FUNDAÇÃO GETÚLIO VARGASESCOLA DE PÓS-GRADUAÇÃO EM ECONOMIA

Lucas Duarte Processi

Arbitrage Opportunities with aDelta-Gamma Neutral Strategy in the

Brazilian Options Market

Rio de Janeiro

2017

Lucas Duarte Processi

Arbitrage Opportunities with aDelta-Gamma Neutral Strategy in the

Brazilian Options Market

Dissertação para obtenção do grau demestre 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. Àminha família pela compreensão e suporte. À minha amada esposa pelo imenso apoio e incentivonas incontáveis horas de estudo e pesquisa.

Abstract

We investigate arbitrage opportunities in the Brazilian options market. Our research consists inbacktesting several delta-gamma neutral portfolios of options traded in B3 exchange to assessthe possibility of obtaining systematic excess returns. Returns sum up to 400% of the dailyinterbank 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 optionsmarket.

This approach is different from other studies because the analysis is taken on several options ondifferent underlying assets, which gives us the opportunity to investigate factors that influencethe magnitude of excess returns. Europeanness is the most relevant factor found.

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

List of Figures

1 Monthly Trade Volume in Brazilian Options Market . . . . . . . . . . . . . . . . 42 Cumulative Percentage of Volume Traded by Maturity . . . . . . . . . . . . . . . 53 Risk-Free Yield Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Brazilian Stock Market Index – IBOVESPA . . . . . . . . . . . . . . . . . . . . . 75 Physical Volatilies Estimated with Univariate GARCH(1,1) processes . . . . . . . 96 Examples of Volatility Smiles/Skews . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Delta-Gamma Neutral Strategy Performance . . . . . . . . . . . . . . . . . . . . . 248 Histogram of Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 Rolling Strategy Returns – 1-year Investment . . . . . . . . . . . . . . . . . . . . 2610 Rolling Strategy Volatility – 1-year Investment . . . . . . . . . . . . . . . . . . . 2711 Rolling Sharpe Ratio – 1-year Investment . . . . . . . . . . . . . . . . . . . . . . 2712 Rolling Accuracy Ratio – 1-year Investment . . . . . . . . . . . . . . . . . . . . . 2813 Actual Returns vs. Expected Returns . . . . . . . . . . . . . . . . . . . . . . . . 2914 Bootstrap Confidence Intervals for Mean Return . . . . . . . . . . . . . . . . . . 2915 Sensitivity Analysis on Brokerage Fees . . . . . . . . . . . . . . . . . . . . . . . . 30

vi

List of Tables

1 B3 Stock Type Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Traded Options by Europeanness and Type in B3’s database . . . . . . . . . . . 43 Data Sources for Calculating Implicit Volatilities . . . . . . . . . . . . . . . . . . 11

4 OLS Estimates of Volatility Convergence Models . . . . . . . . . . . . . . . . . . 165 Wald-Test Statistics – Volatility Convergence Model . . . . . . . . . . . . . . . . 17

6 Daily Returns Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Returns GARCH(1,1) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 Arbitrage Determinants Regression . . . . . . . . . . . . . . . . . . . . . . . . . . 32

vii

Contents

1 Introduction 1

2 Data and Methodology 3

2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Physical Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.2 Implicit Volatilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Volatility Convergence Models 13

3.1 Simple Convergence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.1 Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.2 Estimation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Generalized Convergence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.2 Estimation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Delta-Gamma Neutral Strategy 18

4.1 Strategy Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.1 Delta-Gamma Neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.1.2 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.4 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 be no

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 works of 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

Figure 1.

Extracting underlying stock exchange codes from database involves combining information

contained in two different fields: the first four letters of field NOMRES indicate the company

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 Table 1. 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 3PN Preferred 4PNA Preferred A 5PNB Preferred B 6UNT Unit 11CI 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

2009 2010 2011 2012 2013 2014 2015 2016

24

68

BRL(Billions)

Average

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 denote an

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 Table 2 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 84.8European Put 388,520 33.8 47,229.3 11.0European Call 113,921 9.9 18,113.6 4.2American Put 0 0.0 0.0 0.0TOTAL 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

Maturity (business days)

Cum

ulativepercentageof

volume(%

)

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

(...)

