Can Implied Volatility predict Stock Prices? By P.M. Vermeij Studentnr

Transcription

1 By P.M. Vermeij Studentnr

2 2 Abstract Implied volatility can be a helpful tool for an investor. It shows when expected risks are high or low. It is very interesting to see if implied volatility has some predictive powers and can create outperformance by interpreting implied volatility correctly. This paper will a research document on implied volatility. Implied volatility is explored through the VIX Index. The VIX Index will be investigated in the period to test the predictive powers of this implied volatility index. This will be done by setting up 3 models. The best model will be tested by using a simple but effective trading strategy.

3 3 Introduction The Credit Crisis has caused many problems to investors. Investors invest in different asset categories, because these asset categories such as equities, bonds and hedge funds. Investors can choose a combination of asset categories that fit their risk appetite. An investor with a high risk appetite will invest most of his portfolio in equities and less in bonds and hedge funds and an investor with a low risk appetite will invest more in bonds and hedge funds and less in equities. The expected return of equities is higher than bonds and hedge funds. The loss of value of equities is also expected to be higher than bonds and hedge funds. The value of securities dropped severely during the credit crisis. The value of equities of many companies dropped enormously. Investors of long only equity investment funds lost large sums of money. The Credit Crisis showed that the other asset categories such as corporate bonds and alternative investments did not perform as expected. In the problem description I will show that the different asset categories did not perform as well as they should have. Also implied volatility is introduced. With implied volatility I will try to improve the performance of a portfolio. This will be done by testing the forecast ability of implied volatility. It is my expectation that with implied volatility stock prices can the predicted. This Master thesis contains a problem description, a literature review, a model and a conclusion. At the end of this thesis I hope to prove that implied volatility can predict stock prices.

4 4 Part 1 Problem description This problem description will explain the advantages of implied volatility and the disadvantages of other asset categories than equities. Also what kind of implied volatility indices are available what are the best indices to use. As a reference for equity performance of the S&P 500 index is used. This index represents the 500 largest companies in the US. The S&P 500 Index dropped 56% from its high of 1, on October 9, 2007 to on March 12, Other equity indices dropped similarly. The S&P 500 index is an equity index representing 500 US companies based on market capitalization. All companies included in the S&P 500 are American listed companies with a minimum market capitalization of $ 3 billion and a public float of 50%. At the end of March 2010 the 5 largest Companies in this index were: 1. Exxon Mobil Corp Index Weight 4.86% Market Cap ($ Million) 336, AT&T Inc Index Weight 2.14% Market Cap ($ Million) 148, Johnson & Johnson Index Weight 2.10% Market Cap ($ Million) 145, Microsoft Corp Index Weight 2.03% Market Cap ($ Million) 140, Procter & Gamble Index Weight 1.99% Market Cap ($ Million) 138,012 For fixed income the performance of the Citigroup BIG Index is used. This Citigroup Broad Investment Grade Index (US Big) is the follow up of the Salomon Broad Investment Grade Index and is an American Bond Index. This index comprises treasuries, agency debt, corporate, Non corporate credits, Mortgage backed securities and asset backed securities. However this index does not include Commercial Mortgage backed securities, Inflation linked bonds and high yield bonds. The US Big Index increased in the period from October 2007 and March 2009 by 9.6%. This index can be used for diversified portfolio management. The S&P 500 equity index decreases 56% in the similar period and an investor would have loosed on his equity part and gained on his fixed income part. However when a loss looked at only a credit index is shown. Take for example the Barclays Capital U.S. Credit Index (Capital US Index). The Capital US Index was introduced by Lehman Brothers and is market value weighted inclusive accrued interest. The sectors that are

5 5 included in this index are Industrial, Utility, and Finance, which include both U.S. and non- U.S. corporations. This index dropped in the period October 2007 and October % and is therefore not a good alternative for equities. The Capital U.S. Credit Index consists of investment grade corporate bonds with a maturity longer than 1 year. When looked at the alternative investments these strategies did not show a good result either. Unless their extensive investment possibilities such as equity long short hedge funds which form a large part of the alternative strategies. A good example of an alternative strategy index is the Credit Suisse Tremont Hedge Fund Index (Tremont Index). The Tremont Index is an asset weighted hedge fund index which consists of hedge funds which a minimum capital of $ 50 million. These funds also have to have a track record of 12 months and audited financial statements. The Tremont Index uses the Credit Suisse / Tremont Database which consists of 5000 hedge funds. This index is rebalanced on a monthly basis. The Tremont index performed during %. For the period October 2007 and March 2009 the performance of the Tremont index was also -19%. As discussed investments in equities did not perform well during However a switch to other asset categories would not have brought an investor much more performance. Another idea is to hedge your investment portfolio against price drops. This can be done by buying put options or by writing call options. Also futures can be sold. Futures are available on most indices such as the S&P 500 index. However the question remains when to hedge a portfolio against price drops. If protection is bought and the price of securities increases the paid premium is wasted and performance will stay behind the market performance. Investors need an indicator that gives a signal when to hedge a portfolio or to unhedge a portfolio. If such an indicator is efficient it could generate outperformance. An efficient indicator could be the VIX Index. The VIX index is the implied volatility index of the S&P 500 Index. The VIX index is calculated through options with a remaining expiration time of at least 8 days. The weighted average remaining time of all options including in the calculations is 30 days. The VIX index

