δ푆3) 2 (30) 1 ≈ 푉 (푆) + 푉 ′(푆)(Δ푆) + 푉 ′′(푆)(Δ푆)2 2

1In Equation 12 maturity represents both parameters 푇 and 푡, with (푀 = 푇 − 푡). 2In our portfolios, these financial contracts are options and stocks. Evidently, stocks can be valued withthe simple pricing function 푉푖(푆) = 푆 without loss of generality.

19 where 푂(Δ푆3) is an error term of cubic order. Using Equation 30, if we want to make the portfolio V hedged against price movements of the underlying stock S we must have:

푉 (푆 + Δ푆) ≈ 푉 (푆) (31)

Combining Equations 30 and 31 we get that the conditions to have a hedged portfolio are:

′ 푉 (푆) = Δ푃 = 0 (32)

′′ 푉 (푆) = Γ푃 = 0 (33)

If Condition 32 is satisfied we call the portfolio 푃 a delta neutral portfolio. If Condition 33 is also satisfied we say that the portfolio is delta-gamma neutral, or that it is hedged against stock price movements in first and second derivatives.

Finally, using the definitions stated in Equations 28 and 29 we can say that constructing a delta-gamma neutral portfolio involves finding quantities 푞푖 that satisfy both:

푞1Δ1 + 푞2Δ2 + ... + 푞푛Δ푛 = 0 (34)

푞1Γ1 + 푞2Γ2 + ... + 푞푛Γ푛 = 0 (35)

These two conditions must play a part in the construction of our portfolios, as we see in the next section.

4.1.2 Optimization Problem

We construct different delta-gamma neutral portfolios for each underlying asset, which contains only quantities of call and/or put options and their underlying stock. These quantities are calculated according to a multi-option version of Araújo and Ribeiro(2016)’s methodology.

Suppose that we observe 푛 opening prices for different options of a certain stock. We can then calculate their associated opening implicit volatilities, 휎푡−1. If the stock has a physical volatility of 휎푃 and we believe that a model who belongs to the family of Equation 26 represents the dynamics of implicit volatilities, we can predict closing implied volatility 휎푡 with:

20 퐸(휎푡|Ω푡−1) = 휎푡−1 + 퐸(Δ휎푡|Ω푡−1) (36) (︁ 푃 )︁ = 휎푡−1 + 훾 휎푡−1 − 푓(휎 , 푀, 휏)

If 퐸(Δ휎푡|Ω푡−1) is positive then the option is cheap and we must buy it to profit from arbitrage.

On the other hand, if 퐸(Δ휎푡|Ω푡−1) is negative, we must sell the option because it is considered expensive. In other words, we could define a buy/sell signal 휃 that follows:

⎧ ⎨⎪ 1, if 퐸(Δ휎푡|Ω푡−1) ≥ 0 휃 = (37) ⎩⎪−1, if 퐸(Δ휎푡|Ω푡−1) < 0

We assume that 퐸(푃푡|Ω푡−1) is the expected option price calculated with Black and Scholes 3 (1973) model, using 퐸(휎푡|Ω푡−1) as the underlying stock volatility parameter. Hence, if 푃푡−1 is the option’s opening price we can expect a price movement from 푡 − 1 to 푡:

퐸(Δ푃푡|Ω푡−1) = 퐸(푃푡|Ω푡−1) − 푃푡−1 (38)

To take into account bid-ask spreads in our calculations, we could define a deterministic constant bid-ask spread 푠¯ as the difference between the lowest ask price and the highest bidoffer at any arbitrary time 푡:

퐴푆퐾 퐵퐼퐷 (39) 푠¯ = 푃푡 − 푃푡

Then, an investor who has a position in one option in 푡 − 1 and intends to end it in 푡 would expect to profit:

⎧ ⎨⎪+퐸(Δ푃푡|Ω푡−1) − 푠,¯ if position is long; 훾 = (40) ⎩⎪−퐸(Δ푃푡|Ω푡−1) − 푠,¯ if position is short.

If we observe at the opening of the day at least one cheap and one expensive option of the same underlying 푆, we can construct the maximum profit delta-gamma-cash neutral portfolio that profits from volatility reversion of 푛 options with different opening prices 푝푖, end-of-the-day expected profits 훾푖 and buy/sell signals 휃푖 (푖 = 1, . . . , 푛) by solving the following linear problem:

3 This is, of course, an abuse of notation, as we did not take into account the expected values of other parameters such as time, risk-free rate and underlying stock spot price.