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 Figure 2 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 𝑟𝑡 for maturity 𝑡, which

lies between the two adjacent vertex 𝑖 – with yield 𝑟𝑖 and maturity 𝑇𝑖 – and 𝑖+ 1, then:

(1 + 𝑟𝑡)𝑡 = (1 + 𝑟𝑖)

𝑇𝑖 ×(︁1 + 𝑟𝑖+1

1 + 𝑟𝑖

)︁(︁ 𝑡−𝑇𝑖𝑇𝑖+1−𝑇𝑖

)︁(1)

A sample of yield curves are shown in Figure 3.

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

0.0 0.5 1.0 1.5 2.0 2.5 3.0

78

910

1112

13

Maturity (years)

Risk-free

rate

(%)

dez/2010dez/2012dez/2014dez/2016

Figure 3: Risk-Free Yield Curves

volatilities. We also get Brazilian Stock Market Index (IBOVESPA) daily quotes from the same

source. Figure 4 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 𝑃𝑡 is the price of the

stock – or index – at time 𝑡, then its discrete return 𝑟𝑡 is defined as:

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

6

4050

6070

Index(x1,000pts)

-0.05

0.00

0.05

Returns

2010 2012 2014 2016

Index

0.2

0.3

0.4

0.5

0.6

Volatility

Average

Figure 4: Brazilian Stock Market Index – IBOVESPA

7

𝑟𝑡 =(𝑃𝑡 − 𝑃𝑡−1)

𝑃𝑡−1(2)

Our estimate of current physical volatility is an estimate of the standard deviation of the

process that generates 𝑟𝑡. If 𝑟 is the simple average of returns and 𝜀𝑡 is a gaussian unexpected

shock at time 𝑡 with time-varying conditional variance 𝜎2𝑡 then:

𝑟𝑡 = 𝑟 + 𝜀𝑡, 𝜀𝑡 ∼ N(0, 𝜎2𝑡 ) (3)

The Generalized Autoregressive Conditional Heteroscedasticity – GARCH conditional vari-

ance equation can be stated as:

(︀𝜎2𝑡 |Ω𝑡−1

)︀= 𝜔 + 𝛼𝜀2𝑡−1 + 𝛽

(︀𝜎2𝑡−1|Ω𝑡−2

)︀(4)

We estimate the value of 𝜔, 𝛼 and 𝛽 with the maximum-likelihood method, as proposed

by Engle (1982), for all 120 stocks in our sample. The term Ω𝑡−1 means that we only use data

available until 𝑡−1 to estimate the conditional variance at time 𝑡. Volatilities are then calculated

with the square root of the variance annualized by a factor√252. Some of them are shown in

Figure 5.

𝜎𝑡 =√︁𝜎2𝑡 ×

√252 (5)

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 𝜇 and variance 𝜎2:

𝑑𝑆 = 𝜇𝑆𝑑𝑡+ 𝜎𝑆𝑑𝑍 (6)

where 𝑍 follows Wiener process. This means that 𝑑𝑍 is independent and identically distributed,

following a normal distribution with mean zero and variance 𝑑𝑡. If 𝑉 (.) is a function that

8

2010 2012 2014 2016

0.2

0.4

0.6

0.8

1.0

1.2

Volatilities

PETR4VALE5ITUB4BBAS3

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

9

evaluates the price of a derivative of the stock that follows 6 then, applying Itô’s lemma4 we get:

𝑑𝑉 =(︁𝜕𝑉𝜕𝑆

𝜇𝑆 +𝜕𝑉

𝜕𝑡+

1

2

𝜕2𝑉

𝜕𝑡2𝜎2𝑆2

)︁𝑑𝑡+

𝜕𝑉

𝜕𝑆𝜎𝑆𝑑𝑍 (7)

If we assemble a portfolio Π that is short an option and long 𝜕𝑉𝜕𝑆 stocks:

𝑑Π = −𝑑𝑉 +𝜕𝑉

𝜕𝑆𝑑𝑆 (8)

Joining Equations 6, 7 and 8, we can reestate the dynamics of 𝑑Π:

𝑑Π =(︁− 𝜕𝑉

𝜕𝑡− 1

2𝜎2𝑆2𝜕

2𝑉

𝜕𝑆2

)︁𝑑𝑡 (9)