6 6 was created by the Chicago Board Options Exchange (CBOE) in 1993 and can be considered as the benchmark for stock market volatility. The calculation of the VIX index can be defined as: 2 2 i RT 1 ( ) 2 e Q K i 1 T K T i K 0 Where : is VIX or VIX x T is Time to expiration F is Forward index level derived from index option prices th K is Strike price of i out the money option; a call if K F and a put if K F i i i Interval between strike prices half the dis tan cebetweenthe strike on either side of K i K K i K i 1 K 2 i 1 F 2 K i : (Note: K for the lowest strike is simply the difference between the lowest strike and the next higher strike. Likewise, K for the highest strike is the difference between the highest strike and the next lower strike.) K R Q 0 First strike belowthe forward index level, F Risk freeint erest rate to expiration K The midpoint of the bid ask spread for each option with strike i K i The calculation of the CBOE VIX index was renewed in This renewed VIX index uses a wide range of strike prices. The old VIX index used only at-the-money options. The above mentioned new formula to calculate the VIX index derives expected volatility directly from the prices of a weighted strip of options. The old VIX index used an option model to calculate the VIX index. Another advantage of the new VIX index is that it uses options on the S&P 500 index and the old index uses options on the S&P 100 index. The disadvantage is that the

7 7 new index was introduced in 2003 and that the sample period is rather short. However the new index was calculated back to 1990 and will therefore give a sample period of almost 20 years which will be sufficient for the research in this paper. The VIX index can be regarded as the investor fear gauge. This means that when the VIX index rises the S&P 500 Index tends to fall and when investors have optimistic expectations the VIX index tends to drop. During 18 years the VIX index was moving between 10 and 86. For example during the Russian debt and LTCM crises the index moved at 45 at a high. During the 9/11 WTC terrorist attacks the index ticked 49 as the highest point. During the Credit crisis the VIX index jumped to on November 21, This was also the alltime high of this index. During economic flourishing period the VIX index dropped significantly. For example the all-time low of the index was 9.89 on January 24, 2007, almost 7 months before the highest point of the S&P 500 (October 2007). These examples showed that equity indices fall when volatility expectations are high and therefore shows that the VIX index is indeed the investor fear gauge. The next graph shows that the VIX index tends to be negatively correlated with the S&P 500 Stock index. Graph S&P 500 versus VIX

8 8 In my opinion the VIX index can be used as an indicator when to hedge a portfolio. Graph 1 shows that when the VIX Index is low equity prices tend to rise and when the VIX Index is high equity prices tend to fall. In this paper I want to use the equities as an asset class and to neglect other asset categories such as government bonds, credits and hedge funds. Other asset categories are not needed in a portfolio when the VIX Index is an optimal hedge indicator. In part 2 a literature research is made on volatility, implied volatility, equity prices, forecasting equity prices and trading strategies. The purpose of this part is to develop a method that can indicate a signal when an equity portfolio should be hedged by using the VIX Index. At the end of part 3 I will present several models that issue a signal when a portfolio should be hedged. These models will be tested by using regression analyses through the program Gretl. The models will be tested in the period In this period daily closing prices will be used of the VIX Index and the S&P 500 Index. The VIX Index and the S&P 500 Index will be used for their diversity. The S&P 500 Index includes 500 companies and is therefore a broad index which includes all sectors. Other US Major stock indices such as the Dow Jones Stock Index only contain 30 companies and is not well diversified. The VIX Index is used is this research because it is the volatility index of the

9 9 S&P 500 Index and these indices are closely linked and therefore good to use for research. Other volatility indices are available such as the VDAX index which is the volatility index of the German Dax index. The Dax index is also not well diversified because this index contains only 30 companies. Another advantage of the S&P 500 Index and the VIX Index is that these indices are quoted in US Dollars. After the defined models are created and tested in Gretl, the performance of the best model will be calculated. This calculation is done by a long short trading strategy. The signals when to long or short a portfolio are generated by the best model. In his researchpaper 20 th Century Volatility of UBS Warburg Dillon Read (December 1999) Alexander Ineichen gives a chronological review on volatility. He also tested the correlation between equity returns and volatility and found that implied volatility is negatively correlated with returns. He tested this on the Swiss Stock market because this data was the longest available. He found a negative correlation of at 99% significance level. His conclusion was that there was a weak relationship. He also found that implied volatility is not a leading indicator against returns but tends to lead market movements. Ineichen also concludes that volume on stock markets is a very important factor for implied volatility. In my research I will not use volume and other aspects that can influence implied volatility except for returns on the S&P 500

