The Performance of Put Option Writing,
Long Positions and Combinations
Master Thesis Finance
M.L. van der Vinne 26 June 2015
Table of Contents
INTRODUCTION 3
LITERATURE REVIEW 7
PUT OPTION RETURNS 7
OVERPRICED PUTS PUZZLE 9
Introduction
Quite a handful of research has been conducted in the direction of put option returns. Most of them analyzed the options on the S&P 500 Index, either to analyze the returns or to explain the extraordinary returns that have been found in the past. But unfortunately, the analyses on option returns on different indices than the S&P 500 index are sparse. Furthermore, there is a shortage on studies that compare the performance of put option writing directly to a long position, despite the fact that the payoff characteristics of those strategies are somewhat similar. When the underlying asset goes up in value, investors will receive positive payoffs can be expected for both a strategy that writes European vanilla put options as for a strategy that has a long position in the underlying. And when the underlying asset decreases in value, both strategies can expect a loss.
By comparing the different investment strategies this research takes a different approach than most studies that have been conducted on the subject of option returns. These studies namely focus more on explaining the returns or are determining the returns of a specific strategy, without comparing them to others. And since this research mainly applies the strategies on the AEX Index, also the data that will be used for this research differs from most studies. The AEX index is a relatively small index compared to for instance the S&P 500 Index and fewer data is available on the AEX index. Furthermore, this research will validate the different strategies by comparing the effectiveness of the strategies on the AEX Index to multiple other indices. These different focusses of this research makes that it is an unique combination and can therefore extent existing literature and add value to it. The results of this research could lead to new insights that could be beneficial for investors. It could help investors in determining the optimal portfolio and more insight in the actual trade-offs between options and underlying assets.
This research aims to make a clear comparison between the two strategies and should give more insight on what strategy will give the better results. In order to make this comparison, the following research question has bene formulated.
When do put option writing strategies outperform the long strategy?
Now, the different strategies and the functioning of the different aspects of the strategies will be discussed in more detail. Investing in the index is a straightforward strategy. The investor invests in the index at some point in time, and sells its shares later on, hopefully leaving a profit to the investor. When considering European vanilla put options, like this research, the buyer of the put option has the right to sell the underlying stock at the strike price at the maturity date. The writer, e.g. the investor that shorts the put option, has the obligation to buy the underlying stock for the strike price at the maturity date. The put option writer receives the put option premium for his obligation. The premium that the investor receives can then be reinvested. There are numerous possibilities on what the investor may do with the premium, but to prevent the research from becoming too complex, only two possibilities on what to do with the option premium are taken into account: investing the premium in the risk-free rate and investing the premium in the underlying stock. This leads eventually into three different strategies:
Strategy 1: Go long in the index;
Strategy 2: Write put options on the index and invest the premium in the risk-free rate;
Figure 1
The payoff figures of the different strategies
European vanilla put options come in various shapes and one of the variables that is involved in a put option is the strike price. The strike price of a put option is the amount of money that the holder of the put option receives if it exercises its option. This research will distinct several strike price categories for obtaining the returns of the options. These will be categorized in the difference in percentage with the actual value of the underlying. The strategy of choosing a strike price as a function of the actual spot price can be called an unconditional strategy (Stotz, 2011). In this manner, several categories can be created for various strike prices. In this research strike prices ranging from strike prices that equal 80% of the index value to 115% of the index value will be used, with increments of 5%.
Another variable that is involved in put options is the maturity. The maturity date of a European put option is the date at which it can be exercised. This research will focus on one-month hold-to-expiration option returns and because the options expire the third Friday of each month, the options will have a time-to-maturity of 28 days or 35 days, depending on the month.
R
e
tu
rn
Underlying Asset
Short put + Risk-free
Short put + Long
Long
To understand the strategies better, the Greek letters are studied. The two Greek letters that are discussed in this section are delta and gamma. Delta is defined as the rate of change of the option price with respect to the underlying. (Hull, 2012) For put options, the delta is always negative and lies between minus one and zero. This means that when the underlying security increases in value, the option price will decrease in value. But since this research focuses on writing put options, the delta is positive and therefore lies between zero and one. The relation between the underlying and the option price is not linear. As a short put option gets further in the money, delta approaches one, while as a short put options gets further out of the money, delta approaches zero. The delta of a long position on a security is always one, as when the underlying changes, the price changes with the same percentage. Gamma is defined as the rate of change in delta with respect to the underlying. (Hull, 2012) When gamma is small, the delta changes slowly, but when the delta is highly negative or positive, delta is very sensitive to the price changes in the underlying. Generally, the highest values for gamma of put options are when the price of the underlying is near the strike price of the put options. For deep out-of-the-money or deep in-the-money put options, the gamma approaches zero, as delta is approaching zero ore one. The gamma off a long position on a security is equal to zero, as the delta of constant and equal to one.
Literature review
The literature can be separated into four different parts. At first, a general overview is given on available literature that discusses put option returns. These studies empirically analyzed the option returns and extraordinary returns are often found. Secondly, a brief overview of the so-called overpriced puts puzzle is given. Many studies have tried to solve the overpriced puts puzzle, which is the finding that historical prices of put options have been too high and incompatible with the widely accepted asset-pricing models. (Bondarenko, 2003) This will give a better idea on where the extraordinary put option returns may come from. Consecutively the theoretical framework is described. The different measure metrics such as the excess return and the Sharpe ratio are included in this part. Also, the theoretical return on put options are determined. When each part of the literature review is discussed, some concluding remarks are made and the outcomes of the literature review are interpreted in the light of this research.
Put Option Returns
There are a couple of researchers that have analyzed the returns options. Large returns could have been made in the past by simple strategies that involved put option writing. Below, the findings of these researches are discussed.
Jones (2006) for instance has analyzed daily put option returns of the S&P 500 Index between 1987 and 2000 using a nonlinear multifactor model. He found that deep out-of-the-money put options have statistically significant alphas when compared to his factor model. Excellent Sharpe ratios are obtained by implementing simple put-selling strategies, both in- and out-of-sample. He also found that the priced factors do contribute to the expected returns but are insufficient to explain their magnitudes, particularly for short-term out-of-the-money put options.