21 maximize 푤1훾1 + ... + 푤푛훾푛 (41) 푤1,...,푤푛,푤푆

subject to 푤푖 sign(휃푖) ≥ 0, ∀푖 ∈ 1, . . . , 푛 (42)

푤1 + ... + 푤푛 + 푤푆 = 1 (43)

푤1푝1 + ... + 푤푛푝푛 + 푤푆푝푆 = 0 (44)

푤1Δ1 + ... + 푤푛Δ푛 + 푤푆 = 0 (45)

푤1Γ1 + ... + 푤푛Γ푛 = 0 (46)

Note that 푤1, . . . , 푤푛 are the weights of the 푛 options in the portfolio. 푤푆 is the weight of the underlying stock, and 푝푆 its opening price. Constraints represented in Equation 42 ensure that we must buy cheap and sell expensive options, and constraint 43 guarantees that options and stock weights sum to one. Constraint 44 imposes cash-neutrality. This means that there is no cash transfer to open a portfolio. Constraint 45 states that the portfolio is delta-neutral, while constraint 46 forces gamma-neutrality (notice that the stock has a delta of one and gamma of zero).

At the end of the day, we calculate portfolio returns with:

푟푝 = 푤1푟1 + ... + 푤푛푟푛 + 푤푆푟푆 (47)

푐푙표푠푒 표푝푒푛 (푃푖 − 푃푖 ) − 푠¯푖 푟푖 = 표푝푒푛 , ∀ 푖 ∈ 1, . . . , 푛, 푆 (48) 푃푖 where 푠¯푖 is the bid-ask spread of option (or stock) 푖. In this analysis, we consider that we raise 1 Brazilian real (BRL) at the beginning of the investment period and maintain it in a risk-free cash account while serving as margin deposits. As our portfolios are all cash-neutral, its returns are already excess returns with respect to the return of a risk-free asset. In other words, if we open 퐾 portfolios at a particular day and each one of them achieve a log-return of 푘 as given by Equation 47, then the total log-return 푟푝 perceived in this day will be:

∑︀퐾 푟푘 푟 = 푟 + 푘=1 푝 (49) (푟푓) 퐾 where 푟(푟푓) is the log-return of the 1 BRL invested in the risk-free asset. The second term means that we leverage our investment in another 1 BRL to create 퐾 cash-neutral portfolios. The sum

22 of all long trades in all 퐾 portfolios is 1 BRL at the beginning of the day. The short trades also sum to the same amount of money.

4.2 Results

We use daily data ranging from January/2009 to December/2016 to evaluate the performance of delta-gamma neutral strategies based on all available underlying assets in B3 database. We process separately each underlying asset, using the methodology of the previous sections. We use a rolling window of 3 months to reestimate the model coefficients daily and calculate expected profits to feed the objective function of the problem in Equations 41-46, with a model like Equation 26 using second-degree polynomials of moneyness and maturity, as stated in Section 3.2.2. To engage a more realistic backtest, we only consider options that had more than 100 trades on the previous day. With this filter we try to avoid simulating a trade with options thatare so illiquid that we would have difficulty to trade them in the real world. To avoid outliers and possible database errors, we exclude options that are deeply in-the-money or out-of-the- money, whose absolute log-moneyness (|푙푛(푆/퐾)|) are greater than 1. We also exclude options with negative bid-ask spreads and options with bid-ask spreads greater than its opening price. Finally, to work around rounding problems, we only consider stocks and options which have opening prices above 1 Brazilian real.4 In our simulations, we include brokerage fees of 0.045% in each trade, the highest fee charged on intraday option trades, according to B3’s standard brokerage fee tables.5 Although these tables are regressive on volume and some of our trades may qualify to be charged smaller fees, we maintained this worst-case scenario to make the analysis both simple and conservative. We find that this strategy succeeds on profiting from volatility reversion, even when bid-ask spreads are considered, as shown in Figure7. Over the period of analysis, this strategy earned more than 400% of CDI, Brazilian interbank daily rate, including transaction costs. Over the same period that was studied in Araújo and Ribeiro(2016), we find that the portfolio returns sum approximately 1300% of CDI.6 This means that, despite methodological differences, our results seem to confirm their findings. Overall, our algorithm created 1118 different portfolios, with thousands of different calls and

4We end up with about 15% of the original data. The most restrictive filter is the one that is applied tothe number of trades. 5 http://www.bmfbovespa.com.br/en_us/services/fee-schedules/listed-equities-and-derivatives/ equities/equities-and-investment-funds-fees/equities-options/ 6Araújo and Ribeiro(2016)’s data ranges from October/2012 to March/2013, over which they found returns of 1600% CDI.