As now we have eliminated the stochastic term 𝑑𝑍, we have – at least for a brief moment –

a risk-free portfolio that therefore earns:

𝑑Π = 𝑟Π𝑑𝑡 (10)

where 𝑟 is the instantaneous risk-free rate. Using 9 and 10 we get the Black and Scholes (1973)

partial differential equation (PDE):

𝜕𝑉

𝜕𝑡+

1

2𝜎2𝑆2𝜕

2𝑉

𝜕𝑆2+ 𝑟𝑆

𝜕𝑉

𝜕𝑆− 𝑟𝑉 = 0 (11)

Black and Scholes (1973) solved this PDE for European options, developing a closed formula

to price them:

𝑉 (𝜔, 𝑆, 𝑡,𝐾, 𝑇, 𝑟, 𝜎) = 𝜔[︀𝑆𝑁(𝜔𝑑1)−𝐾𝑒𝑟(𝑇−𝑡)𝑁(𝜔𝑑2)

]︀(12)

𝑑1 =𝑙𝑛( 𝑆

𝐾 ) + (𝑟 + 𝜎2

2 )(𝑇 − 𝑡)

𝜎√𝑇 − 𝑡

(13)

𝑑2 = 𝑑1 − 𝜎√𝑇 − 𝑡 (14)

where 𝑆 is the underlying stock price at time 𝑡, 𝑇 is the time at which the option expires, 𝑟 is the

continuous risk-free rate and 𝜎 is the underlying asset volatility. 𝜔 is a parameter that equals 1

for call options and −1 for puts. 𝑁(.) is the standard normal cumulative distribution function.

If we observe at time 𝑡 a certain stock option which is traded with a price 𝑃 *, we can find a

volatility 𝜎* that is implicit in the price of this option, assuming that Black and Scholes (1973)

4Itô’s lemma states that if 𝑑𝑆 = 𝑎(𝑆,𝑡)𝑑𝑡+ 𝑏(𝑆,𝑡)𝑑𝑍 and 𝐹 (𝑆,𝑡) is a double differentiable scalar function, then

𝑑𝐹 =(︀𝜕𝐹𝜕𝑡

+ 𝑎 𝜕𝐹𝜕𝑆

+ 𝑏2

2𝜕2𝐹𝜕𝑆2

)︀𝑑𝑡+ 𝑏 𝜕𝐹

𝜕𝑆𝑑𝑍, where dZ is a Wiener process.

10

Parameter Source Field𝜔 COTAHIST CD_BDI𝑆 Bloomberg PX_OPEN or PX_LAST1

𝑡 COTAHIST DT_PREGAO𝐾 COTAHIST PREEXE𝑇 COTAHIST DATVEN2

𝑟 B3 and CETIP —3

𝑃 * COTAHIST PREABE or PREULT

1 Not adjusted by dividends and capital changes. 2 𝑇 − 𝑡 in Brazilian work-

ing 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 𝜎* such that:

𝑉 (𝜔, 𝑆, 𝑡,𝐾, 𝑇, 𝑟, 𝜎*) = 𝑃 * (15)

Being 𝑉 (.) a strictly increasing function of 𝜎, it is guaranteed that 15 has a unique solution

for positive prices. Because there is no easy way to isolate 𝜎 in Equation 12, the problem of

finding the implicit volatility must be solved numerically. To speed up calculations, as we have

to evaluate a great number of implicit volatilities, we use Newton’s method to find the root of

function 𝑓(𝜎) shown in Equation 16, with a quadratic convergence rate. Evidently, comparing

with Equation 15, the 𝜎 that satisfies 𝑓(𝜎) = 0 is the implicit volatility 𝜎*.

𝑓(𝜎) = 𝑉 (𝜔, 𝑆, 𝑡,𝐾, 𝑇, 𝑟, 𝜎)− 𝑃 * (16)

So, we begin the process by choosing an arbitrary guess for 𝜎𝑖 and then we iteratively update

our guess with the formula shown in Equation 17, until we reach a value 𝜎𝑖 that yields a 𝑓(𝜎𝑖)

sufficiently close to zero.5

𝜎𝑖+1 = 𝜎𝑖 +𝑓(𝜎𝑖)

𝑓 ′(𝜎𝑖)= 𝜎𝑖 +

𝑉 (𝜔, 𝑆, 𝑡,𝐾, 𝑇, 𝑟, 𝜎)− 𝑃 *

𝜕𝑉𝜕𝜎

(17)

Using Equation 17 we calculated implicit volatilities for all opening and closing prices by

joining B3 and Bloomberg databases. The sources are listed in Table 3.