Santa-Clara and Saretto (2009) come to similar results when analyzing the returns on S&P 500 Index option portfolios traded between January 1985 and May 2001. Portfolios that existed of naked and covered positions in put options, delta-hedged puts, and combinations of calls and puts, such as straddles and strangles have been considered in this research. It was found that the returns on selling of put option was not as large as stated in previous researches when the margin account was taken into consideration, but were still large and statistically significant in every metric. It was also argued by the authors that individual investors would have difficulties in achieving similar returns due to the margin requirements and possible margin calls.
combination with an investment in either the risk-free rate or the S&P 500 Index itself. All trading positions are closed out one month after investing. It was found that the strategies yield high returns in both absolute and risk-adjusted levels. The author argues that the most likely explanation for the high returns is mispricing of the options in the market.
In the research of Bakshi and Kapadia (2003), delta-hedged portfolios were constructed using long positions in put options and short positions in the underlying. The returns of these portfolios was calculated using data on the S&P 500 Index from January 1988 to December 1995. It was found that the underperformance of these portfolios is significant and robust.
Another study that have analyzed expected put option returns is conducted by Driessen and Maenhout (2004), where different strategies that involving S&P 500 index options were analyzed from 1987 to 2001. The strategies included out-of-the-money put options and at-the-money straddles. It was found that optimal portfolios for constant relative risk averse investors consist of a combination of short put options and long equity positions. Moreover, it was found that investors, regardless of their risk aversion, find it always optimal to short put options and straddles. Only when the loss aversion of an investor is combined with highly distorted probability assessments, it is possible to obtain positive portfolio weights for puts and straddles. The authors have done different robustness tests and therefore the option positions can be considered statistically and economically significant, even when to corrections for transaction costs, margin requirements and Peso problems are taken into account.
In another research on put option returns, Bondarenko (2003) has computed the monthly hold-to-maturity returns of the S&P 500 Index future options from August 1987 to December 2000. It was shown that trading strategies that involve writing at-the-money put options and out-of-the-money put options would earn high returns that are statistically significant and inconsistent with the single-factor equilibrium models. The average monthly excess returns of at-the-money put options and out-of-the-money put options are -39% and -95% respectively, while the Jensen’s alpha for at-the-money put options is approximately -23% per month. A positive relation was found between the option returns and the strike price of the options, i.e. the return on put options is increasing with the strike price of the put options. The results of the research of Bondarenko are robust to transaction costs, risk adjustments, peso problems and the underlying equity premium.
too low to be consistent with the Black-Scholes/CAPM model. They conclude that options in fact are not mispriced, but that the evidence implies that additional priced risk factors need to be taken into account.
Another study that analyzed the returns of option writing strategies is the study conducted by Bollen and Whaley. (2004) Data was used on the S&P 500 Index from June 1988 to December 2008. The strategies involve option writing and then delta-hedge the position using the underlying security of the option. These strategies yield significant returns when they were applied on an index, but not if they were applied on individual stocks. It was found that the strategy which yielded the highest profit contained short positions in out-of-the-money put options. Also when considering only options, the authors state that writing out-of-the-money put options yielded significant abnormal positive returns Broadie, Chernov and Johannes (2009) have analyzed the returns of monthly hold-to-maturity returns of the S&P 500 Index options between August 1987 and June 2005. Constructing long positions in put options it was found that the average monthly returns for out-of-the-money puts were -57% while at-the-money puts on average yielded -30% every month. Both results were statistically different from zero using t-statistics as the probability-values were close to zero. In their research, Broadie, Chernov and Johannes also compared their statistics in the sub-sample to the results of Bondarenko (2003). The results of both studies are very similar but despite the similarities also some differences were found. The results of Broadie, Chernov and Johannes were slightly more negative than the results of Bondarenko. The only exception to this is that the deepest out-of-the-money category in the study of Bondarenko yielded a lower return. It was also found that the average put option returns were unstable as they were extremely negative in the late 1990s during the dot-com bubble, but were positive and large from late 2000 to early 2003. Therefore the authors advised to use longer sample periods when investigating option returns.
Overpriced Puts Puzzle
Besides his analysis on daily put option returns of the S&P 500 Index between 1987 and 2000 using a nonlinear multifactor model as discussed in the previous section, Jones (2006) also tries to explain the realized option returns. It was found that two- and three-factor models, where the first two factors can be interpreted as market and volatility factors, are most successful in explaining the expected and realized option returns. Further it was found that volatility risk and possibly jump risk are priced in the cross section of index option but that these systematic risks are insufficient to explain the average option returns. Moreover it was found that short-term deep-out-of-the-money put options seem to be overpriced when comparing them with other options, with excess returns that regularly reached 0.5% per day. Furthermore the authors argue that some unknown state variables might solve the overpriced put puzzle.
Another study that have tried to explain the returns on the S&P 500 Index options was conducted by Cao and Huang (2007). The authors have analyzed common factors that affect the returns on the S&P 500 Index options. Daily data from June 1988 to May 1994 on the S&P index options was used to calculate the daily returns on options with a constant moneyness and maturity. It was found that on average 93% of the total variation in option returns can be explained by only three factors. 87% of the total variation can be explained by a factor that represents the underlying security while two volatility factors can explain 4% and 2% of the total variation in option returns.
Garleanu, Pedersen and Poteshman (2009) have observed that proprietary traders and customers of brokers have a net long position in the S&P 500 Index options with large net positions in OTM puts while dealers have negative net positions in index options. Therefore the authors suggest the demand to options could help explain the overpriced puts puzzle. The authors conclude that the high prices of index options and its skew patterns can partly but not completely be explained by the demand.
Besides the analysis that Coval and Shumway (2001) made on expected option returns, they have also studied the reasoning behind the expected option returns. Therefore the authors determined the daily returns of options on the U.S. Treasury bond, Eurodollar, Nikkei 225 Index and the Deutsche Mark using the closing prices from October 1988 to August 1999 and in case of the Nikkei the closing prices from September 1994 to August 1999.The assumption was made that if the only systematic volatility in the economy is market volatility, then only assets with volatilities that are positively correlated with that of the market should earn a risk premium. The results that were found were interpreted by the authors that market volatility plays a role in pricing options.
S&P 500 Index option prices. Three different proxies are used for investor sentiment. For the first investor sentiment proxy the author uses a popular sentiment index based on weekly surveys of approximately 150 investment newsletter writers. Then, each newsletter is marked as bullish, bearish or neutral based on the expectations of future market movements. The second proxy is based on trading activity in S&P 500 Index futures. The proxy is calculated by scaling the non-commercial contracts minus the number of short non-commercial contracts by the total open interest in S&P 500 futures. For the third investor sentiment proxy, the Sharpe’s (2002) valuation errors of the index are used. This proxy consists of the log dividend payout, the residuals of the log price-earnings ratio of the S&P 500 Index regressed on earnings growth expectations, the inflation expectations and several other variables. It was found that when investors are more bearish, they would have a stronger demand and be willing to pay more for options that pay off when the index level is low. As the results still hold after taking a set of rational factors that may be related to the sentiment proxies and variables related to index risk-neutral skewness are taken into account. Therefore, the author concludes that investor sentiment is an important determinant of index option prices.