23 ovdwt oiieojciefunction. objective positive a with solved onunderlying movements of theinfluence neutrality. delta-gamma toeliminate imposing by wemanaged stocks that confirms This ). tails 0.001 Fatter ( 8. Figure in level. displayed confidence visually as strategy a market, to stock much lead are the also that in tails over exhibit observed market and stock those asymmetric the than positively of also fatter that are times returns (IBOVESPA) three Strategy stock nearly period. is and same obtained the (CDI) volatility strategy the rate of that interest statistics see summary We short-term reports markets. Brazilian 6 Table with fact, comparison In volatility. in high returns its of because , 90% than was analysis of period the . whole -0.35 over the was investment over ratio risk-free IBOVESPA’s strategy a while the , by of rate 0.20 ratio obtained annual Sharpe 10.6% growth annualized of The compound CAGR period. a a same in against resulting 28% returns, about BBAS3. of positive and (CAGR) obtained BBDC4 portfolios ITUB4, the OGXP3, of PETR4, VALE5, 51% stocks: underlying 6 on puts

8 7 Drawdown Daily Return Cumulative Return cudbe could 41 -46 returns. Equations daily in of problem simulation the historical which a in with times calculated of VaR number the represents portfolios of number The ecoetewno faayi ae nifrainaalblt ftevrosdt sources data various the of availability information on based analysis of window the chose We more of drawdown maximum a exhibits strategy the that see also can we 7, Figure in However,

-1.0 -0.6 -0.2 -0.4 -0.2 0.0 0.2 0 5 10 15 2009-04-03 CDI STRATEGY 8 h eao h taeywt epc otesokmre ne snal zero nearly is index market stock the to respect with strategy the of beta The 2010-04-01 iue7 et-am eta taeyPerformance Strategy Neutral Delta-Gamma 7: Figure Value-at-Risk 2011-04-01 VR hti lotfv ie BVSAsVRata1% VaR IBOVESPA’s times five almost is that (VaR) 2012-04-02 2013-04-01 2014-04-01 2015-04-01 2016-04-01 7 About 24 Strategy CDI IBOVESPA Min. -43.17% p.d. 0.03% p.d. -8.43% p.d. 1st Qu. 0.03% p.d. 0.03% p.d. -0.87% p.d. Median 0.04% p.d. 0.04% p.d. 0.02% p.d. Mean 0.23% p.d. 0.04% p.d. 0.02% p.d. 3rd Qu. 0.05% p.d. 0.05% p.d. 0.90% p.d. Max. 36.52% p.d. 0.05% p.d. 6.39% p.d. Annualized Std. Dev 80.72% p.a. 0.13% p.a. 23.61% p.a. Annualized Sharpe Ratio 0.20 -0.00 -0.35 Skewness 0.05 0.07 0.00 Exc. Kurtosis 11.12 -1.04 1.37 VaR (0.1%) -27.84% p.d. 0.03% p.d. -5.07% p.d. VaR (1%) -15.99% p.d. 0.03% p.d. -3.56% p.d. VaR (5%) -7.44% p.d. 0.03% p.d. -2.39% p.d. VaR (10%) -4.07% p.d. 0.03% p.d. -1.82% p.d.

Table 6: Daily Returns Summary Statistics

used in this study. However, choosing different beginning and ending dates can alter drastically the results that we have found. For instance, Figure9 shows strategy rolling returns, i.e. the annual returns that would be perceived by an investor that executes our strategy for 1 year, ending on each of the dates represented in the horizontal axis. Annualized Returns can be as high as 1200% per year, but also can be lower than stock and risk-free market returns, or even negative depending on the initial date we pick. Even strategy volatility can vary substantially depending on the window we adopt, as pictured in Figure 10, between a peak of 120% per year and a base of 40%. However, we can say that the portfolio volatility is consistently above stock market volatility throughout the entire period.

