How to price an option is a question that seemed resolved since Fischer Black, Myron Scholes and Robert Merton introduced in 1973 their paper “The Pricing of Options and Corporate Liabilities” (Black & Scholes, 1973). Many people thought that it was possible to predict the price of an option with the Black-Scholes formula. The Black-Scholes formula uses several assumptions. The assumptions made by the Black-Scholes formula are the following, there are no dividends paid during the life of the option, the options are European options thus can only be exercised at expiration, markets are efficient, there is no commission paid, the risk free rate is known, the volatility is known and constant and the underlying follows a lognormal distribution. After studying this method it seemed interesting to research if it is really possible to determine the price of options with use of the Black-Scholes formula. It is interesting to research this because for some of the assumptions made in the Black-Scholes formula there are some points of discussion. The main point of discussion is that the volatility is constant and another point of discussion is that the underlying follows a lognormal distribution. That the volatility is constant is a point of discussion because since the Wall Street crash in 1987 it is known that the volatility isn’t always constant. (Haugh, 2009) Another point of discussion by pricing put options with the Black-Scholes formula is the assumption that investors are risk neutral (Sundaram, 2008). In the theory about people’s risk behavior it is certainly not clear that all people are risk neutral. The reason for people to be risk averse is that people find losses more powerful than gains. (Ert & Erev, 2010) It is thus discussable if the Black-Scholes formula can determine option prices, because they use some assumptions that can be discussed.
This is not a discussion of recent years, because since the day that the Black-Scholes paper is introduced there has been increasing evidence and doubts that the Black-Scholes formula has several flaws. By these thoughts it is interesting if the price of an option can really be determined with the Black-Scholes formula. So if the volatility is constant over put options with the same underlying but with different strike prices.
With the information provided above the following research question can be defined: is it possible to determine the price of a put option, with use of the Black-Scholes formula? This question will be answered for 3 months AEX put options in the period 2009 to 2012. With put options are here European put options meant and not American put options. Why there is chosen for these put options and period can be read in chapter three Data. For answering this research question is the following hypothesis used, H_0: there is no significance difference between the real time value and the theoretical time value and the alternative hypothesis is, H_A:there is significance difference between the real time value and the theoretical time value.
Before it is even possible to answer the research question there are some subjects where literature research need to be done. This is required to have enough knowledge of the subject, and gives you, as reader enough background knowledge of this subject. The literature research is done for subjects such as what is a put option, how a put option is priced, what is the Black-Scholes formula, what is a normal distribution and what risk aversion and volatility are.
After this introduction follows the theoretical framework of this research, in this theoretical framework are the subjects that are introduced in the introduction treated. First is explained what a put option is, as second is the Black-Scholes formula given and explained to make sure that it is clear how this formula works and what the assumptions behind this formula are, thereafter is explained what the normal, standard and lognormal distribution are and at the end of this chapter is explained what risk aversion and volatility are. In the third chapter is explained why there is data used, which data is used, why that particular data is used and with which programs this data is collected and analyzed. In chapter four is explained how the research is done, the first paragraph shows how the data is analyzed, the second paragraph shows the comparison between two variables and at the end of this chapter are the results of the chi square test shown. With the results of chapter four is in chapter five a conclusion made, with this conclusion is an answer on the research question given. Chapter six is the discussion chapter of this thesis, in this chapter are possible explanations for the results given. In addition, there is also a restriction of this research and a suggestion for a follow-up research given.
2. Theoretical framework
In this chapter are the subjects, that are introduced in the introduction, one by one treated to make sure that these important definitions and theories are clear. This is done to understand the subject of the research question better. The first two subjects in this chapter explains what a put option is and how a put option premium is established. The third subject explains what a normal, standard normal and a lognormal distribution are. The last subject explains what risk aversion and volatility are.
2.1 Put options
For clarity as stated in the introduction this research looks at European put options and not at American put options. The difference between these two types of options is that European options could only be exercised on expiration and American options could be exercised at any time before expiration. (Merton, Brennan, & Schwarts, 1977) European put options are used for this research because this is one of the assumptions made in the Black-Scholes formula. (Black & Scholes, 1973)
A put option is an instrument of the financial markets. In the financial markets there are a couple of instruments available which provide payoffs that depends on the values of other assets, like commodity prices, bond and stock prices and market index values. For this reason these instruments are called derivative assets, because their values derive from the values of other assets. (Bodie, Kane, & Marcus, Derivative Markets, 2014) This is also the case by a put option.
