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

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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 Arbitrage Opportunities with a Delta-Gamma Neutral Strategy in the Brazilian Options Market 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. This approach is different from other studies because the analysis is taken on several optionson different underlying assets, which gives us the opportunity to investigate factors that influence the 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................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
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