10 10 Part 2 Literature research In this literature research I will explore the empirical articles that I have found on implied volatility and trading strategies. According to most literature a buy and hold strategy works the best in portfolio management. It creates a higher performance and less risk in your portfolio. In this part I will explore the articles available on Implied Volatility and the available trading strategies. This part will be a basis for the model that will be created in the next part 3. Literature on Implied volatility Latane and Rendleman (1976) concluded in their article that 4 ingredients of the B+S model are directly observable and that the standard deviation cannot easily be found. The standard deviation must be used to calculate if an option is not under- or overpriced and not correctly priced. This standard deviation is based on historical standard deviations of the underlying stock. Latane and Rendleman (1976) derived the implied standard deviations (ISD) from the B+S model. These ISDs are the standard deviations of continuous price relative returns and are implied from call options. Latane and Rendleman (1976) assumed that these call options are price efficiently according the B+S model. The authors weighted these ISDs and used these outcomes in their paper. Latane and Rendleman(1976) tested their model on 24 companies of which the options were traded on the Chicago Board Options Exchange for 38 weeks. They concluded that their model was a better predictor of future standard deviations. The standards were highly correlated with the standard deviations obtained from price movements on the underlying equities. Their main conclusion was that the weighted implied standard deviations were a better predictor for future volatility than predictions based on historical volatility.

11 11 Latane and Rendleman (1976) compared the implied volatility with the historical volatility. The historical volatility is calculated using historical returns. The advantage of this method is that it is rather simple. The disadvantage of historical volatility is that all historical returns have the same weight in the calculation of the standard deviation and volatility. An event of one year ago has the same weight of an event that has happened yesterday. Therefore more complicated calculations processes were developed. Several models were developed to estimate a better volatility than the historical volatility. Many of these models were based on ARCH (Autoregressive Conditional Heteroskedastic) or GARCH (Generalised Autoregressive Conditional Heteroskedastic) models developed by Engle (1982). These models take into account volatility clustering and the mean reverting of volatility. Clustering and mean reverting are properties of volatility. These properties will be discussed later in this paper. When implied volatility is compared to historical volatility implied volatility is usual higher than a forecast based on historical volatility. There are three main causes why implied volatility is higher than predictions based on historical volatility. Markets are not perfectly priced on the short term and short term data do not try to explain the markets. To compensate this imperfectness investors are willing to pay a risk premium. Also investors are willing to pay a risk premium when call options offer an upside potential at a low price or put options offer insurance for a downside at a low price. The above mentioned theory that predict future returns can be tested with a similar method as Lamont (1998) tested the predictability of stock returns through dividends. The author suggested that dividend is a measurement of future earnings and future earnings are the basis of today s stock price and therefore the dividend is a forecast of tomorrow s stock price. Lamont (1998) found that current earnings and dividends have a predictive role. Lamont (1998) also found that mean reversion in stock prices plays a very important role. This mean reversion was seen as rational time-varying discount rates or as irrational movements in prices.

12 12 The normal way to define volatility is through the standard deviation of historical returns. This is of course the historic volatility and differs from implied volatility. The historic volatility is calculated from historic returns of stocks or indices. The disadvantage of this calculation is that all older data has the same weight in the calculation than more recent data. Therefore implied volatility is a better predictor due to it is calculated through option prices and expected future returns. Research by Poon and Granger (2002) showed that predictions based on implied volatility were higher than predictions based on historic volatility. Dumas et al (1998) tested some implied volatility functions and concluded that simpler is better. Dumas et al (1998) tried to prove that the B+S model was outdated and that newer models were better to estimate future volatility. The authors suggested that there are deficiencies in the B+S model due to the constant volatility. They also concluded that the typical pattern in implied volatility, the so called volatility smile, that arises from the B+S Model is not really a smile but merely a sneer. This sneer results from the fact that in-themoney and out-of-the-money options have higher implied volatilities than at-the-money options with the same time left to expiration. For this reason Dumas et al (1998) introduced the Implied Tree Approach and the Deterministic Volatility Function (DVF) Model in their research. These models were different from the B+S Model in a way that they anticipated on future volatility implied from option prices instead of using a model. The DVF model also disregarded outliers and was also very more complicated than the B+S Model. The conclusion of the authors was that simpler model worked better. They found that the B+S Model was a better model to use. In their tests the B+S Model provided a better Hedge Ratio. Also the DVF model produced prediction errors that grow in time. The conclusions from this article for my research is that the use of the B+S Model is sufficient and no other complex implied volatility models are needed to perform a better research. Schwert (1989) wrote an important article on Stock market volatility. The title of the article was also the most important conclusion of this article. Schwert (1989) tested in his article Why Does Stock Market Volatility Change Over Time? the relation of economic factors