Despite of Black and Scholes (1973) model assumption of constant volatility, when we calcu-

late 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 𝜕𝑉𝜕𝜎

is the derivative called vega, whose closed formula is 𝜈 = 𝑆𝜑(𝑑1)√︀

(𝑇 − 𝑡), if 𝜑(.) is the standardnormal probability density function.

11

20 22 24 26 28 30 32

0.0

0.2

0.4

0.6

0.8

1.0

(a) PETR4: 2011-fev-14

Strike price

Volatility

Maturity5 bd23 bd43 bd

20 22 24 26

0.00

0.10

0.20

0.30

(b) PETR4: 2012-mar-12

Strike priceVolatility

Maturity5 bd24 bd48 bd

16 18 20 22 24

0.0

0.5

1.0

1.5

(c) PETR4: 2013-nov-11

Strike price

Volatility

Maturity4 bd24 bd47 bd

18 20 22 24 26 28

0.0

0.2

0.4

0.6

(d) PETR4: 2012-fev-24

Strike price

Volatility

Maturity16 bd35 bd59 bd

Notes: Vertical and horizontal dotted lines represent, respectively, underlying spotprices and physical volatilities. Maturities in business days.

Figure 6: Examples of Volatility Smiles/Skews

smile. That is why the lines shown in Figure 6 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 au-

toregressive AR(1) process, such that:

𝜎𝑡 = 𝛽0 + 𝛽1𝜎𝑡−1 + 𝛽2𝜎𝑃 + 𝜀𝑡, 𝜀𝑡 ∼ N(0, 𝜎2

𝑡 ) (18)

where 𝜎𝑡 is the implicit volatility observed in time 𝑡, 𝜎𝑃 is the physical volatility and 𝜀𝑡 is white

noise.1 If we substract 𝜎𝑡−1 from both sides then:

1For more on mean-reversion property of autoregressive models, see Tsay (2010).

13

Δ𝜎𝑡 = 𝜎𝑡 − 𝜎𝑡−1 = 𝛽0 + (𝛽1 − 1)𝜎𝑡−1 + 𝛽2𝜎𝑃 + 𝜀𝑡

= 𝛽0 + 𝛾𝜎𝑡−1 + 𝛽2𝜎𝑃 + 𝜀𝑡

(19)

where 𝛾 = 𝛽1 − 1. This process mean reverts exactly to physical volatility (𝜎𝑃 ) in the long-run

if the following conditions are satisfied:

𝛽0 = 0 (20)

−1 < 𝛾 < 0 (21)

𝛽2 = −𝛾 (22)

Using these conditions, Equation 19 can be restated as:

Δ𝜎𝑡 = 𝛾(𝜎𝑡−1 − 𝜎𝑃 ) + 𝜀𝑡 (23)

and we have that 𝜎𝑡 mean reverts if 𝛾 < 0. If implicit volatility 𝜎𝑡−1 is higher than physical

volatility 𝜎𝑃 then 𝛾(𝜎𝑡−1−𝜎𝑃 ) < 0, which tends to bring implicit volatility back to its mean. On

the other hand, if implicit volatility is lower than physical volatility, 𝜎𝑃 then 𝛾(𝜎𝑡−1 − 𝜎𝑃 ) > 0,

which creates an upward trend proportional to the difference between current volatility and its

long-term level 𝜎𝑃 .

