Bachelor Thesis Finance

The accuracy of the Black Scholes model in

pricing AEX index call options:

Literature and empirical study

Name: Ali Ezzahti

ANR: S558900

Supervisor: Jiajia Cui

Date: 29 June 2007

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Table of Contents

Introduction 3

1 What determines the price of an option? 5

1.1 Main factors 6

1.2 Lesser known factors 7

2 How can options be priced? 7

2.1 Black Scholes Model 8

2.2 Binomial model 9

2.3 Risk neutral valuation 10

3 What if the assumptions of the Black Scholes model do not hold? 12

3.1 Volatility 12

3.2 Geometric Brownian motion 13

3.3 Constant risk-free interest rate 13

3.4 No transaction costs 14

3.5 Elasticity of variance and other properties 14

Validity of a pricing model 15

4 How large are the pricing errors with use of the Black Scholes model? 17

4.1 Whaley 17

4.2 Rubinstein 18

4.3 Dan Galai, Macbeth and Merville Rubinstein 19

5 How accurate is the Black Scholes model in pricing AEX index

call options? 20 5.1 In the case of historical volatility 20

5.2 In the case of implied volatility 23

6 What are the causes in the eventually pricing errors, and how can they be

reduced? 27

6.1 Dividend payments 27

6.2 Risk-free interest rate 27

6.3 Lognormal distribution 28

Conclusion 30

Appendix 31

References 33

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Introduction

Many investors invest in the AEX-index options. Forecasting the option price is a very

important issue for these investors.The option prices will reflect the "likelihood" of the

option finishing "in the money” (2).There are a lot of options, and also different kinds of

options. The prices of these options are determined by a lot of factors. In chapter 1 you

will get an insight in these factors and how important each factor is. Also there will be

showed which kind of important options there exist.

The factors that are mentioned in chapter 1 are used in option pricing techniques. There

are three important option pricing techniques. The Black Scholes model, Binomial model

and the Risk Neutral valuation model will be highlightened in chapter 2. The main

differences between these option pricing techniques will be mentioned, and there will be

concluded which of these three techniques is best to use.

The Black scholes model, the most popular and common used model, can be better used

to price options. But the option price you get with the use of the Black Scholes model is

not identical with the actual option price. A possible reason for this is that the

assumptions that the Black Scholes model make can not be true in reality. A lot of

researcher state that some of the assumptions are not true, and suggest other possible

ways to get a better option price. This will be considered in chapter 3.

The validity of the Black Scholes model has been researched by a lot of researchers.

Because you have different kind of options, this also leads to different kind of results

from these researchers. In chapter 4 there will be given the results of the empirical

research on the pricing errors that arise with the use of the Black Scholes model.

There has not been done research on the validity of the Black Scholes model in the

pricing of the AEX-index options. Volatility is an important factor in the pricing of the

AEX-index option. But researchers still not agree about which kind of volatility

(historical or implied) is best to use (1) In chapter 5 there will be done data analyze where

the validity of the Black Scholes formula will be tested. There will be showed how much

pricing error consists in the pricing of the AEX-index options with the use of Black

Scholes model. Further there will also be analyzed which kind of volatility is best to use.

These results shall be compared with the results of other empirical research discussed in

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chapter 3. How much are these results in line? In chapter 6 there will be finally analyzed

in which ways the pricing error errors can be reduced. The possible reasons for these

pricing errors will also be mentioned.

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1 What determines the price of an option?

Options are types of financial derivatives contracts, where the future payoffs to the buyer

and seller of the contract are determined by the price of the underlying. The underlying

can be the stock price, interest rates, currency rates etc. You have two kinds of options,

call options and put options. A call option is an agreement that gives the holder (owner)

the right (not the obligation) to purchase the underlying security at a predetermined price

called the strike price either at the call’s expiration date (European-style options) or

anytime during the life of the option (American-style options).

The value of an option consists of two components, the intrinsic value and the time value

(time premium). The intrinsic value is the difference between the exercise price of the

option (the strike price, K) and the current value of the underlying instrument (spot price,

S). The time value is the difference between the value of the option and the intrinsic

value. Time value derives from what might happen in the future. The option’s time value

captures the possibility that the option may increase in value due to volatility in the

underlying asset. The time value depends on the time until the expiration date and the

volatility of the underlying price. In time the time value declines and is zero at the

expiration date.