13 13 with the stock market volatility (historic). These economic factors were PPI Inflation rates, Industrial production growth rates and monetary base growth rates. The author tried to find that volatility that arises from the mentioned economic factors was correlated with the volatility that aroused from the stock market. Schwert (1989) tried to forecast the stock market volatility with the help of the volatility of the economic factors. Schwert (1989) found no evidence for his research. However he found some evidence that financial volatility helps to predict macroeconomic volatility. But Schwert (1989) was not able unravel the volatility puzzle. Schwert (1989) found that the volatility of some economic factor were very high in the great depression period of , but he also found that none of these economic factors increased as large as the stock return volatility did. This volatility was up to three times higher in the great depression than in a normal situation. Schwert (1989) also found that volatility is higher during other recessions. Schwert (1989) also noted in his paper that Karpoff found in 1987 a positive correlation between trading volume and volatility. Schwert (1989) researched this correlation again and also found that when stock market volatility is higher when trading activity is higher. Finally Schwert (1989) found in his article that volatility and leverage were correlated. When operating leverage is higher also volatility tends to rise. An interpretation of Schwert (1989) was that operating leverage during recessions is higher than in normal economic situations. Schwert (1989) introduced herewith the Leverage effect. This effect will be explained later. Schmalensee and Trippi (1978) also based their article on the B+S model. Schmalensee and Trippi (1978) conclude that the B+S Model was very accurate but they also see that in the real world volatility is changing over time. The authors tested the constant volatility of the B+S Model against the behaviour of options series on six companies. These companies were Avon Products, Eastman Kodak, IBM, Texas Instruments and Xerox. The options of these companies were listed on the Chicago Board of Options Exchange (CBOE). Option prices of 56 weeks were used in the test. Schmalensee and Trippi (1978) tested if the constant volatility in the B+S Model was an item in the option prices of the CBOE. For this result they found no

14 14 evidence. However Schmalensee and Trippi (1978) found some other interesting relationships in their research. Schmalensee and Trippi (1978) also tested the relationship between expectations of investors, the relation between historic and implied volatility, the movement of the stock prices of the six companies and the implied volatility and the correlation between the volatility of the six stocks. The authors found no evidence that investors can predict prices on the basis of average figures of volatility. They also found no evidence that the historic volatility is correlated to the implied volatility. However Schmalensee and Trippi (1978) found evidence that investors can predict price movements with the knowledge that if implied volatility is high stock prices are likely to drop and if implied volatility is low stock prices are likely to rise. Schmalensee and Trippi (1978) also found that the movement of the implied volatility of the options of the six companies were closely correlated to each other. From their article the authors can conclude that macro economic factors influence implied volatility. Company news is not a mayor influencer of implied volatility. Another important paper for my research is Asymmetric Volatility and Risk in Equity Markets by Bekaert and Wu (2000). Bekaert and Wu (2000) investigated the volatility feedback. This effect takes the leverage effect a step further. The leverage effect is one of the explanations for the negative correlation between implied volatility and the stock prices. The authors claimed that the leverage effect is the reason for a rise in volatility when the stock price decreases due to a deterioration of the balance sheet of a company. In other words high volatility is according to the authors an anticipation of worse company balance sheet. Bekaert and Wu (2000) claimed that other studies on this issue did not come to the same conclusions. For example French, Schwert and Stambaugh found a positive relation between expected return and volatility. Turner, Startz, and Nelson came to the conclusion that this relation was negative. Bekaert and Wu (2000) created a model with the help of a GARCH model and used a riskless debt model for individual firms as done by Christie in This model tried to explain the

15 15 asymmetry in volatility and expected return on firm level. The authors called this the Leverage Effect and this model was based on the next table. Volatility Feedback News Firm Level Shocks Market Level hocks Market volatility rises Firm volatility rises correlation with Market rises Market ERP rises Not priced through CAPM method Firm ERP rises ERP = Expected Return Premium Market spreads rises Leverage Improves changes Firms share price declines Volatility Feedback Bekaert and Wu (2000) proved the function of the Leverage effect and volatility by researching stocks that were listed in the Nikkei 225 and they found support for the volatility feedback. They combine the CAPM Model with a GARCH model and construct four portfolios. The portfolios were the market portfolio the Nikkei 225 index, a high leveraged portfolio, a medium portfolio and a low leveraged portfolio. The authors proved in their model that the high leveraged portfolio drops in value when the implied volatility rises. Jeff Fleming (1998) used in his article implied volatility to predict future volatility. Fleming (1998) expected that implied volatility was an unbiased volatility forecast. However he found that implied volatility was not correlated with realised future volatility and therefore cannot be used to predict future volatility. The author s main conclusion was that implied volatility was very biased towards realised volatility. Fleming (1998) suggested that this was due to market