Theory
This part of the literature research will focus on the theory that lies underneath the research topic. At first the different components of the different strategies will be briefly discussed and afterwards the equations regarding the returns of the different strategies will be determined.
As this thesis focusses on one-month hold-to-expiration short put returns, it is possible to specify the return of such put options. According to Broady, Chernov and Johannes (2009), the return of a put option is defined as
𝑟𝑡,𝑇𝑝 =(𝐾 − 𝑆𝑡+𝑇)
+
𝑃𝑡,𝑇(𝐾, 𝑆𝑡) − 1
(1)
where 𝑥+≡ max(𝑥, 0) and 𝑃𝑡,𝑇(𝐾, 𝑆𝑡) is the time-t price of a put option written on 𝑆𝑡, struck at 𝐾,
and expiring at time 𝑡 + 𝑇. By adjusting the equation above, it possible to define the return of a short put option. The return of a short put option is defined as
𝑟𝑡,𝑇𝑠𝑝 = 1 −(𝐾 − 𝑆𝑡+𝑇)
+
𝑃𝑡,𝑇(𝐾, 𝑆𝑡)
(2)
where 𝑥+≡ max(𝑥, 0) and 𝑃𝑡,𝑇(𝐾, 𝑆𝑡) is the time-t price of a put option written on 𝑆𝑡, struck at 𝐾,
and expiring at time 𝑡 + 𝑇. Please note that no term is included where the received option premium 𝑃𝑡,𝑇(𝐾, 𝑆𝑡) is reinvested.
𝑟𝑡,𝑇𝑖 = ln (𝑆𝑡+𝑇
𝑆𝑡 )
(3)
Where 𝑆𝑡+𝑇 is the spot price of the index at time 𝑡 + 𝑇 and 𝑆𝑡 is the spot price of the index at time 𝑡.
The last ingredient that is needed before the formulas can be determined that calculate the returns is the risk-free rate. However, the risk-free rate is considered to be equal to a long-term government bond and is therefore stated as 𝑟𝑓.
Taking the above into account, the performance of the different strategies as described in the introduction are as follows. For the long position in the index, the return is equal to eqution (3):
𝑟𝑡,𝑇𝑙𝑜𝑛𝑔 = ln (𝑆𝑡+𝑇 𝑆𝑡 )
(4)
For put option writing and investing the received premium in the risk-free rate, equation (2) needs to be combined with the risk-free rate. As a result, the first term of equation (2) will now be larger than 1, the risk-free rate will be added to the formula. The return for this strategy is now specified as follows:
𝑟𝑡,𝑇𝑠𝑝𝑟𝑓= (1 + 𝑟𝑓) −(𝐾 − 𝑆𝑡+𝑇)+ 𝑃𝑡,𝑇(𝐾, 𝑆𝑡)
(5)
And lastly, when put options are written on the index and the received premium is invested in the index, a comparable transformation is made as for equation (5). The return for this strategy can therefore be specified as follows:
𝑟𝑡,𝑇𝑠𝑝𝑙𝑜 = (1 + ln (𝑆𝑡+𝑇 𝑆𝑡 )) − (𝐾 − 𝑆𝑡+𝑇)+ 𝑃𝑡,𝑇(𝐾, 𝑆𝑡) (6) Theoretical returns
Coval and Shumway (2001) have also made an analytical analysis on the theoretical expected option returns. In order to derive the results, the authors make the assumption that a stochastic discount factor exists that prices all assets according to the relation described in equation 7.
E[R𝑖∙ 𝑚] = 1 (7)
Where E is the expectation operator, 𝑅𝑖 denotes the gross return on any asset and where 𝑚 denotes
price.”. Furthermore, it was stated that the theoretical expected put returns in theory can either me positive or negative depending on the strike price.
Sharpe Ratio
Another metric to measure the performance of strategies is the Sharpe ratio. The Sharpe ratio indicates what the excess return is per unit of risk in a strategy (Sharpe W. , 1994). The Sharpe ratio is defined as
𝑆ℎ𝑎𝑟𝑝𝑒𝑅𝑎𝑡𝑖𝑜 = 𝐸[𝑟 − 𝑟𝑓] √𝑉𝑎𝑟[𝑟 − 𝑟𝑓]
(8)
Where 𝐸[𝑟 − 𝑟𝑓] is the mean of the portfolio return minus the risk-free rate, and 𝑉𝑎𝑟[𝑟 − 𝑟𝑓] is the
variance of the mean of the portfolio return minus the risk-free rate. As the Sharpe ratio calculates the return per unit of risk, the higher the Sharpe ratio, the better the investment is considered to be. The Sharpe ratio is a very popular tool to compare different investment strategies. However, the Sharpe ratio can only be applied on returns that are normally distributed. When the distribution is skewed or has other abnormalities such as heavier tails, it can be problematic for the Sharpe ratio. (Ledoit & Wolf, 2008)
Conclusion
So how is this research affected by the literature research that has been conducted? And how do these results affect the expectations of the different strategies? When the implementing the strategies that this research considers on the S&P 500 Index, it can probably be said that the strategies involving put option writing would outperform the strategy that consists out of a long position on the Index, as the listed empirical literature showed extraordinary returns on put options on multiple samples. But as all the listed literature focusses on the S&P 500 Index, it is not certain whether the same holds for other indices like the AEX Index. This research focuses on the returns of the strategies when implemented on the AEX Index and therefore can determine the differences between the S&P 500 and the AEX to some extent. However, it is likely that the AEX Index shows approximately the same behavior as the S&P 500 index. Therefore it is to be expected that the strategies that include put option writing both will outperform the strategy that has a long position in the index only, specifically in terms of average monthly returns. Furthermore it is likely that the AEX Index will outperform the risk-free rate as in general investors are compensated for the additional risk on the AEX by receiving a higher return. This however may be depend on the sample period. In terms of Sharpe ratios this might lie a little different. It is doubtful whether the extra return, which is expected at the strategy that combines selling put options with a long position in the index, comes at the cost of a lower Sharpe ratio as the variance could relatively increase more than the expected return. The section of the literature review that describes different papers that try to explain the overpriced puts puzzle. Some factors that help explaining the high option prices were determined, such as the demand and preferences of investors. Even though it is uncertain whether the overpriced puts puzzle also exists for put options on the AEX Index, these findings may possibly explain the returns of the different strategies, especially when put option prices are indeed overpriced.