Figure 1.1 Value of a call option

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When the option does not have positive value, it’s referred as out the money. When an

option is out the money then it will not be exercised, so the option will never have a value

less than zero.

Value of call option = Max [ (S – K), 0 ] -c

Value of put option = Max [ (K – S), 0 ] -c

The price (c) that you must pay for the option declines the value of an option.

1.1 Main factors

There are many factors that influence the option price or option premium. The following

factors are the more influential and better-known factors that determine the option price:

• Price of the underlying; this influences the intrinsic value of the option.

• Strike price of the option; this also influence the intrinsic value of the option.

The price of the underlying and the strike price of the option do also influence the time

value of the option. All other factors being equal, the closer an option’s strike price is to

the price of the stock, the greater the chance the stock will move sufficiently to give the

option intrinsic value before expiration. Consequently this also gives greater time value.

The time value tends to have the greatest value with at- the-money options, while deeply-

in-the-money or deeply-out-the-money has the least time value.

• Time remaining before the expiration; the longer the time remaining, the greater the

time value. If there is a longer period ahead, there is also more time for the stock to

potentially reach a price level that would yield a significant profit for the option holder.

• Volatility of the underlying; with high volatility the probability of the option ending in-

the-money is greater. Higher volatility implies also greater time value.

• Dividend payout of the underlying (if there is a payout); as dividends rise, prices

decrease for calls and increase for puts.

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• Risk-free interest rate; if the interest rate increases then the price of call options

increases and the price of put options decreases. The risk-free interest rate has the

opposite influence to that of dividends on option prices.

1.2 Lesser known factors

The following factors are the lesser-known factors that can determine the option price:

• Skewness; this is a measure of symmetry, or more precisely, the lack of symmetry. A

distribution, or data set, is symmetric if it looks the same to the left and right of the centre

point (normal distribution). With a normal distribution the coefficient of the skewness is

0.

• Kurtosis; this is a measure of whether the data are peaked or flat relative to a normal

distribution. The coefficient of the kurtosis in the case of a normal distribution is 3.

• Cycles, seasonal effects and events; this generate news at some point in the future. An

example of these is earnings report and financial news about a certain firm.

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2 How can options be priced?

There are a lot of techniques with where you can price options with. I will highlighten

three important option pricing techniques. The three option pricing techniques I will

analyze are; Black Scholes Model, Risk-neutral valuation and The Binomial model.

Further I will look to the difference between these techniques and conclude which one is

best to use.

2.1 Black Scholes Model

The most popular and common used option pricing technique is the Black Scholes

Model, discovered by Fischer Black and Myron Scholes (1973). The Black Scholes

Model is used to calculate a theoretical price of an option, for both call and put options.

There are some assumptions that this model uses. Here follows the assumptions that the

Black Scholes Model uses to determine the theoretical price of an option.

• The underlying asset pays no dividends during the life of the option; but you can adjust

the formula to take into account the payout of dividends.

• European exercise terms are used.

• Markets are considered to be efficient.

• There are no commissions being charged and also no transaction costs.

• The risk-free interest rate and asset volatility remain constant and known functions of

time over the life of the option.

• Returns are log normally distributed.

The Black Scholes formula uses the five key determinants of an option’s price;

underlying price (S), strike price (X), volatility (v), time to expiration (T-t), short-term

risk-free interest rate (r).

The original Black Scholes formula for calculating the theoretical option price (OP) is as

follow:

Where:

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The meaning of the variables is provided in the appendix.

The Black Scholes Model doesn’t only provide the theoretical price of an option but also

other useful products. These products are called Greeks; Delta, Gamma, Theta, Vega and

Rho.

• Delta; also known as the hedge ratio, measures how much an option’s premium is likely

to change in response to a small change in the price of the underlying.

• Gamma; the rate at which Delta changes with movement in the underlying price.

• Theta; rate which measures the calculated option value's sensitivity to small changes in

time until maturity.

• Vega; measures the option’s sensitivity to changes in the volatility.

• Rho; measures the sensitivity of option prices to interest rates.

The main advantage of the Black Scholes model is that you can very quickly calculate the

theoretical price of an option. But it is not very appropriate to calculate American options

because the formula can determine the price only at one time in a moment (expiration

date).