16 16 inefficiencies, but he could not prove that the bias was large enough to prove the existence of large trading profits. In his conclusion he suggested that implied volatility is very useful tool. The unpredictability of realised volatility could be used as an index of market sentiment, for evaluating asset pricing model and most importantly predicting stock market returns. Christensen and Prabhala (1998) also suggested in their article The relation between implied and realized volatility that implied volatility predicted the realized volatility. The authors suggested that if option markets operate efficient implied volatility is able to predict the realized volatility. In the literature study in the article of Christensen and Prabhala (1998) stipulated that the implied volatility is too biased to predict the realized volatility. However they found that implied volatility was less biased than expected due to the fact that the authors used a longer period for the implied volatility to examine. The authors used the implied volatility of the S&P 100 in the period November 1983 up to and including May According to Christensen and Prabhala (1998) implied volatility was more biased before a market crash than after the market crash. Christensen and Prabhala (1998) concluded in their paper that implied volatility did predict realized volatility on a monthly basis. This article is also interesting because the authors concluded that after the stock market crash of October 1987 implied volatility was a better predictor of realized volatility than before the stock market crash. The authors claimed that the reason for this was that the implied volatility of options on the S&P 100 had a greater errors-in-variable problem. Corrado and Miller (2006) checked in their article the relation between excess return and historical and implied volatility. The authors tested the relation between expected and realized excess return from the S&P 500 index from January 1995 to December To estimate the excess return correctly a Reward-to-Risk model of Merton (1980) was used. This Return-to- Risk estimator combined with historical or option-implied volatility was used to predict excess return for the S&P 500. These excess returns were compared to monthly realize excess returns.

17 17 A model portfolio was created based on research and contains stocks, bonds and money market funds. To estimate risk the CBOE Implied Volatility index was used. By using implied volatility and other risk measures Corrado and Miller (2006) calculate expected excess returns and compared these returns with the realized returns. There model was not adopted for the period January 1994 up to December However in the period 1994 up to 1998 the authors found a positive and significant relation between expected and realized excess returns when they used as a risk measure the option implied volatility. The authors found that stock market volatility was positively priced but they also concluded that this explained only a small portion to the total return variation. This outcome is very useful for my research. Harvey and Whaley (1991) tested in their article the usefulness of implied volatility. They expected to see that implied volatility was a good estimate of the market s expectation of the equity s future volatility. In their sample they used a 253 day sample period and calculated the implied volatility from one at the money put option and one at the money call option. In their conclusion comes forward that the implied volatility is not a good estimator of the future volatility. The authors claimed that there are three difficulties. These difficulties were the non simultaneous price problem, Bid / Ask Price effect and the infrequent trading of stocks. The non simultaneous price problem was that the stock market closed at 3:00 pm and that the option market closed at 3:15 pm. The closing prices were therefore difficult to compare. Due to the Bid / Ask Price effect, the closing price was sometime near the bid price and sometimes near the ask price. There was no pattern in this and therefore difficult to compare. Due to infrequent trading of certain stocks they may not have been traded near the closing of the market. Due to these problems Harvey and Whaley could not conclude that implied volatility was a good estimator to future volatility.

18 18 Literature on trading strategies Bushee and Smith Raedy (2005) found evidence that trading strategies delivered a profit in their article. These trading strategies were based on publicly known events, figures and other public information. They did prove that if restrictions were in place against short selling or interest percentages limits the trading strategies would not work. Also funds that have a larger diversified portfolio tended to outperform the smaller diversified portfolios. Bushee and Smith Raedy (2005) investigated 7 trading strategies in their article. These trading strategies are book-to-market strategy, the cash flow-to-price strategy, the return momentum strategy, the operating accrual strategy, the return reversal strategy, the size effect strategy and the post-earnings-announcement drift with a fourth quarter reversal. These strategies have been chosen because these were well documented. Bushee and Smith Raedy (2005) tested these strategies in the real world by constructing portfolios and testing the performance by real data. The authors used factors in their portfolios such short selling restrictions, maximum stake size, explicit trading costs and price pressures to block trades. The authors have collected daily data from January 3, 1990 until December 31, With this data a portfolio was constructed and buys and sells were created according to the trading strategy. Also corporate actions such as dividends and stock delisting were used to model the portfolio. The conclusions of this paper were that in the real world almost all strategies perform outperformance. Only the trading strategy with short positions did not perform outperformance, the long positions trading portfolios did perform outperformance. According to the authors this also was due to the increasing period of prices of the stock markets. The authors also found some other interesting facts in their tests. These were that equally weighted portfolios perform better than value weighted portfolios. They also find that portfolios with a larger number of companies performed better than portfolios with a smaller number of companies. Claessen and Mittnik (2002) concluded in their article that implied volatility was a useful tool to predict future volatility. However they concluded that although implied volatility contained a lot of information the efficient market hypothesis was applicable for their research. This