Data
Unfortunately, the amount of data is limited by the Datastream Database. There is only data available as of May 2006 on options on the AEX Index. Therefore, only data from May 2006 to May 2015 is used. Gathering the option data is a long and slow process. It was unfortunately not possible to select a specific category for an entire level of moneyness as the University of Groningen does not have the appropriate license for these types of data. Therefore, for each date, the options have to be manually selected. And because only options exist for certain strike prices and not for moneyness levels, two options, one above each moneyness strike price and one below, have to be selected for in order to calculate the option price for a specific level of moneyness later. This implicates that in total 1744 different put options have been manually selected, ranging over 109 different dates. For each level of moneyness the sample implies that it should have two times 109 observations. Unfortunately, the Datastream Database does not provide all the options that are needed to calculate the price for a specific level of moneyness. The only moneyness levels that are complete in terms of data are the moneyness levels of 95% and 100%. These option prices and corresponding strike prices are later transformed to implied volatilities, as interpolating on the implied volatility will obtain more accurate strike prices. For some options it was not possible to calculate the implied volatilities. For these options, the implied volatility is obtained directly from the Datastream Database.
The remaining option prices are calculated by interpolating the implied volatility of the options that have a strike price near the missing strike prices. The exact interpolation process is described in the methodology section. But to perform this interpolation the three closest option strike prices and their implied volatility have been determined using the Datastream Database.
Methodology
The methodology consists out of a couple of sections. At first the methodology that has been used to calculate the option prices of the different strike prices of the moneyness levels is described. Secondly, the methodology is describes that approximates the observations that are missing. Afterwards, the descriptive statistics that have been used to analyze the returns are stated. Then the methodology that has been used to validate the obtained results are outlined and lastly the justification section tries to explain why the used methodologies are appropriate.
Option Prices
The option strike prices that have been calculated using the historical AEX index values and a percentage of moneyness do not match with the strike prices as traded actively on the market. Therefore, the option prices have to be estimated using the strike prices that lie just above and just below the desired moneyness strike prices. An relatively accurate way of approximating the option prices, is by interpolating on the implied volatility. But since the data that has been gathered does not include the implied volatility, but only the strike prices and option prices, the implied volatility needs to be calculated. This has been done by using the BLSIMPV function in MATLAB, a function that uses the Black-Scholes model to compute the implied volatility of an underlying asset from the market value. The exact script that has been used is stated in the appendix. For the option prices that MATLAB did not calculate, the implied volatility is obtained directly from the Datastream Database. Then the implied volatilities are interpolated. The formula that was used for this transformation is stated in equation 9.
𝜎𝑁𝑃= 𝜎𝐿𝐵+ (𝜎𝑈𝐵− 𝜎𝐿𝐵) ×𝑋𝑋𝑁𝑃− 𝑋𝐿𝐵
𝑈𝐵− 𝑋𝐿𝐵
(9)
Where 𝜎𝑁𝑃 is the implied volatility of the new put option, 𝜎𝐿𝐵 is the implied volatility of the lower
bound put option, 𝜎𝑈𝐵 is the implied volatility of the upper bound put option, 𝑋𝑁𝑃 is the strike price
of the new put option, 𝑋𝐿𝐵 is the strike price of the lower bound put option and , 𝑋𝑈𝐵 is the strike
price of the lower bound put option.
When the implied volatility of the new put option has been determined, the Black-Scholes option pricing formula is used to make the transformation to the put option price. As the implied volatility and the other inputs that are needed for the Black-Scholes formula are now all known, it is possible to determine an option price. The Black-Scholes formula is stated in equation 10 to 12 (Hull, 2012).
where 𝑑1 =ln ( 𝑆 𝑋) + (𝑟 +𝜎 2 2 ) × 𝑇 𝜎√𝑇 (11) and where 𝑑2 =ln ( 𝑆 𝑋) + (𝑟 −𝜎 2 2 ) × 𝑇 𝜎√𝑇 (12)
Where 𝑃 is the put options price, 𝑋 is the strike price of the put option, 𝑆 is the current stock price of the underlying security, 𝑟 is the risk-free interest rate, 𝑇 is the time to maturity, 𝜎 is the implied volatility and 𝑁(𝑥) is the cumulative normal distribution function.
Missing observations
To correct for the missing observations, the option prices of the remaining missing options are interpolated using the implied volatility. The three nearest options in terms of strike price are used and their corresponding implied volatility. This data is the input for the Ordinary Least Squares method that estimates linear line in such a way that the errors are minimized. When the linear line is estimated, the implied volatility can then simply be obtained by filling in the strike price of the option in the equation as determined by the Ordinary Least Squares method. The line that is the output of the Ordinary Least Squares method is described in equations 13 to 16.
𝑉 = 𝐴 + 𝐵 × 𝑋 (13) where 𝐴 = 𝑚𝑒𝑎𝑛(𝑉) − 𝐵 × 𝑚𝑒𝑎𝑛(𝑋) (14) and where 𝐵 = 𝑅 ×𝑠𝑡𝑑𝑒𝑣(𝑋) 𝑠𝑡𝑑𝑒𝑣(𝑉) (15) where 𝑅 = 𝑁 ∑(𝑉 × 𝑋) − ∑ 𝑉 × ∑ 𝑋 √(𝑁 × ∑(𝑉2) − (∑ 𝑉)2) × (𝑁 × ∑(𝑋2) − (∑ 𝑋)2) (13)
When the implied volatility has been determined of the missing options, the same Black-Scholes transformation is made as described for the other option prices. These transformation is described in equation 10 to 12.
Descriptive Statistics
A variety of statistics are calculated in this research. The monthly returns are for instance calculated according to equations 4, 5 and 6. Then the mean, median, minimum and maximum monthly returns are calculated. Also the variance and the standard deviation are determined. To analyze the distribution of the returns, the skewness and the kurtosis are reported. Furthermore the t-statistic associated with a null hypothesis that the monthly returns have a mean return of zero is calculated just like the corresponding p-values. These statistics are also determined for a null hypothesis that the monthly returns have a mean return equal to the risk-free rate. Also, the Sharpe ratio of each portfolio is determined according to equation 7. These statistics are calculated for every strategy and for every moneyness level. Therefore, the descriptive statistics of in total twenty strategies are determined and reported. To test whether one strategy outperforms the other, Welch’s t-test has been used, the t-statistics are determined for the different indices with a null hypothesis of equal means and with an alternative hypothesis that the mean of one strategy is greater than the mean of the other strategy. To evaluate the robustness of the different strategies to some extent, the sample is cut in half and all the statistics mentioned above are again calculated. The first sample includes the first 55 observations while the second sample includes the last 54 observations. The idea behind this is that when the statistics for both periods show great similarities, the strategies are robust.