Rolling Sharpe ratios over 1-year periods are seen in Figure 11. The series lies between Sharpe ratios of almost 12.0 and -0.5. We can conclude that although the stock market provides a lower volatility, investing in this delta-gamma neutral strategy is often preferable to investing in IBOVESPA, from the standpoint of risk and reward – as measured by Sharpe ratio.

We define accuracy ratio as the number of portfolios that earned positive returns overthe total number of portfolios. It is a straightforward measure of how accurate is our model in predicting end-of-day volatilities. Figure 12 shows us the accuracy ratio calculated for rolling windows of 1-year size. Sometimes the model predicts volatilities with an accuracy of almost 65%, but sometimes its performance decreases to levels far below 40%, suggesting that the model’s adequacy should be evaluated periodically if we were to implement this strategy in the real world.

25 Strategy IBOVESPA 30 30 25 25 20 20 15 15 Density Density 10 10 5 5 0 0

-0.2 -0.1 0.0 0.1 0.2 -0.2 -0.1 0.0 0.1 0.2

Returns Returns

Figure 8: Histogram of Returns

Strategy CDI 1200 IBOVESPA Average (CAGR) 800 600 400 Return (% p.a.) 200 0

2011 2012 2013 2014 2015 2016 2017

End of 1-year Investment

Figure 9: Rolling Strategy Returns – 1-year Investment

26 Strategy 1.2 CDI IBOVESPA Averages 1.0 0.8 0.6 0.4 Annualized Std. Dev. 0.2 0.0 2011 2012 2013 2014 2015 2016 2017

End of 1-year Investment

Figure 10: Rolling Strategy Volatility – 1-year Investment

Strategy 12 IBOVESPA Overall 10 8 6 4 Sharpe Ratio 2 0 -2

2011 2012 2013 2014 2015 2016 2017

End of 1-year Investment

Figure 11: Rolling Sharpe Ratio – 1-year Investment

27 65 Average 60 55 50 45 Acuracy Ratio (%) 40

2011 2012 2013 2014 2015 2016 2017

Figure 12: Rolling Accuracy Ratio – 1-year Investment

4.3 Discussion

In fact if we plot earned returns against expected returns – as calculated with Equation 41, we see that they differ significantly, as shown in Figure 13. Coefficients for the linear model represented by the blue line can be seen in Model (4) of Table8. The hypothesis that actual returns equal expected returns in average (OLS slope = 1) is rejected at a 1% confidence level. As great as they might seem at first sight, the returns obtained by the strategy werenot found statistically greater than zero. Bootstrapped confidence intervals for their geometric mean can be seen on Figure 14, and they include zero at 1% and 5% levels.9 The t-statistic for testing mean return’s significance is 1.5, leading to a conclusion that it is not possible to reject thenull hypothesis that mean return is unconditionally greater than zero. If we suppose a conditional variance process we find that conditional mean returns are also not significantly greater than zero at 1% level. In Table7, we report the results of a process to test if mean return 휇 is significant subject to a GARCH(1,1) conditional variance. The t-statistic of 2.07 does not allow us to reject the null hypothesis that returns are conditionally equal to zero at 1%, but certainly this result is a stronger evidence in favor of the existence of arbitrage. The similitude of results obtained from unconditional and conditional approaches suggests that

9We resample daily returns with replacement and calculate the statistic of interest – geometric mean – 100,000 times. Bootstraped confidence intervals come from the empirical distribution of the realizations obtained ineach simulation.