In the literature is a put option described as follows: “a put option gives its holder the right to sell an asset for a specified exercise price on a specified expiration date”. (Bodie, Kane, & Marcus, Derivative Markets, 2014) For example a July put option on a company X with an exercise price of €180 gives the owner of the put option the right to sell the stock of company X to the put writer at a price of €180 on expiration date, even if the market price of company X is lower than €180. Thus the profit of a put option increases when the asset value falls. (Bodie, Kane, & Marcus, Derivative Markets, 2014) To make this clearer, a figure with explanations is added below in figure 1.
Figure 1, put option
For a put option it is possible to be in the money, at the money and out the money. A put option is in the money when the present spot price is less than the strike price, this is in figure 1 shown on the left side of the arrow on the bottom. A put option is at the money when the present spot price equals the strike price, this is shown by the arrow on the bottom in figure 1. A put option is out of the money when the present spot price exceeds the strike price, this is in figure 1 on the right side of the arrow on the bottom. (Madura, 2010)
2.2 Pricing of a put option
To know how an option and especially a put option is priced it is common to make use of the Black-Scholes formula. It is common to make use of the Black-Scholes formula because that is probably the best-known method of option pricing (Folger, W.D.) Beside that, this is also the only option pricing formula explained in the book “investments” (Bodie, Kane, & Marcus, Option Valuation, 2014), which is an indication of the importance of the Black-Scholes formula.
Before looking at the Black-Scholes formula it is important to explain some basic principles of a put option price. A put option price or a put option premium is primarily influenced by three factors, the difference between the spot price and the strike/exercise price, the time to maturity and the volatility of the underlying. The difference between the spot price and the strike/exercise price is called the internal value and is important because the lower the difference is, the more valuable the put option will be. This put option will be more valuable because there is a higher probability that the put option will be exercised. The time until expiration date influenced the put option premium because the longer the time to expiration, the greater the put option premium will be. This is also known as the time value or external value of a put option. The premium will be higher when the put option has a longer period because a longer time to expiration creates a higher probability that the underlying will move into a range where it is feasible to exercise the put option. The volatility influenced the premium of the put option because the greater the volatility is, the higher the probability is that the put option would be exercised. (Madura, 2010)
The Black-Scholes formula for calculating the premium of an option was introduced in 1973 in a paper entitled, “The Pricing of Options and Corporate Liabilities”. This paper was published in the Journal of Political Economy. The formula, developed by three economists Fischer Black, Myron Scholes and Robert Merton, is perhaps the world’s best-known options pricing model, this formula is shown on the next page. The Black-Scholes model is used to calculate the theoretical price of European put and call options, ignoring any dividends paid during the lifetime of the option (Folger, W.D.).
As introduced in the introduction there are several important assumptions and components for the Black-Scholes formula. The assumptions and components that are stated in the paper the pricing of Options and Corporate Liabilities are listed below. (Black & Scholes, 1973)
The assumptions made in the Black-Scholes model are:
The options are European and can only be exercised at expiration
No dividends are paid out during the life of the option
Efficient markets
No commissions
The risk-free rate is known and constant
The volatility of the underlying is known and constant
Follows a lognormal distribution that means that, returns of the underlying are normally distributed.
The components used in the Black-Scholes model are shown below in bold:
Current underlying price (S)
The current underlying price is the current price of the underlying stock.
Options strike price (X)
The options strike price is the price for which the holder of the put option the stock can sell to the writer of the put option at the expiration date.
Time until expiration (T-t)
The time until expiration is how long it takes before the expiration date is reached.
Implied volatility (σ)
Implied volatility is the estimated volatility of the security’s price.
Risk-free interest rate (r)
The risk-free interest rate is the rate for which you can borrow risk free.
Standard normal distribution (N)
The standard normal distribution is explained in the second part of paragraph 2.3
With the components explained above it is possible to use the Black-Scholes formula to price a put option with the formula that is shown below. The first formula is for pricing a put option, and to use this formula, d_1 and d_2 need to be calculated first.
p=Xe^(-rT) N(-d_2 )-SN(〖-d〗_1)
d_1= ln〖S/X)+(r+σ^2/2)T〗/(σ√T)
d_2= ln〖(S/X)+ (r-σ^2/2)T〗/(σ√T)
2.3 Normal distribution
One of the assumptions of the Black-Scholes method is that the returns of the underlying are lognormal distributed. So it is of great importance to know what a normal distribution, standard normal distribution and a lognormal distribution is.
