Chapter -7 CONCLUSION

Transcription

1 Chapter -7 CONCLUSION

2 Chapter 7 CONCLUSION Options are one of the key financial derivatives. Subsequent to the Black-Scholes option pricing model, some other popular approaches were also developed to value options. These include binomial tree, Cox-Ross-Rubinstein, Jump Diffusion, Monte-Carlo Simulation, and Finite Difference. However, despite all academic evidence that refuted the assumptions underlying the Black-Scholes model, it is still the most popular amongst investors. The underlying assumptions of the Black-Scholes model have been empirically proven to be inconsistent with observed realities over time. In particular, the Black-Scholes model has been found to include systematic biases. Further, some researchers have found trends in over-pricing or under pricing. Macbeth and Merville (1979) found that the Black- Scholes model under prices in-the-money options and overprices out-of-the-money options. The Black-Scholes model uses spot price of the underlying asset, strike price, the risk free rate, time to expiry of option and volatility. Among these, volatility is the only unobservable input. There are multiple ways to calculate volatility. These include historical volatility, implied volatility, GARCH based models, ARIMA models etc. Each of these models of calculating volatility yields a different measure and therefore a different theoretical option price. There have been multiple studies that compare the effectiveness of these volatility models from the perspective of pricing options. These include Becker and 116

3 Adam (2008), Bluhm, Hagen, and Jun Yu.(2001),Boudoukh, Matthew, Whitelaw (1997) etc. There are numerous other studies on this topic. However, none of these studies has been able to conclusively suggest one volatility model that is consistently better than the others. Further, as per our knowledge, there has been no academic research on evaluating a machine learning approach to finding volatility in the Indian markets. We therefore consider the effectiveness of the Black-Scholes model in pricing options using machine learning volatility. CHAPTER SUMMARY The entire research is divided into chapters which are summarized below. Chapter 1: This presents an introduction to financial derivatives, options and the market for such derivatives. It also briefly touches upon the history of options trading in India. Further, it covers some core concepts necessary to discuss option pricing. It then moves on to describing two models, the binomial option pricing model and the Black-Scholes option pricing model. Chapter 2: This presents the scope of the present research. It focuses on the need for this research and what are the key dimensions defining this research. Chapter 3: This chapter reviews the literature relevant to this study. It presents the empirical studies on various subjects related to the present research. These studies were reviewed and have acted as the reference points for this research. These studies helped understand the nuances of the topic and have contributed fundamentally to this research. There has been a great deal of focus on derivatives in general and option pricing in particular and it has therefore seen some very high quality academic research. This chapter 117

4 presents some of these studies which cover contextually relevant issues such as theoretical option pricing framework, Review the performance of alternate volatility models tested in the literature and learn about the performance of Black and Scholes option pricing model in different markets according to these studies. Chapter 4: This chapter outlines the different methodologies used. It covers Granger causality, the different conventional volatility models and the machine learning approach to volatility used in this study. It also covers a primer on the methodology to calculate MIBOR, though that is not strictly a part of this research. Chapter 5: This presents the data used in this research. Data plays a key part of any empirical research and this is no different. This chapter summarizes the nature of data and outlines the source and summarizes the observations. Chapter 6: This chapter outlines the analysis and the results of this research based on the data collected and methodologies used. It summarizes the results and presents the errors measures and comparison charts. Hypothesis and results: This research attempts to answer the following key questions: - What is the direction of causality between stock returns and option returns? o Null Hypothesis: Spot return does not cause option return and option return does not cause spot return o The null hypothesis was rejected after conducting a Granger causality test. The test revealed that there exists a bi-directional causality among spot returns and option returns. - Is the machine learning model effective for option pricing? 118

5 o Null Hypothesis: A machine learning model performs better than the Black- Scholes model for option pricing o The null hypothesis was rejected. The machine learning option prices were compared with the observed prices. Similarly, the Black-Scholes prices were also compared with the observed prices. The mean square error test showed that the machine learning option pricing did worse than the Black- Scholes pricing. - Is there a way to improve the volatility parameter of the Black-Scholes model to better its efficacy in predicting option prices? o Null Hypothesis: A volatility measure using machine learning performs better in pricing options than implied volatility, historical volatility and EGARCH volatility o The null hypothesis was accepted. The volatility was measured using conventional models such as implied volatility, historical volatility and EGARCH volatility. The Black-Scholes formula was then deployed to use each of these volatility models and forecast option prices. Similarly, a machine learning algorithm was deployed to predict volatility. The Black- Scholes formula was then deployed again to use the machine learning volatility and forecast option prices. The option price forecasts based on the conventional models as well as the machine learning approach were then compared with the observed prices. The efficacy was measured by calculating the errors (Mean Absolute Error and Root Mean Squared Error). The option price forecasts based performed better than the conventional volatility models. 119

6 Limitations of this research This study has made an effort to embrace the three key volatility models; historical, implied and GARCH based (EGARCH) and compared the effectiveness of a Machine Learning volatility to each of them. However, there is still a scope for the study to be limited because; - This research considers crude oil prices, volatility in US markets and risk free rate in India to be the predictors of volatility. However, there may be many more predictors - This research is limited to one year (2013). The economic cycles and thereby forecasts experience swings. A model which works for a particular time period may not work in others. - The study is limited to call options. Though puts are expected to behave similarly, it is a definite gap. - The study covers options on the index (NIFTY). However, there has been no coverage of individual stocks. There may have been trends in individual stocks that this study might have missed. - This study has been limited to Indian markets only. There has been no study of the foreign markets. India accounts for a very minor portion of the global option liquidity. Therefore, this study would not be conclusive without looking at the major option markets such as the US or EU. 120

7 Areas For Further Research A research is more meaningful when it can identify applications and potential for further research. This research considers a relatively new area of forecasting, called machine learning, to estimate volatility. When the accuracy of forecasting volatility is improved, it is expected that the Black-Scholes option price estimates shall become closer to observed prices. Some of the forthcoming research can be directed towards; - Evaluate if there are other predictors of volatility and identify them. Carry out research to find if the other predictors further improve the volatility estimates. - Study the potential of machine learning to consider cyclical patterns in volatility. Every swing in the economy brings in new dynamics and patterns. Further research is needed to identify if these can be captured early using machine learning so that the volatility estimates are more accurate. - The mean absolute error and root mean squared error have been used to measure errors. A few more ways of measuring errors may be used to compare models. 121