19 19 would suggest that no profits can be made by using Implied Volatility as a trading strategy. The authors investigated the possibilities using eight different methods to predict the implied Volatility of the Dax index options markets. The authors found also evidence that when using historical data this had no predictable information. Dajiang Guo (1999) tesed in his article the dynamic volatility trading strategies in the currency option market. Guo (1999) used currency options on the US Dollar and the German Mark for his model. These options were traded on the Philadelphia Stock Exchange. The author used a GARCH volatility method and an Implied Stochastic Volatility Regression (ISVR) model. Guo (1999) concluded that both models performed significantly before using transactions costs. Other remarkable conclusions were that investors that use Delta Neutral and Straddle trading strategies can lock-in considerable profits. Guo (1999) also concluded that ISVR models showed a better performance than using the GARCH model. This last conclusion is negative for my models. Chan et al (2006) did see the importance of volatility modeling for asset pricing, portfolio selection, option valuation and risk management. In their article they tested the benefits of using volatility for trading and hedging of portfolios. The article of Chan et al (2006) is very helpful in my research and the construction of the implied volatility model. Chan (2006) performed some in and out of sample tests based on several methods to construct volatility. The models were tested on several option trading strategies such as delta neutral straddles and option writing strategies. The authors found a significant average daily returns of 3.23% after adjusting for trading costs. It was also concluded that the combination of IV and ARFIMA models of realized volatility had the highest Sharpe ratios. Also it was concluded that this combination of realized volatility created the least risk exposure for financial institutions that write call options and delta hedge them. The article of Chan (2006) showed evidence that outperformance can obtained from using implied volatility in a portfolio. The conclusion of the literature investigation is two sided. Many authors find that the market and also the volatility work efficiently and that no performance can be obtained from the market information such as implied volatility. The articles of Claessen and Mittnik (2002),

20 20 Guo (1999) provided evidence for this. Other difficulties to create a good model were difference in time of data, biased data etc.. Also trading costs were important. If evidence is provided the outperformance was small and could not make good the transaction costs that were not included in many literature. However some literature for example from Chan et al (2006), Bekaert and Wu (2001) were more positive for my model. These articles are very useful for my model. In the next part I will explain how the model will be set-up and tested.

21 21 Part 3 Setting up the model In part 2 articles by Latane and Rendleman (1978) showed that using implied volatility is better than historical volatility. Schmalensee and Trippi (1976) found evidence that investors can predict stock prices by using implied volatility. Schmalensee and Trippi (1976) also found that the implied volatility of the equity of the six companies studied was highly correlated with each other and that company news does influence implied volatility. Another important article by Bekaert and Wu (2000) will be the basis for my model. They claimed that due to a leverage effect a rise in volatility will lead to a decrease of the stock prices and a drop in volatility will lead to a rise of stock prices. They studied the companies in the Nikkei 225. Christensen and Prabhala (1998) found out that implied volatility is more biased before a market crash than after a market crash. This implies that the implied volatility is higher before a market crash. Another conclusion of these authors is that implied volatility cannot predict realized volatility. Corrado and Miller (2006) concluded in their article that volatility was highly correlated with stock prices. However they found that this was only a small portion of the total return variation. And finally according to Chan et al (2006) using implied volatility in trading strategies could lead to substantial outperformance. There are also some negative aspects from the literature I studied using implied volatility and the predictable value of it. Engle (1982) concluded in his article that Implied Volatility tends to be higher than the realized volatility. Also many authors concluded in their investigations that markets operated efficiently. This would mean that no outperformance can be reached using implied volatility. Another problem for using implied volatility would be transaction costs. Little academic studies on implied volatility use transactions costs. Of course these transaction costs will influence performance negatively. Using implied volatility can create outperformance for an investor, because it can predict performances on the stock market. However this should be very well tested for a longer period because stock markets also tend to be efficient and almost no outperformance can be obtained. The goal of the model is to test if implied volatility can be a helpful tool to create outperformance. I will test if implied volatility can predict the movement of the stock prices.