28 Expected = Actual OLS regression 50 0 Actual Returns (%) -50

0 20 40 60 80 100

Expected Returns (%)

Figure 13: Actual Returns vs. Expected Returns

1% CI 5% CI 300 250 200 150 Density 100 50 0

-0.004 -0.002 0.000 0.002 0.004 0.006

Mean return

Figure 14: Bootstrap Confidence Intervals for Mean Return

29 Total Return Standard Brokerage Fee 1200 800 600 400 Return (% CDI) 200 0

0.0 0.1 0.2 0.3 0.4 0.5

Brokerage fee (%)

Figure 15: Sensitivity Analysis on Brokerage Fees returns’ mean and variance regimes do not seem to be related to each other. Although we do not find statistical significance for arbitrage opportunities, it is possible that they have economic relevance. Depending on the risk aversion of an investor, he could choose to invest in this strategy to earn a CAGR of 28% with a long-term volatility of 94% p.a.10

Estimate Std. Error t value Pr(>|t|) 휇 0.00218 0.00105 2.07471 0.03801** 휔 0.00001 0.00000 4.25198 0.00002*** 훼 0.02257 0.00552 4.09190 0.00004*** 훽 0.97457 0.00182 534.30122 0.00000***

Notes: *p<0.1; **p<0.05; ***p<0.01.

Mean Model: 푟푡 = 휇 + 휀푡, 휀푡 is white noise. 2 2 2 Variance Model: 휎푡 = 휔 + 훼휎푡−1 + 훽푟푡 Table 7: Returns GARCH(1,1) Model

We also find that modifying brokerage fees can alter significantly our results. Figure 15 is a sensitivity analysis conducted on percentage brokerage fees. For instance, in a brokerage-free world, our strategy would have gained about three times what it has achieved with B3’s standard brokerage fee. On the other hand, a fee of 0.16% can bring total returns down to zero over the period of analysis.

10GARCH long-term volatility calculated with 휔/[1 − (훼 + 훽)].

30 Finally, once many of the options traded in Brazil are not very liquid investments, we believe that the size of the portfolios created in implementing our strategy in the real world would not be attractive to institutional investors. In fact, if we use traded volumes observed in the market to limit the size of our portfolios, we find that a median portfolio would have a theoretical maximum size of 750 thousand Brazilian real (about 200 thousand US dollars). This means that if we are able to trade, for instance, 10% of the total quantity traded by the market, the maximum amount invested in this strategy would not exceed 75 thousand Brazilian real (about 20 thousand US dollars) half of the days. The fact that we can only assemble small portfolios can explain the existence of arbitrage, even if we do not find statistical significance in our research.

4.4 Arbitrage Determinants

Although we do not find statistically significant evidence of arbitrage, we can still analyze factors that influence the magnitude of excess returns produced by our strategy. We use a modelbased on OLS and LASSO regression of excess returns against candidate factors:

퐸 (50) 푟푡 = 훽0 + 훽1퐹1푡 + 훽2퐹2푡 + ... + 훽푛퐹푛푡 + 휀푖푡 where 퐹푥푡 is the value of factor 푥 at time 푡 and 휀푖푡 is white noise. We can then assess the relevance of each factor by testing the hypothesis 퐻0 : 훽푥 = 0, being relevant those in which the null hypothesis is rejected. We test several candidate factors, such as maturity, moneyness, liquidity, implicit volatility, price, bid-ask spread, call indicator and ‘europeanness’. These option-related factors are calculated by weighted-summing the appropriate quantities on all options contained in the portfolio. For instance, if we have a portfolio in which puts sum 30% of the opening value of all its options, we assign to it a call indicator of 0.7. We also assess the influence of underlying-related factors, such as physical volatility and returns, and market-related factors, such as risk-free rate, market physical volatility and market returns. In Table8, we display several specifications of arbitrage models. In column 1 the negative significant coefficient of stock market physical volatility means that the strategy usually presents higher excess returns in times of low stock market volatility. We can also see that maturity has a significant negative impact in excess returns, i.e. portfolio with shorter options seem to perform better in this strategy. However, maturity is correlated with some other characteristics of our options such as europeanness and option type (put/call).