Coefficient 𝛾 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

𝑑0. As 𝐸[𝜀𝑡] = 0, we expect that 𝜎𝑡 takes a step of size |𝛾|𝑑0 in the direction of 𝜎𝑃 , reducing the

gap to 𝑑0−|𝛾|𝑑0 = 𝑑0(1−|𝛾|). In the next period, the step size expectation will be |𝛾|[𝑑0(1−|𝛾|)],

and the new distance will be 𝑑0(1− |𝛾|)− |𝛾|𝑑0(1− |𝛾|) = 𝑑0(1− |𝛾|)2. If we keep on calculating

expected distances, we will notice that the distance 𝑑ℎ after ℎ steps is governed by:

𝑑ℎ = 𝑑0(1− |𝛾|)ℎ (24)

Using 𝑑ℎ = 12𝑑0 we can calculate the half-life for our mean-revert process by isolating ℎ:

1

2𝑑0 = 𝑑0(1− |𝛾|)ℎ → ℎ = − 𝑙𝑜𝑔(2)

𝑙𝑜𝑔(1− |𝛾|)(25)

14

3.1.2 Estimation and Results

We estimate the coefficients of Equation 19 with opening and closing implicit volatilities calcu-

lated 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 Table 4. The estimated

coefficient for opening implicit volatility, 𝛾 = −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 a standard

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 Table 5. 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

Figure 6. 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 (𝑀) and maturity (𝜏):

Δ𝜎𝑡 = 𝛾(︁𝜎𝑡−1 − 𝑓(𝜎𝑃 ,𝑀, 𝜏)

)︁+ 𝜀𝑡 (26)

The function 𝑓(.) 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: 𝑀 = 𝑆/𝐾.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 dailyclosing (Closing IV) and opening (Opening IV) prices. Physical Volatility (PV) is a GARCH(1,1) volatilitycalibrated with returns of the underlying stock. Moneyness and maturity terms are orthogonalized. We run theseOLS regressions with daily data on all options traded in B3 between january/2006 and december/2016. Column2 shows that IV’s exhibit mean-reversion, but their long-term levels are not given exactly by PV’s. The half-lifeimplied by this model is 1.03 days. Columns 3-5 shows that adding moneyness and maturity polynomials helpsexplaining IV long-term levels. Using AIC and BIC criteria we opt to use models with second-order polynomialsin moneyness and maturity, like the one in column 4, to predict option closing prices.

Dependent Variable: Delta IV = Closing IV - Opening IV

Δ𝜎𝑡

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

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

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

Note: *p<0.1; **p<0.05; ***p<0.01The 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)𝛽0 = 0 5993.45 0***𝛾 = 0 143370.62 0***𝛾 = 0 and 𝛾 + 𝛽2 = 0 71698.50 0***𝛾 + 𝛽2 = 0 11494.71 0***𝛽0 = 0 and 𝛾 = 0 and 𝛾 + 𝛽2 = 0 48450.08 0***Note: OLS Model: Δ𝜎𝑡 = 𝛽0 + 𝛾𝜎𝑡−1 + 𝛽2𝜎𝑃 + 𝜀𝑡

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 Table 4 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 𝑅2’s.

We calculate Akaike and Bayesian Information Criterion – AIC and BIC – for various poly-

nomial 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 Table 4.

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 𝜎𝑡−1 coefficient in column 4 is compatible with a half-life of 1.03 days. This means

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 con-

vergence, 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 deterministic path.1

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 𝑃 of 𝑛 financial contracts that can be priced adequately with 𝑛 pricing

functions 𝑉𝑖(𝑆), 𝑖 = 1, . . . , 𝑛, where 𝑆 is the spot price of a stock.2 If 𝑞𝑖 is the quantity of asset

𝑖 held, then the value of 𝑃 is:

𝑉 (𝑆) =𝑛∑︁

𝑖=1

𝑞𝑖𝑉𝑖(𝑆) (27)

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 stock price 𝑆:

Δ𝑃 = 𝑉 ′(𝑆) =𝑛∑︁

𝑖=1

𝑞𝑖𝑉′𝑖 (𝑆) =

𝑛∑︁𝑖=1

𝑞𝑖Δ𝑖 (28)

Γ𝑃 = 𝑉 ′′(𝑆) =𝑛∑︁

𝑖=1

𝑞𝑖𝑉′′𝑖 (𝑆) =

𝑛∑︁𝑖=1

𝑞𝑖Γ𝑖 (29)

where 𝑉 ′(𝑆) is the first derivative of 𝑉 (𝑆) with respect to stock price 𝑆, usually referred to as

delta (Δ); and 𝑉 ′′(𝑆) is the second derivative of 𝑉 (𝑆) with respect to 𝑆, usually called gamma

(Γ).