Normal distribution
In the normal distribution are all the values plot in a curve. This curve has several characteristics. The total area under the curve is 1 (100%), the curve is symmetrical so that the mean, median and mode fall together and thus are the values equally likely to plot either above or below the mean, the curve is bell shaped, the greatest proportion of the values lies close to the mean, and almost all the values (99,7%) lie within 3 standard deviations of the mean. (Gordon, 2006)
Standard normal distribution
To say something over the normal distribution it is normal to use the table of the standard normal distribution. The standard normal distribution is a normal distribution with a mean of zero and a standard deviation of 1. By a standard normal distribution 68% of the observations lie within one standard deviation of the mean, 95% of the observations lie within two standard deviations of the mean and 99,7% of the observations lie within three standard deviations of the mean. (Studenmund A. , 2014)
Lognormal distribution
A lognormal distribution, also called a Galton distribution, is a probability distribution with a normally distributed logarithm. A random variable is lognormal distributed if its logarithm is normally distributed. (Baudewyn, 2010) Translated to the Black-Scholes formula this means that there is assumed that the returns on the underlying are normally distributed and thus that the prices of the underlying follow a lognormal distribution.
2.4 Risk aversion and volatility
In the introduction is introduced that there has been increasing evidence that the Black-Scholes formula has several flaws and thus that it is discussable that the Black-Scholes formula can determine the prices of put options. One of these flaws is that in the Black-Scholes formula is assumed that people are risk neutral and that the volatility is constant, these assumptions doesn’t match with the theory.
Risk aversion
In the Black-Scholes formula is assumed that people are risk neutral, but this is definitely not always the case. There are people who are risk averse, which follows the Cambridge dictionary means that these people are unwilling to take risks or want to avoid risks as much as possible (Cambridge, W.D.). According to the paper Anomalies: Risk Aversion, risk aversion can be explained by two concepts, loss aversion and mental accounting. “Loss aversion is the tendency to feel pain of a loss more acutely than the pleasure of an equal-sized gain” (Rabin & Thaler, 2001) Loss aversion is also incorporated in Kahneman and Tversky’s prospect theory, they stated that decision makers are roughly twice as sensitive to perceived losses than to gains (Rabin & Thaler, 2001) and that losses loom larger than gains (Kahneman & Tversky, 1979). It is therefore likely that the people, who are risk averse, fear a crisis or a sudden crash of a stock so much that they overrate the chance that this would happen. Mental accounting is also important for the explanation of risk aversion. The explanation in the paper Anomalies: Risk Aversion why mental accounting is important for risk aversion is as follows, “because small-scale risk aversion seems to derive from the tendency to assess risks in isolation rather than in broader perspective. If small-scale better-than-fair gambles were evaluated in broader perspective, people would be more likely to accept them. They would realize that by taking a series of such bets the gains would tent to outweigh the losses in the long run.” (Rabin & Thaler, 2001) So people’s tendency to assess risk in isolation ensures that they are losing the broader perspective and thus they are therefore more likely to become risk averse.
Volatility
By option pricing is volatility a variable that shows the extent to which the return of the underlying asset will fluctuate between now and the expiration date of the option. For the Black-Scholes formula is, as stated before, volatility or rather the implied volatility an important component. The implied volatility is a measure of the estimation of the future variability for the asset underlying option contract. As described before the assumption that is made in the Black-Scholes formula is that the volatility is constant, but since the Wall Street crash in 1987 it is known that the volatility isn’t constant. (Haugh, 2009) After research the terms volatility smile and volatility skew were found and these terms supported the thoughts that the implied volatility isn’t always constant. A volatility smile is a geographical pattern of implied volatility for a series of options that has the same expiration date.
When plotted against the strike prices, these implied volatilities can create a line that slopes upward on either end, this is why it is called a volatility smile. Based on the Black-Scholes option pricing theory a volatility smiles should never occur. Based on the Black-Scholes option pricing theory there will always occur a completely flat volatility curve. (Ross, 2015) This volatility smile was first introduced by Shimko in Risk 6, with the title Bounds of probability (Shimko, 1993). A volatility skew is almost the same as a volatility smile but then creates the implied volatility only a line that slopes upward on one end and not on both ends.