31 Table 8: Arbitrage Determinants Regression

Excess returns OLS OLS OLS OLS LASSO (1) (2) (3) (4) (5) Maturity −0.002*** 0.0005 0.0003 (0.001) (0.001) (0.001) Moneyness 0.008 0.671*** (0.177) (0.196) Number of trades −0.012* −0.003 −0.004 (0.007) (0.007) (0.007) Quantity traded −0.002 −0.002 −0.001 (0.004) (0.004) (0.004) Traded volume 0.002 0.001 0.001 (0.002) (0.002) (0.002) Opening IV 0.022 0.135 0.127 (0.092) (0.095) (0.095) Opening price 0.009 0.010 0.014* (0.008) (0.008) (0.008) Percentual bid-ask 0.008 −0.085* −0.088** (0.043) (0.045) (0.045) Stock PV −0.121 −0.234** −0.226** (0.094) (0.100) (0.100) Stock lagged return 0.250 0.280 0.294 (0.258) (0.253) (0.253) Risk-free rate 0.682** 1.121*** 1.102*** (0.316) (0.339) (0.338) Stock market PV −0.229* −0.211* −0.216* (0.122) (0.120) (0.120) Stock market return 0.135 −0.377 −0.274 (0.431) (0.429) (0.422) Is european 0.082 0.084 0.047 (0.052) (0.052) Is call −0.190*** −0.193*** (0.055) (0.055) Is in-the-money 0.052*** (0.018) Absolute Moneyness 0.331 (0.230) Expected return 0.028 (0.038) Constant 0.073 −0.579*** 0.064 0.009 0.007 (0.187) (0.210) (0.061) (0.009) Observations 1,118 1,118 1,118 1,118 1,118 R2 0.032 0.076 0.076 0.0005 0.029 Adjusted R2 0.021 0.063 0.062 -0.0004 0.027 Residual Std. Error 0.201 0.197 0.197 0.203 0.200 F Statistic 2.838*** 6.036*** 5.641*** 0.544 16.401***

Note: *p<0.1; **p<0.05; ***p<0.01 PV: Physical Volatility; IV: Implicit Volatility.

32 We include these variables in column 2 and conclude that maturity was capturing the effect of option type, which implies that portfolios with put options usually achieve higher returns. Underlying stock’s physical volatility also gets a negative significant coefficient, meaning that higher systematic and specific volatilities are associated with lower strategy returns. Moneyness, which in our database is positively correlated with calls, turned out to have a positive impact on returns after the inclusion of the latter. Column 3 decomposes moneyness into two distinct variables: ‘Is in the money’, a boolean variable that assumes true if the option has positive intrinsic value; and ‘Absolute moneyness‘, a variable that measures absolute distance to the moneyness of an at-the-money option (푆 = 퐾). As all other option-specific measures, they are weighted-summed across all options ofeach portfolio. We then conclude that the direction of moneyness (in-the-money or out-of-the-money) is more important than its absolute magnitude to determine strategy returns. We then use all variables included in column 3 to estimate column 5 with a LASSO structured model, using penalized maximum likelihood, as proposed by Tibshirani(1996), to shrink the model by dropping insignificant variables. The selection that resulted from LASSO algorithm chooses europeanness as the most significant relationship that determine return levels. Europeanness is a characteristic that is highly correlated with other variables in our database. For instance, there are no American puts traded in B3 in the period of analysis. European options are also negatively correlated with all measures of liquidity, and often are associated with higher bid-ask spreads. Is it reasonable to think that some of the effect on returns caused by these variables may be shrunk into variable Europeanness by LASSO model.

33 Chapter 5

Conclusion

In this thesis, we investigate arbitrage opportunities in the Brazilian options market. We analyze time series of implicit volatilities on several options and conclude that there is mean-reversion in the process that drives them. Even with solid evidence of convergence, it is not guaranteed that market agents can explore this mean-reversion characteristic to set a strategy that profits from volatility reversion. We then implement and backtest several delta-gamma neutral portfolios, finding returns that sum up to 400% of CDI throughout eight years of data. However, the findings are strongly dependent on the window of analysis, and formal tests state that it is not possible to reject the null hypothesis of zero excess returns. This statement is consistent with Fama(1970)’s concept of efficient markets, in which there are no arbitrage opportunities exploitable by using information available to the market, such as volatility mean-reversion patterns. We conducted a wider but less realistic analysis than the one executed by Araújo and Ribeiro (2016). While they have browsed PETR4 options intraday data, we looked into a broader time series window and at all underlyings traded in B3. In spite of methodological differences, we got similar results when we used the same subset of data and similar assumptions. The variance of our data gives us the chance analyze factors that influence the levels of excess returns found. We tested these returns against candidate factors and conclude that the most relevant is europeanness. Future research on this subject could investigate economic theory to bring up hypotheses that explain the influence of these variables in our results. More empirical experiments could also be undertaken in order to investigate if other strategies could exploit successfully volatility mean-reversion or other stylized facts present in the Brazilian option market’s data.

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