Suppose that stock price moves to 𝑆 +Δ𝑆. The new value of 𝑃 can be approximated by a

Taylor expansion of second degree:

𝑉 (𝑆 +Δ𝑆) = 𝑉 (𝑆) + 𝑉 ′(𝑆)(Δ𝑆) +1

2𝑉 ′′(𝑆)(Δ𝑆)2 +𝑂(Δ𝑆3)

≈ 𝑉 (𝑆) + 𝑉 ′(𝑆)(Δ𝑆) +1

2𝑉 ′′(𝑆)(Δ𝑆)2

(30)

1In Equation 12 maturity represents both parameters 𝑇 and 𝑡, with (𝑀 = 𝑇 − 𝑡).2In our portfolios, these financial contracts are options and stocks. Evidently, stocks can be valued with the

simple pricing function 𝑉𝑖(𝑆) = 𝑆 without loss of generality.

19

where 𝑂(Δ𝑆3) 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:

𝑉 (𝑆 +Δ𝑆) ≈ 𝑉 (𝑆) (31)

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

𝑉 ′(𝑆) = Δ𝑃 = 0 (32)

𝑉 ′′(𝑆) = Γ𝑃 = 0 (33)

If Condition 32 is satisfied we call the portfolio 𝑃 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 𝑞𝑖 that satisfy both:

𝑞1Δ1 + 𝑞2Δ2 + . . .+ 𝑞𝑛Δ𝑛 = 0 (34)

𝑞1Γ1 + 𝑞2Γ2 + . . .+ 𝑞𝑛Γ𝑛 = 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 𝑛 opening prices for different options of a certain stock. We can

then calculate their associated opening implicit volatilities, 𝜎𝑡−1. If the stock has a physical

volatility of 𝜎𝑃 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 𝜎𝑡 with:

20

𝐸(𝜎𝑡|Ω𝑡−1) = 𝜎𝑡−1 + 𝐸(Δ𝜎𝑡|Ω𝑡−1)

= 𝜎𝑡−1 + 𝛾(︁𝜎𝑡−1 − 𝑓(𝜎𝑃 ,𝑀, 𝜏)

)︁ (36)

If 𝐸(Δ𝜎𝑡|Ω𝑡−1) is positive then the option is cheap and we must buy it to profit from arbitrage.

On the other hand, if 𝐸(Δ𝜎𝑡|Ω𝑡−1) is negative, we must sell the option because it is considered

expensive. In other words, we could define a buy/sell signal 𝜃 that follows:

𝜃 =

⎧⎪⎨⎪⎩1, if 𝐸(Δ𝜎𝑡|Ω𝑡−1) ≥ 0

−1, if 𝐸(Δ𝜎𝑡|Ω𝑡−1) < 0

(37)

We assume that 𝐸(𝑃𝑡|Ω𝑡−1) is the expected option price calculated with Black and Scholes

(1973) model, using 𝐸(𝜎𝑡|Ω𝑡−1) as the underlying stock volatility parameter.3 Hence, if 𝑃𝑡−1 is

the option’s opening price we can expect a price movement from 𝑡− 1 to 𝑡:

𝐸(Δ𝑃𝑡|Ω𝑡−1) = 𝐸(𝑃𝑡|Ω𝑡−1)− 𝑃𝑡−1 (38)

To take into account bid-ask spreads in our calculations, we could define a deterministic

constant bid-ask spread 𝑠 as the difference between the lowest ask price and the highest bid offer

at any arbitrary time 𝑡:

𝑠 = 𝑃𝐴𝑆𝐾𝑡 − 𝑃𝐵𝐼𝐷

𝑡 (39)

Then, an investor who has a position in one option in 𝑡− 1 and intends to end it in 𝑡 would

expect to profit:

𝛾 =

⎧⎪⎨⎪⎩+𝐸(Δ𝑃𝑡|Ω𝑡−1)− 𝑠, if position is long;

−𝐸(Δ𝑃𝑡|Ω𝑡−1)− 𝑠, if position is short.(40)

If we observe at the opening of the day at least one cheap and one expensive option of the

same underlying 𝑆, we can construct the maximum profit delta-gamma-cash neutral portfolio

that profits from volatility reversion of 𝑛 options with different opening prices 𝑝𝑖, end-of-the-day

expected profits 𝛾𝑖 and buy/sell signals 𝜃𝑖 (𝑖 = 1, . . . , 𝑛) 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 otherparameters such as time, risk-free rate and underlying stock spot price.