Validation
into some benchmarks that can easily be compared to the results obtained in this study. Comparing the results of this research to the obtained results of the DAX index and the Eurostoxx index may validate the correctness of the research while the results on the S&P 500 index can easily be compared to the literature that was found on option returns and may show that the appropriate methodology was applied. Furthermore a comparison of all options
Justification
The first method of estimating the option prices is using the price of one option that has a slightly higher strike price and the price one option that has a slighter lower strike price. Interpolating linear on the implied volatility is a commonly used method and will give only small errors, therefore the option prices are transformed to implied volatility. The implied volatility of the new option is then based on a linear relationship between these data points. Inserting the determined implied volatility along with the other variables in the Black Scholes formula is a widely accepted method of calculating option prices. This approach therefore can deliver option prices with respectable accuracy, especially for the option prices that lie near 100% moneyness, as there are often option prices available where the increments between the strike prices are very small.
For approximating the remaining options prices, the implied volatility of neighboring options in terms of strike price have been used to estimate the implied volatility of the missing options. Due to the volatility smile, a linear approximation of the implied volatility will always be an underestimation. This would result in underestimated results as this leads to lower options prices. Therefore this approach is considered to be safe as it is not boosting the results. Again, the implied volatility is used for the interpolation rather than the option prices itself as this method delivers more accurate option prices. Inserting the determined implied volatility along with the other variables in the Black Scholes formula is a widely accepted method of calculating option prices.
Results
Strategy 1
In the first strategy, the investor has a long position on the AEX Index. The results of the strategy are summarized in table 1. It can be seen that the strategy yields an monthly return with a mean of 0.02%. The T-statistic and the corresponding P-value of this strategy indicate that the null hypothesis of a monthly mean return equal to zero cannot be rejected. The same holds for the null hypothesis of a monthly return equal to the risk-free rate. The negative Sharpe Ratio of -0.026 simply indicates that the risk-free rate does outperform this strategy in this sample in the sample. The robustness has been evaluated as the sample is cut in half and the statistics are again calculated. Due to the largeness of the table, the results for this strategy can be found in table 8 in the appendix. The first sample yields a negative mean of minus -0.60 while the seconds sample yields an average monthly return of 0.66%. This difference is mainly due to the crash in 2008.
Table 1
The different performance and descriptive statistics of the first identified strategy: A long position in the AEX Index
Statistic Result Mean return 0.02% Median 1.37% Minimum -41.45% Maximum 13.65% Variance 0.5% Standard Deviation 7.3% Skewness -1.9 Kurtosis 8.7
T-Statistic zero mean 0.04
P-value zero mean 0.97
T-statistic rf mean 0.27
P-value rf mean 0.79
Strategy 2
Table 2
The different performance and descriptive statistics of the second identified strategy: shorting put options and investing the received premium in the risk-free rate. The statistics have been calculated for different levels of moneyness ranging from 80% to 115% with increments of 5%
Statistic Result Moneyness 80% 85% 90% 95% 100% 105% 110% 115% Mean return 45.86% 60.93% 12.46% -42.40% -11.53% 3.47% 4.22% 3.04% Median 100.22% 100.22% 100.20% 100.16% 100.11% 33.44% 18.62% 12.96% Minimum -5824.05% -3109.46% -2728.00% -2259.98% -790.59% -484.82% -309.73% -216.75% Maximum 100.43% 100.43% 100.43% 100.43% 100.43% 100.40% 100.40% 97.91% Variance 3219.9% 1044.8% 1858.4% 1801.4% 408.7% 103.1% 41.1% 20.1% Standard Deviation 567.4% 323.2% 431.1% 424.4% 202.2% 101.6% 64.1% 45.1% Skewness -10.4 -9.2 -5.1 -3.4 -2.1 -1.7 -1.4 -1.2 Kurtosis 109.0 88.5 26.0 11.9 4.1 4.4 4.4 4.4
T-Statistic zero mean 0.84 1.97 0.30 1.04 0.60 0.36 0.69 0.68
P-value zero mean 0.40 0.05 0.76 0.30 0.55 0.72 0.49 0.50
T-statistic rf mean 0.84 1.96 0.30 1.05 0.61 0.33 0.65 0.65
P-value rf mean 0.40 0.05 0.77 0.30 0.55 0.74 0.52 0.52
Strategy 3
Table 3
The different performance and descriptive statistics of the third identified strategy: shorting put options and investing the received premium in the AEX Index. The statistics have been calculated for different levels of moneyness ranging from 80% to 115% with increments of 5%
Statistic Result Moneyness 80% 85% 90% 95% 100% 105% 110% 115% Mean return 45.67% 60.74% 12.27% -42.59% -11.72% 3.28% 4.03% 2.85% Median 101.37% 101.37% 101.37% 101.37% 101.37% 34.10% 19.21% 13.54% Minimum -5865.81% -3151.22% -2742.80% -2269.58% -832.35% -526.58% -351.49% -258.51% Maximum 113.65% 113.65% 113.65% 113.65% 113.65% 113.65% 113.65% 111.34% Variance 3266.0% 1073.7% 1895.6% 1842.2% 433.9% 117.9% 50.9% 27.5% Standard Deviation 571.5% 327.7% 435.7% 429.2% 208.3% 108.6% 71.3% 52.4% Skewness -10.4 -9.2 -5.1 -3.4 -2.1 -1.7 -1.4 -1.3 Kurtosis 109.0 88.3 25.9 11.7 4.1 4.7 4.8 4.9
T-Statistic zero mean 0.83 1.94 0.29 1.04 0.59 0.31 0.59 0.57
P-value zero mean 0.41 0.06 0.77 0.30 0.56 0.75 0.56 0.57
T-statistic rf mean 0.83 1.93 0.29 1.04 0.60 0.29 0.56 0.52
P-value rf mean 0.41 0.06 0.77 0.30 0.55 0.77 0.58 0.60
Comparison
The first comparison that will be made is between the different short put writing strategies. When looking at average monthly returns of these strategies it can be seen that strategy 2 outperforms strategy 3 for all moneyness levels. The reason behind this is simple; the risk-free rate outperforms the AEX in the sample period. When looking at the variances it is obtained that the variance of strategy 2 is slightly lower than the variance of strategy 3 but still considerably high, especially for the in-the-money moneyness levels. Looking at the distributions of the strategies, it can be seen that the skewness of low moneyness levels of strategy 2 are slightly higher than the skewness of those moneyness levels of strategy 3 while for high moneyness levels this is just the other way around. The Sharpe ratios of the different strategies also differ. The fact that the risk-free rate does outperform the AEX in the sample period while the risk-free rate also has a lower variance than the AEX explains this difference. Both the lower variance as the higher returns of the risk-free rate result in a boost of the Sharpe ratio of strategy 2 compared with strategy 3. For almost every level of moneyness, the Sharpe ratio of strategy 2 is higher than the Sharpe ratio of strategy 3. The only exception is the moneyness level of 100%. Where investing the premium in the AEX seem to reduce the variance of the portfolio.