22 22 The basic thought will be that when implied volatility is high a portfolio should be hedged and when implied volatility is low a portfolio should not be hedged. In this part a regression analysis will be made to see if differences of the daily implied volatility and the average volatility can predict stock prices. This would be a result of the leverage effect. I already showed that the implied volatility of listed stocks is highly correlated therefore I will use the daily performance of the S&P 500 index. The VIX index is the implied volatility index of the S&P 500 index and will also be used in or this model. The last price of this index will be used. If the VIX index is high than a market portfolio will be hedged and if the VIX is low than a market portfolio will not be hedged. To determine at what moment a market portfolio should be hedged the deviation of the average implied volatility is used. This average volatility will be calculated in three different ways. The first method used is the deviation of the average implied volatility for the whole period. The second method used is the deviation of the average implied volatility of the full year before the daily performance is estimated. A full year runs from January 1 until December 31. The third method used is the average deviation from a rolling period on 1 year before the daily performance is estimated. These three models will be tested in regression analysis. The best model will be simulated by using real data to determine in the real world if outperformance can be obtained by using implied volatility. A portfolio will be created to test the best model. This portfolio will be a simple long-short portfolio. If the daily implied volatility is higher than the average implied volatility Spiders on the S&P 500 are sold (short position) and if the daily implied volatility is lower than the average implied volatility than Spiders on the S&P 500 are bought (long position). In the portfolio Spiders on the S&P 500 are used to take a long or short position. Portfolio managers normally hedge a portfolio by using derivates such as futures or options. Using derivatives has some difficulties. First of all future and option data are not free obtainable. For example futures on the S&P 500 are listed on the Chicago Mercantile Exchange (CME).

23 23 Historic have to be purchased from the CME and the raw data that can be obtained is not easy to use. Derivatives also have specified contract month and expire at a certain date. At the end of a lifecycle of a future a position have to be rolled over or renewed to a new contract month. This is not very useful for a test for a longer period. Therefore in my research I will use Spiders. Spiders on the S&P 500 or SPDR s are short for S&P depositary receipt. These are Exchange Traded Funds (ETF) and are management by State Street Global Advisors. The value of a Spider is approximately 1/10 of the value of the S&P 500 but are an exact copy of this index. Spiders are listed on the American Stock Exchange (AMEX) and can be sold short and bought long. Another advantage is that Spiders provide regular dividend payments. Although these dividends are not incorporated in my investigation. The last advantage is that they can easily be bought by retail investors for low regular brokerage commissions. The period in which the regression and the portfolio performance will be calculated is a period from January 1, 1994 until December 31, The advantage for this longer period is that a number of recessions and periods of increasing stock prices are included. In my investigation daily observations will be used for this period. This period includes for example the fall of the LTCM fund (Russian Ruble crisis), the Internet bubble, and the Subprime crisis. Of course all recovery periods are also included. The regression analysis will be performed in Gretl. This is an open source statistical software package that is equal to Eviews and SPSS. The performance of the S&P 500 was in the period more than 94%. At January 1, 1994 the index was and on December 31, 2008 the index was The S&P 500 reached its highest position on October 9, 2007 at and its lowest position on April 4, 1994 at The VIX Index reached its lowest position on January 24, 2007 at 9.89 and its highest position on November 20, 2008 at At January 1, 1994 the Spiders on the S&P 500 (SPY) was worth $ and on December 31, 2008 these was worth $ This also means a performance of more than 94% (Dividend excluded). The SPY reached its highest position on October 9, 2007 at $ and its lowest position on April 4, 1994 at $

24 24 Model 1 This model will forecast the daily performance based on difference of the VIX Index and the total average VIX position on the day before. This regression analysis has a total of 3776 observations. I expect the model has some explanatory results, but it will not be very good. The average VIX index has been taken over the whole period of I would expect that the risk appetite of investors will change over time. Therefore this regression model will be least explanatory of my 3 models. The graph below will prove this. VIX Index In the table below the three first observations are given. SP500 is the % change of the closing of the S&P 500 of the day before. Vixdif is the difference of the closing of the VIX index of the day before minus the total average VIX number. In this model the total average VIX number is and this represents the average of all 3776 observations of the VIX index.

25 25 Date SP500 Vixdif The First SP500 observation of is the percentage change of 0.31% the S&P 500 of the day before. The closing of the S&P 500 Index was on January 3, 1994 and on January 4, 1994 making a percentage change of 0.31%. The Vixdif is the difference of the closing of the VIX Index of the day before minus the average VIX Index. The average VIX Index over the whole period is The first VIX Index closing was (January 3, 1994). Therefore the first Vixdif observation is If these data is analyzed in a regression analysis for the period , the following outcome is obtained. Model 1: OLS, using observations 1994/01/ /12/30 (T = 3776) Dependent variable: SP500 Coefficient Std. Error t-ratio p-value const Vixdif e e * Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F( ) P-value(F) Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson The regression model will be %Change S&P500 = Change VIX Index As expected this model is not very good. R-squared is only , which means that this model is only 0.08% descriptive the other 99.2% comes from other variables and residuals. The model in itself is performed well. No signs of autocorrelation. The graph of the residuals prove my conclusion

26 26 Regression residuals (= observed - fitted SP500) 0,15 0,1 0,05 residual 0-0,05-0, Model 2 Model 2 will be more explanatory than model 1 according to my expectations. The average VIX index will be more flexible in this model. Model 2 will use an annual average VIX index. The annual average VIX index runs from January 1 until December 31 the full year before the observation take place. For example: In 1995 every daily % change of the S&P 500 index is compared with the VIX Index of the day before minus the average VIX Index position of In the graph below the average VIX Index position is stated per year.