21

maximize𝑤1,...,𝑤𝑛,𝑤𝑆

𝑤1𝛾1 + . . .+ 𝑤𝑛𝛾𝑛 (41)

subject to 𝑤𝑖 sign(𝜃𝑖) ≥ 0, ∀𝑖 ∈ 1, . . . , 𝑛 (42)

𝑤1 + . . .+ 𝑤𝑛 + 𝑤𝑆 = 1 (43)

𝑤1𝑝1 + . . .+ 𝑤𝑛𝑝𝑛 + 𝑤𝑆𝑝𝑆 = 0 (44)

𝑤1Δ1 + . . .+ 𝑤𝑛Δ𝑛 + 𝑤𝑆 = 0 (45)

𝑤1Γ1 + . . .+ 𝑤𝑛Γ𝑛 = 0 (46)

Note that 𝑤1, . . . , 𝑤𝑛 are the weights of the 𝑛 options in the portfolio. 𝑤𝑆 is the weight of the

underlying stock, and 𝑝𝑆 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:

𝑟𝑝 = 𝑤1𝑟1 + . . .+ 𝑤𝑛𝑟𝑛 + 𝑤𝑆𝑟𝑆 (47)

𝑟𝑖 =(𝑃 𝑐𝑙𝑜𝑠𝑒

𝑖 − 𝑃 𝑜𝑝𝑒𝑛𝑖 )− 𝑠𝑖

𝑃 𝑜𝑝𝑒𝑛𝑖

, ∀ 𝑖 ∈ 1, . . . , 𝑛, 𝑆 (48)

where 𝑠𝑖 is the bid-ask spread of option (or stock) 𝑖.

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 𝐾 portfolios at a particular day and

each one of them achieve a log-return of 𝑟𝑘𝑝 as given by Equation 47, then the total log-return

perceived in this day will be:

𝑟 = 𝑟(𝑟𝑓) +

∑︀𝐾𝑘=1 𝑟

𝑘𝑝

𝐾(49)

where 𝑟(𝑟𝑓) 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 𝐾 cash-neutral portfolios. The sum

22

of all long trades in all 𝐾 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 that are

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 (|𝑙𝑛(𝑆/𝐾)|) 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 Figure 7. 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 to thenumber 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

05

1015

STRATEGYCDI

Cum

ulativeReturn

-0.4

-0.2

0.0

0.2

Daily

Return

2009-04-03 2010-04-01 2011-04-01 2012-04-02 2013-04-01 2014-04-01 2015-04-01 2016-04-01

-1.0

-0.6

-0.2

Drawdown

Figure 7: Delta-Gamma Neutral Strategy Performance

puts on 6 underlying stocks: VALE5, PETR4, OGXP3, ITUB4, BBDC4 and BBAS3.7 About

51% of the portfolios obtained positive returns, resulting in a compound growth annual rate

(CAGR) of about 28% against a CAGR of 10.6% obtained by a risk-free investment over the

same period. The annualized Sharpe ratio of the strategy over the whole period of analysis was

0.20 , while IBOVESPA’s ratio was -0.35 .

However, in Figure 7, we can also see that the strategy exhibits a maximum drawdown of more

than 90% , because of its high volatility. In fact, Table 6 reports summary statistics of strategy

returns in comparison with Brazilian short-term interest rate (CDI) and stock (IBOVESPA)

markets. We see that the volatility obtained is nearly three times that of the stock market over

the same period. Strategy returns are also positively asymmetric and exhibit tails that are much

fatter than those observed in the stock market, as visually displayed in Figure 8. Fatter tails

also lead to a strategy Value-at-Risk (VaR) that is almost five times IBOVESPA’s VaR at a 1%

confidence level.8 The beta of the strategy with respect to the stock market index is nearly zero

( 0.001 ). This confirms that we managed to eliminate the influence of movements on underlying

stocks by imposing delta-gamma neutrality.