minus 11.53% while strategy 1 yields a slightly positive return equal to 0.02%. Therefore, the investor should prefer strategy 1 over strategy 2 with a moneyness level of 100% when just looking at the returns. However, due to the P-value of 0.28, there is actually no statistical evidence that strategy 1 actually outperforms strategy 2 with a moneyness level of 100%.
At lower moneyness levels of strategy 2, the average monthly returns increase and become as big as 60.39% for a moneyness level of 85%. Comparing this to the average monthly return of 0.02% which is the result of strategy 1, it is not strange that investor would prefer strategy 2 over strategy 1 when just looking at the returns. However, this high return comes at the cost of a high variance and therefore a high risk. Despite the high variance, attractive Sharpe ratios are found for low moneyness levels. The highest Sharpe ratio, which is equal to 0.193, was found at a moneyness level of 85%. Looking at table 4, it can be seen that the T-statistic is equal to 1.93 and the corresponding P-value equals 0.03. This implicates that strategy 2 with a moneyness level of 85% statistically outperforms strategy 1 with a significance level of 0.05. This is the only moneyness level that statistically outperforms strategy 1. So this strategy 2 with a moneyness level of 85% is considered to be better investment than strategy 1.
Table 4
The T statistic and the P-value for
Figure 2
The returns on strategy 1 in the sample period
Figure 3
The returns on strategy 2 for a moneyness level of 85%
Figure 4
The returns on strategy 2 for a moneyness level of 100%
Figure 5
The returns on strategy 2 for a moneyness level 115%
Validation
In order to validate the results to some extent, other indices have been analyzed. This has only been done only for at-the-money options due to the data availability. The results are summarized in table 5 while the correlations between the indices are stated in table 6. The t-statistic and corresponding P-value associated with a null hypothesis of equal means against the alternative hypothesis that the strategy applied on one index outperforms the other are reported in table 7.
As can be seen in table 5, the put option writing strategies of both the DAX Index as the Eurostoxx Index yield negative average monthly returns that lie in the same order of magnitude as the returns on the AEX. Even though the monthly average return on long positions on those Indices differs only up to 0.56%. Also in terms of Sharpe ratio, variance and other statistics as the T-statistic and Skewness, the results on the AEX show similarities with the results on the DAX Index and the Eurostoxx Index. The means of the returns of the different strategies for all indices are not statistically different from zero or the risk-free rate. The t-statistics, as reported in table 7, associated with equal means for strategy 2 and strategy 3 on the AEX, DAX and Eurostoxx are also considerably low, indicating that there is no statistical significant difference between the means of the returns of the strategies applied on these indices. Remarkably, the average monthly returns on the put option writing strategies on the S&P 500 Index are completely different. The put option writing strategies on the S&P 500 Index yield average monthly returns of greater than 13%. Not surprisingly, this number is comparable to the average monthly returns as described in the literature. It also can be noticed that strategy 3 outperforms strategy 2 when just looking at the returns of the strategies, as the indices both outperform the risk-free rate. The T-statistic and the corresponding P-value of a return equal to zero for all strategies applied on the S&P 500 Index lie around 0.8 and 0.4 respectively. Therefore it is not possible to reject the null hypothesis that returns are equal to zero.
Table 5
The different performance and descriptive statistics of the different strategies applied on different indices: the AEX Index, the Eurostoxx Index, the DAX Index and the S&P 500 Index. The statistics have been calculated for a level of moneyness of 100%.
Statistic Result
Index AEX Eurostoxx DAX S&P 500
Strategy Long SP+RF SP+Long Long SP+RF SP+Long Long SP+RF SP+Long Long SP+RF SP+Long
Mean 0.02% -11.53% -11.72% -0.08% -7.03% -7.33% 0.58% -5.33% -4.97% 0.44% 13.43% 13.81% Median 1.37% 100.11% 101.37% 1.34% 100.12% 101.34% 1.39% 100.12% 101.39% 1.64% 100.04% 101.64% Minimum -41.45% -790.59% -832.35% -25.07% -595.21% -616.90% -27.58% -937.72% -965.55% -28.85% -684.40% -700.28% Maximum 13.65% 100.43% 113.65% 13.23% 100.40% 113.23% 13.93% 100.40% 113.93% 12.35% 100.10% 112.35% Variance 0.53% 408.70% 433.93% 0.43% 322.66% 342.84% 0.46% 379.48% 401.88% 0.30% 294.95% 310.47% Standard Deviation 7.30% 202.16% 208.31% 6.58% 179.63% 185.16% 6.75% 194.80% 200.47% 5.50% 171.74% 176.20% Skewness -1.91 -2.1 -2.1 -1.0 -1.9 -1.9 -1.3 -2.5 -2.5 -1.9 -2.4 -2.4 Kurtosis 8.7 4.1 4.1 2.0 2.8 2.8 3.7 7.0 6.9 7.7 5.6 5.5
T-statistic zero mean 0.04 0.60 0.59 0.12 0.41 0.41 0.89 0.29 0.26 0.84 0.82 0.82
P-value zero mean 0.97 0.55 0.56 0.90 0.68 0.68 0.37 0.78 0.80 0.40 0.42 0.42
T-statistic rf mean 0.27 0.61 0.60 0.46 0.42 0.42 0.56 0.28 -0.26 0.36 0.79 0.80
P-value rf mean 0.79 0.55 0.55 0.64 0.68 0.68 0.58 0.78 0.80 0.72 0.43 0.43
Sharpe Ratio -0.03 -0.06 -0.06 -0.04 -0.04 -0.04 0.05 -0.03 -0.02 0.03 0.08 0.08
Table 6
The correlations between the four different indices.