One of the most common explanations that explain the existence of a volatility smile is that implied volatilities are higher for strike prices far from the current underlying price because the Black-Scholes model assumes normally distributed returns. The value of these options is the result of the small chance that the underlying moves past these far from the current underlying price. However, the chance that these large moves from the current price happen are more likely than the normal distribution would imply. Thus the options far away from the current underlying have more value than the Black-Scholes formula ascribes to these options. (Wu, 2016) The existence of a volatility skew can be explained with the suggestion Rubinstein made in is his paper Implied Binominal Trees. In this paper is stated that the fear for a market crash is the main reason why the volatility isn’t constant (Rubinstein, 1994). The idea behind this is that deep out of the money put options would be nearly worthless if stock prices evolve smoothly. These put options are nearly worthless because the probability of the stock falling by a large amount, whereby the put option moves in a short time into the money, would be very small. But a possibility of a sudden large downward jump that could move the puts into the money, as in a market crash, would impart greater value to these options. Thus the market might price these options as though there is a bigger chance of a large drop in the stock price than would be suggested by the Black-Scholes assumptions. The result of these thoughts of people is a higher option price and thus a greater volatility than derived from the Black-Scholes model. The fear for extreme market crashes explains thus why the demand for out the money puts is higher than expected and why the volatility of out the money puts is higher. In figure 2 is a graph shown to provide some extra explanation and a graphical representation of the volatility skew and volatility smile.
Figure 2: Volatility skew and volatility smile
3. Data
To answer the research question it is necessary to use data. In this chapter is explained which data is used, why this data is used, how this data is used and with which programs are used.
For this research is the data of 3 month AEX put options from the period 2009 to 2012 used. The decision to look for AEX put options is made due to availability of the data and for being the best-known stock market index in the Netherlands, which is a country with a lot of trading activities. The decision to use the 3 months put options is made because these put options are very liquid. It is important that the options are liquid because the bid-ask spread is then smaller, so there is a small bid-ask spread for these options (Wyatt, 2011). Beside that, there was also enough data available for these put options, which made it possible to do a thorough research. The first reason to choose for the period 2009 to 2012 is that the financial crises already started, so the first shock of the crisis wouldn’t affect the research. The second reason is that there was enough data available.
The data is collected with data platform Datastream. “Datastream is a global financial and macroeconomic data platform covering equities, stock market indices, currencies, company fundamentals, fixed income securities and key economic indicators for 175 countries and 60 markets” (Datastream, 2017). This data platform was available at Radboud University. The exact data that is collected from Datastream are the opening price and the underlying price. The definition of the opening price in Datastream is the first traded price of the day and if an option is not traded on a specific day, the market price is used for the opening price. (Datastream, 2017)
Risk-free interest rate
For the risk-free interest rate is the three months Euribor rates chosen. These rates are chosen because with use of the Euribor rates it is easy to find risk-free rates with the same duration. In the paper on the use of risk-free rates in the discounting of insurance cash flows Ir. Gil Delcour stated that the biggest problem to use the Euribor as risk-free rate is that the Euribor is only available for maturities up to 1 year (Ir Delcour, 2011). To use the Euribor as risk-free rate in the Black-Scholes formula this isn’t a problem, because the put options have duration of three months so the three months Euribor need to be used and there are no rates needed with maturities longer than one year. To collect these rates the site of home finance is used . On this site it is possible to find for each individual day the historical 3 months Euribor rates. The course of the interest rates can be seen in appendix 3.
Chi square test and null hypothesis
To test if the data is significant the chi square test is used. This test determines whether there is an association between the variables, in other words it test if the variables are significant different from each other (Kent State University, 2017).This chi square test calculated a p-value, this p-value helps to determine the significance of the results. A small p-value, typically smaller or equal to 0,05 indicate strong evidence against the null hypothesis, so the null hypothesis will be rejected then. (Rumsey, W.D.) A null hypothesis is typically a statement of the values that the researcher does not expect. The null hypothesis attempts thus to show that there is no variation between variables if there is variation expected. The null hypothesis holds until statistical evidence reject the null hypothesis, the alternative hypothesis will be accepted then. The alternative hypothesis is a statement of the values that the research does expect. The notation used to specify the null hypothesis and the alternative hypothesis is “H_0:” and “H_A:” and these notations are then followed by a statement. (Studenmund A. , 2014)
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