We chose the window of analysis based on information availability of the various data sources

7The number of portfolios represents the number of times in which the problem in Equations 41-46 could besolved with a positive objective function.

8VaR calculated with a historical simulation of daily returns.

24

Strategy CDI IBOVESPAMin. -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 -0.35Skewness 0.05 0.07 0.00Exc. Kurtosis 11.12 -1.04 1.37VaR (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, Figure 9 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 over the

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

Returns

Density

-0.2 -0.1 0.0 0.1 0.2

05

1015

2025

30

IBOVESPA

Returns

Density

-0.2 -0.1 0.0 0.1 0.2

05

1015

2025

30

Figure 8: Histogram of Returns

2011 2012 2013 2014 2015 2016 2017

0200

400

600

800

1200

End of 1-year Investment

Return(%

p.a.)

StrategyCDIIBOVESPAAverage (CAGR)

Figure 9: Rolling Strategy Returns – 1-year Investment

26

2011 2012 2013 2014 2015 2016 2017

0.0

0.2

0.4

0.6

0.8

1.0

1.2

End of 1-year Investment

Annualized

Std.

Dev.

StrategyCDIIBOVESPAAverages

Figure 10: Rolling Strategy Volatility – 1-year Investment

2011 2012 2013 2014 2015 2016 2017

-20

24

68

1012

End of 1-year Investment

SharpeRatio

StrategyIBOVESPAOverall

Figure 11: Rolling Sharpe Ratio – 1-year Investment

27

2011 2012 2013 2014 2015 2016 2017

4045

5055

6065

Acuracy

Ratio(%

)

Average

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 Table 8. 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 were not

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 the null

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 Table 7, we report the results of a process to

test if mean return 𝜇 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 – 100,000times. Bootstraped confidence intervals come from the empirical distribution of the realizations obtained in eachsimulation.

28

0 20 40 60 80 100

-50

050

Expected Returns (%)

ActualReturns

(%)

Expected = ActualOLS regression

Figure 13: Actual Returns vs. Expected Returns

Mean return

Density

-0.004 -0.002 0.000 0.002 0.004 0.006

050

100

150

200

250

300

1% CI5% CI

Figure 14: Bootstrap Confidence Intervals for Mean Return

29

0.0 0.1 0.2 0.3 0.4 0.5

0200

400

600

800

1200

Brokerage fee (%)

Return(%

CDI)

Total ReturnStandard 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|)𝜇 0.00218 0.00105 2.07471 0.03801**𝜔 0.00001 0.00000 4.25198 0.00002***𝛼 0.02257 0.00552 4.09190 0.00004***𝛽 0.97457 0.00182 534.30122 0.00000***

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

Mean Model : 𝑟𝑡 = 𝜇+ 𝜀𝑡, 𝜀𝑡 is white noise.

Variance Model : 𝜎2𝑡 = 𝜔 + 𝛼𝜎2

𝑡−1 + 𝛽𝑟2𝑡

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 𝜔/[1− (𝛼+ 𝛽)].

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 model based

on OLS and LASSO regression of excess returns against candidate factors:

𝑟𝐸𝑡 = 𝛽0 + 𝛽1𝐹1𝑡 + 𝛽2𝐹2𝑡 + . . .+ 𝛽𝑛𝐹𝑛𝑡 + 𝜀𝑖𝑡 (50)

where 𝐹𝑥𝑡 is the value of factor 𝑥 at time 𝑡 and 𝜀𝑖𝑡 is white noise. We can then assess the

relevance of each factor by testing the hypothesis 𝐻0 : 𝛽𝑥 = 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 calcu-

lated 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 Table 8, 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 1,118R2 0.032 0.076 0.076 0.0005 0.029Adjusted R2 0.021 0.063 0.062 -0.0004 0.027Residual Std. Error 0.201 0.197 0.197 0.203 0.200F Statistic 2.838*** 6.036*** 5.641*** 0.544 16.401***

Note: *p<0.1; **p<0.05; ***p<0.01PV: 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 (𝑆 =

𝐾). As all other option-specific measures, they are weighted-summed across all options of each

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.

34

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