27 27 Average VIX index position per year For example the first three trading days of 1995 will give the following results Date SP500 Vixdif For 1995 the average position of 1994 (13.93) will be used. If a regression is made of this model, I expect a better forecast than model 1. This model has 3525 observations. The First SP500 observation of is the percentage change of -0.03% the S&P 500 of the day before. The closing of the S&P 500 Index was on January 3, 1995 and on December 31, The Vixdif is the difference of the closing of the VIX Index of the day before minus the yearly average VIX Index. The average VIX Index over the 1995 is The VIX Index closing was (December 30, 1994). Therefore the first Vixdif observation is The following outcome is obtained.

28 28 Model 2: OLS, using observations 1995/01/ /12/30 (T = 3525) Dependent variable: SP500 Coefficient Std. Error t-ratio p-value const Vixdif e e ** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F( ) P-value(F) Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson ,15 Regression residuals (= observed - fitted SP500) 0,1 0,05 residual 0-0,05-0, The regression model will be %Change S&P500 = Change VIX Index An unexpected outcome is received from the regression in model 2. The model turns out to be somewhat more explanatory than model 1. The Model fits excellent. However

29 29 this model explains only 0.12%. This is neglectable and is quite the opposite of my expectations. Model 3 Unless the unexpected outcome of model 2. I still expect a better outcome in model 3. Model 3 is an improvement of model 2. The annual average VIX index is replaced by a moving average VIX. The first observation will also be January 1, The VIX Index position of the December 31, 1994 will minus the average VIX index position of In my opinion this model incorporates much better the changed risk appetite of the investors. Therefore this should better predict the outcome of the daily change of the S&P 500 index. The moving average VIX Index will be a trend line for the actual VIX Index. The graph below will prove this. The first moving average VIX Index position is If a regression is made of this model, I expect a better forecast than model 1 or 2. This model has 3525 observations. The First SP500 observation of is the percentage change of -0.03% the S&P 500 of the day before. The closing of the S&P 500 Index was on January 3, 1995 and on December 31, The Vixdif is the difference of the closing of the VIX Index of the day

30 30 before minus the moving average VIX Index. The average VIX Index over the 1995 is The VIX Index closing was (December 30, 1994). Therefore the first Vixdif observation is The following outcome is obtained. Model 3: OLS. using observations 1995/01/ /12/30 (T = 3525) Dependent variable: SP500 Coefficient Std. Error t-ratio p-value const Vixdif e e *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F( ) P-value(F) Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson ,15 Regression residuals (= observed - fitted SP500) 0,1 0,05 residual 0-0,05-0, The regression model will be %Change S&P500 = Change VIX Index The result of the outcome of this model is very disappointing but is in line with

31 31 the outcome of model 2. Model 3 is a little bit better than model 2. However model 3 is only 0.23% explanatory according to the R-squared. After performing regression analysis on the three mentioned models we can conclude that the VIX index has no predictable value. The method that is used to predict the movement of the S&P 500 index in this investigation with the VIX index has no effect. The R-Squared in the three models that have been set-up and tested are far below 1% and are neglectable. On forehand it seemed suitable that the higher the VIX Index the more unsecure investors are and as a result of these stock prices will fall. My investigations proved that markets are efficient and that no extra performance can be obtained using the VIX Index. As mentioned before I will build a portfolio in which the VIX index will be used. This portfolio will use a long short strategy and will be based on model 3. Trading strategy & portfolio construction Despite the fact that models 1, 2 and 3 did not have any predictive power mode, model 3 will be tested on a portfolio. A long-short portfolio will be created and simulated with realized data in the period January 1, 1995 December 31, With this long-short portfolio I will try to beat the performance of the S&P 500 Index. The long-short portfolio will only exist out of Spiders on the S&P 500 Index. Spiders are ETF s on the S&P 500 and can be bought and sold. The VIX Index will also be used in the period January 1, 1995 December 31, The average VIX Index will be a moving average VIX Index. The moving average VIX Index will be created of 1 year back. The first average VIX Index observation will be calculated on the close of the VIX index between January 1, 1994 December 31, Every new average will be calculated based on a new latest VIX index observation and the first observation will be removed from the average VIX observation. The trading strategy will be either 100 shares Spider on the S&P 500 Index (SPY) long or short. If the actual VIX index is higher than the average VIX Index investors are expected to be bearish and a short position will be kept of 100 shares SPY. If the actual VIX index is lower than the average VIX Index investors are expect to be bullish and a long position will