AEX Eurostoxx DAX
Eurostoxx 0.96
DAX 0.53 0.35
Table 7
The T-statistics and (P-value) of the null-hypothesis that the strategy 2 and 3 of 100% moneyness applied both indices have an equal mean against the alternative hypothesis that one strategy outperforms the other. The strategies that are used are in the t-statistic are always compared to the
same strategy but then applied on the other index.
AEX Eurostoxx DAX
SP+RF SP+Long SP+RF SP+Long SP+RF SP+Long
Eurostoxx 0.17 (0.43) 0.16 (0.43) DAX 0.23 (0.41) 0.24 (0.40) 0.07 (0.47) 0.09 (0.46) S&P 500 0.98 (0.16) 0.98 (0.17) 0.86 (0.20) 0.86 (0.19) 0.75 (0.23) 0.73 (0.23)
Conclusion
This research aims to make a comparison between two fundamental investment strategies: going long in an asset and writing put options on the same asset. The payoff characteristics of these investment strategies are somewhat comparable. When the underlying goes up in value, both strategies yield a positive return and when the underlying decreases in value, a negative payoff can be expected. As the writer of put options receives a premium which can be reinvested, this research considers two options on where this premium can be invested: the risk-free rate and the underlying stock itself. Therefore, three different strategies are taken into account: Strategy 1, a long position in the index; Strategy 2 which combines put option writing with an investment in the risk-free rate and Strategy 3, an investment in the index along with put option writing. The research focuses on one-month hold-to-expiration European vanilla put options with a moneyness level ranging from 80% to 115% with increments of 5%. At the day of maturity, new put options expiring the following month will be sold. This implies that the investor that follows strategy 2 or three will have a short position in put options at all times. To evaluate and compare the different strategies, the Sharpe ratio and the returns are calculated. Also the T-statistics are determined to evaluate whether the strategies are statistically different from zero and the risk-free rate. This research mainly applies the strategies on data on the AEX from May 2006 to May 2015. Furthermore, the strategies are also applied on the Eurostoxx Index, the DAX Index and the S&P 500 Index using a moneyness level of 100% for the put option writing strategies.
the risk-free rate outperforms the index in the sample. Furthermore it is noticeable that there is a trend in the returns and moneyness levels: the higher the moneyness level, the lower the average return. Surprisingly there are two main exception to this trend, these are the moneyness levels of 95% and 100%, where both strategies yield serious negative monthly returns. When analyzing the t-statistics of the strategy 2 and 3, it is found that for every moneyness level of both strategies the null hypothesis that the mean returns are equal to zero and equal to the risk-free rate against an alternative hypothesis that the mean returns are not equal to zero and the risk-free rate cannot be rejected on a significance level of 0.05. However, on a significance level of 0.10, these null hypotheses can be rejected for both strategies with a moneyness level of 85%. The highest Sharpe Ratio is found at strategy 2 with a moneyness level of 85%. The put option writing strategies also seem to be the most robust, as in both samples these strategies yield positive results while this is not the case for other moneyness levels.
Comparing strategy 1 to the put option writing strategies, it can be seen that all put option writing strategies with other moneyness levels than 95% and 100% outperform strategy 1 in terms of average returns and Sharpe ratio. However, only at a moneyness level of 85% do both strategy 2 and strategy 3 statistically outperform strategy 1 on a significance level of 0.05. For all other moneyness levels the null hypotheses that strategy 2 or 3 has the same mean return as strategy 1 cannot be rejected on a significance level of 0.05.
Looking at the returns of strategy 2 and 3 for a moneyness level of 100% on the other indices, Eurostoxx, DAX and the S&P 500 Index, it can be seen that the returns on the European indices Eurostoxx and DAX are both negative and lie in the same order of magnitude as the AEX Index while the return on the S&P 500 Index yields considerable positive monthly returns. It is remarkable that that the returns of the same strategies applied on different indices, that are also highly correlated, differ so much. However, the T-statistic and corresponding P-value that have been determined imply that applying a put option writing strategy on the S&P 500 Index does not statistically significant outperform that strategy when applied on one of the European indices.
positive returns that are decreasing with the strike price. In that sense the findings of this research support theory and other empirical researches. However, there are also some findings that do not correspond with the literature. The most remarkable difference between the literature and this research are the returns for put option writing strategies with a moneyness level of 100% on different indices. The empirical literature mostly report positive returns for put option writing strategies on the S&P 500 index, just like this research. However, put option writing strategies applied on the AEX, Eurostoxx and DAX index yield negative results, even though the correlations with the S&P 500 Index can be fairly high. Especially the correlation between the DAX Index and the S&P 500 Index of 0.97 is considerably high. While the returns on the same sample period are substantially different, implying that the options are priced differently. Despite the statistic that put option writing strategies do applied on the S&P 500 Index does not statistically significant outperform these strategies when applied on the European Indices, the big differences are remarkable. Furthermore, the same holds for a moneyness level of 95%. Although this research does not evaluate other indices for this moneyness level, empirical research reports significant positive results whereas this research reports insignificant negative returns.
It is hard to tell where these similarities and differences come from. As the trading market is a global market, one may argue that the difference in option prices may appear due to approximately the same factors, such as demand, volatility factors and investors preferences. However, this is naturally uncertain as explaining the differences and similarities was not the focus of this research.
Limitations
difference between the scientific and the investors perspective. From an investors point of view, writing many put options may probably not be realistic due to the low demand for options deep out-of-the-money or deep in-the-money. Furthermore, offering many put options on the market may affect the option price. And when writing deep out-of-the-money put options, it may be impossible to yield the high returns on a great amount of money, as put option writing may receive only €0.05 per put option.
Future research
An interesting direction for this research would be to continue to focus on the difference in performance of put option writing strategies applied on different indices. The differences that have been found in this research, even though there are not statistically significant, are remarkable and elaborating on these differences more would be a nice direction for future research. Another direction for future research may be focusing on other strategies rather than one-month hold-to-expiration put options. There might be different results obtained for put options with a longer maturity, or for strategies where the options are closed out early.
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Appendix
Robustness tables
Table 8
The different performance and descriptive statistics of the first strategy, calculated for two different sample periods. Statistic Result First Sample Mean return -0.60% Median 1.17% Minimum -41.45% Maximum 13.65% Variance 0.77% Standard Deviation 8.78% Skewness -1.90 Kurtosis 7.5
T-Statistic zero mean 0.50 P-value zero mean 0.62 T-statistic rf mean 0.75 P-value rf mean 0.46 Sharpe Ratio -0.093 Second Sample Mean return 0.66% Median 1.65% Minimum -18.39% Maximum 9.09% Variance 0.29% Standard Deviation 5.42% Skewness -0.86 Kurtosis 1.6
T-Statistic zero mean 0.89 P-value zero mean 0.38 T-statistic rf mean 0.70 P-value rf mean 0.48
Table 9
The different performance and descriptive statistics of the second identified strategy: put option writing and investing the premium in the risk-free rate. The statistics have been calculated for different levels of moneyness ranging from 80% to 115% with increments of 5% and have been calculated for two different sample periods.
Statistic Result First Sample Moneyness 80% 85% 90% 95% 100% 105% 110% 115% Mean return -7.42% 41.93% -39.81% -73.41% -23.73% -5.41% -2.14% -0.88% Median 100.29% 100.29% 100.28% 100.25% 100.23% 29.13% 12.70% 8.32% Minimum -5824.05% -3109.46% -2728.00% -1888.70% -790.59% -484.82% -309.73% -216.75% Maximum 100.43% 100.43% 100.43% 100.43% 100.43% 100.40% 100.40% 97.91% Variance 6381.42% 1873.18% 3032.09% 2013.57% 507.43% 137.35% 55.37% 28.64% Standard Deviation 798.84% 432.80% 550.64% 448.73% 225.26% 117.20% 74.41% 53.52% Skewness -7.42 -7.42 -4.10 -2.84 -2.07 -1.76 -1.44 -1.15 Kurtosis 55.0 55.0 16.0 7.5 3.6 4.1 4.2 3.7 T-Statistic zero mean 0.07 0.72 0.54 1.21 0.78 0.34 0.21 0.12
P-value zero mean 0.95 0.48 0.59 0.23 0.44 0.73 0.83 0.90 T-statistic rf mean 0.07 0.71 0.54 1.22 0.79 0.36 0.24 0.16 P-value rf mean 0.94 0.48 0.59 0.23 0.43 0.72 0.81 0.87 Sharpe Ratio -0.010 0.097 -0.073 -0.166 -0.107 -0.048 -0.032 -0.021 Second Sample Moneyness 80% 85% 90% 95% 100% 105% 110% 115% Mean return 100.14% 80.27% 65.70% -10.81% 0.90% 12.51% 10.69% 7.03% Median 100.13% 100.13% 100.13% 100.12% 100.09% 36.63% 19.10% 13.01% Minimum 100.01% -972.32% -1759.42% -2259.98% -622.47% -275.48% -155.74% -105.92% Maximum 100.32% 100.32% 100.32% 100.32% 100.28% 100.18% 95.19% 63.67% Variance 0.00% 212.99% 640.36% 1599.04% 312.70% 68.56% 26.41% 11.98% Standard Deviation 0.06% 145.94% 253.05% 399.88% 176.83% 82.80% 51.39% 34.62% Skewness 0.48 -7.35 -7.35 -4.38 -2.18 -1.18 -0.59 -0.61 Kurtosis 0.6 54.0 54.0 20.7 4.4 1.6 0.9 1.0 T-Statistic zero mean 11500.68 4.04 1.91 0.20 0.04 1.11 1.53 1.49
Table 10
The different performance and descriptive statistics of the third strategy: put option writing and investing the premium in the AEX Index. The statistics have been calculated for moneyness levels ranging from 80% to 115% with increments of 5% and have been calculated for two different sample periods.
Statistic Result First Sample Moneyness 80% 85% 90% 95% 100% 105% 110% 115% Mean return -8.31% 41.05% -40.69% -74.30% -24.62% -6.30% -3.03% -1.77% Median 101.17% 101.17% 101.17% 101.17% 101.17% 29.82% 13.39% 9.01% Minimum -5865.81% -3151.22% -2742.80% -1902.24% -832.35% -526.58% -351.49% -258.51% Maximum 113.65% 113.65% 113.65% 113.65% 113.65% 113.65% 113.65% 111.34% Variance 6471.87% 1922.54% 3091.49% 2068.67% 541.31% 157.66% 69.04% 38.74% Standard Deviation 804.48% 438.47% 556.01% 454.83% 232.66% 125.56% 83.09% 62.24% Skewness -7.42 -7.41 -4.09 -2.82 -2.06 -1.78 -1.48 -1.25 Kurtosis 55.0 55.0 15.9 7.4 3.7 4.3 4.5 4.2 T-Statistic zero mean 0.08 0.69 0.54 1.21 0.78 0.37 0.27 0.21
P-value zero mean 0.94 0.49 0.59 0.23 0.44 0.71 0.79 0.83 T-statistic rf mean 0.08 0.69 0.55 1.22 0.79 0.39 0.30 0.25 P-value rf mean 0.94 0.49 0.59 0.23 0.43 0.70 0.77 0.81 Sharpe Ratio -0.011 0.094 -0.074 -0.165 -0.108 -0.052 -0.039 -0.032 Second Sample Moneyness 80% 85% 90% 95% 100% 105% 110% 115% Mean return 100.66% 80.79% 66.22% -10.29% 1.42% 13.03% 11.21% 7.55% Median 101.65% 101.65% 101.65% 101.65% 101.65% 37.88% 20.32% 14.23% Minimum 81.61% -990.95% -1778.06% -2269.58% -632.07% -294.11% -174.38% -124.55% Maximum 109.09% 109.09% 109.09% 109.09% 109.09% 109.09% 104.15% 72.57% Variance 0.29% 221.04% 654.10% 1625.20% 329.23% 77.64% 32.26% 16.02% Standard Deviation 5.42% 148.67% 255.75% 403.14% 181.45% 88.11% 56.80% 40.03% Skewness -0.86 -7.34 -7.34 -4.36 -2.16 -1.17 -0.62 -0.64 Kurtosis 1.6 53.9 54.0 20.4 4.3 1.6 1.0 1.0 T-Statistic zero mean 136.58 3.99 1.90 0.19 0.06 1.09 1.45 1.39
Excel file
The data analysis of this thesis has mainly been performed in Microsoft Excel. The entire excel file is available enclosed along with the final submission of this Master thesis.
MATLAB file
The MATLAB file that has been used to calculate the implied volatilities of the options is enclosed in a separate file. As the MATLAB file uses the Excel file, which has been restructured later on, the following references are different:
- The file ‘thesisMATLAB.xlsx’ stated in the MATLAB file refers to the Excel document, which is renamed to ‘Master Thesis Finance Excel M.L.van.der.Vinne.xlsx’.