2.2 Binomial model

There is also an opportunity to calculate the option price with the binomial model (Cox,

Ross and Rubenstein, 1979). An important assumption that the binomial model makes is

that the stock’s return can only take one of two values every period. So the distribution of

the stock return is independent, with this assumption the stock price follows the binomial

distribution. The binomial model breaks down the time to expiration into potentially a

very large number of time intervals. In each time period, from present to expiration the

stock price will be calculated. At each time period it is assumed that the stock price will

move up or down by a certain amount (in percent). This amount can be calculated using

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the volatility and time to expiration. With this procedure you can produce the binomial

distribution, a tree with the underlying stock prices. With this tree you can get al possible

paths that the stock price could take during the life of the option. At the expiration date

(time period at the end of the tree) you can get the option price because al the possible

final stock prices are known. This option price (at the end of the tree) equals the intrinsic

value because at expiration there is no time value. Then the option prices can be

calculated at each step working back from expiration to the present.

Within the binomial model there is also the opportunity to take into account dividends

payout. The stock price should be adjusted; to do this you should extract the present value

of the dividends from the stock price. Dividends are cash outflow within a company; this

means that a dividend payout decreases the value of a company. So a dividend payout

also decreases a stock price.

To get a better insight in how this model works I will give an example of a call options

that last one period .See the appendix for this example.

This example gives an insight for how you can calculate the price of an option with one

period, with the same method you can also calculated where there are a lot of time

periods. You can use computer programmes to calculate the option price quickly.

The advantage of the binomial model is that it's possible to check at every point in an

option's life (every time period of the binomial tree) for the possibility of early exercise.

Further you can calculate the option price at every moment during the life of the option.

So the big advantage that the binomial model has over the Black-Scholes model is that it

can be used to price American options. But the binomial model is very slow, there must

be made a lot of calculations before the option price can be determined.

However these two models have the same underlying assumptions regarding stock prices;

stock prices follow a stochastic process. As a result of this, pricing of European options

with the binomial model (if the time periods increase infinite) converges to the same

prices with the Black-Scholes model (Jarrow Robert A, 1983). Thus, the best way you to

price options of European style is the Black Scholes model.

2.4 Risk neutral valuation

Option pricing techniques like Black Scholes model, binomial model and risk neutral

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valuation is derivered from the option pricing theory of arbitrage-free pricing. This

fundamental theory states that there is no arbitrage only if there exist a risk neutral

measure that is equivalent to the original probability measure.

The Black Scholes model assumes that the value of an option does not depend on the

expected rate of return (µ) of the underlying. The input of the Black and Scholes formula

are parameters that do not measure the risk preference, so al the parameters (volatility,

exercise price etc) are independent of risk preferences.

These observations led financial economists Cox and Ross to develop an important tool

known as risk-neutral valuation method. The principle of the risk-neutral valuation states

that when pricing options, it is valid to assume that the world is risk neutral (all

individuals are indifferent to risk). Also that the resulting option prices are correct, not

only in a risk-neutral world but also in the real world. Cox and Ross give the reasoning as

follows:

If the stocks prices follow a random motion, then the option values must be the same as

the values predicted by the Black-Scholes formula so that there are no arbitrage

opportunities. This formula should be valid where it does not matter what the average

investor’s view is towards risk because the formula does not use the parameter of the

expected return of the underlying asset. As long as the investment world satisfies the

main basic assumptions of the Black and Scholes formula, the option values given by the

formula will be true (Hull and John C, 2000).

Cox and Ross have derived the option valuation formula where they assumed a risk-

neutral investment world. You can characterise a risk neutral world as a place where the

investors require no risk premium for their investments. In other words, the investors

always demand only the risk-free rate of interest as the average expected return on their

investments.

The option prices can be determined by discounting the expected value by the risk-free

rate. The expected value is calculated using the intrinsic values from the option going up

or the option going down.

So the present option price must represent its expected value discounted at the risk free

rate. See the appendix for the formula.

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3 What if the assumptions of the Black Scholes model do not hold?

In chapter 2 there are mentioned the assumptions of the Black Scholes model. Pricing

options should be accurate if these assumptions hold. The most important assumption

shall be discussed.

3.1 Volatility

An important input to calculate the price of an option is volatility. This is also the most

critical parameter for option pricing because option prices are very sensitive to changes in

volatility. Volatility however cannot be directly observed; it is not given and can only be

estimated. The input of volatility in the Black and Scholes model is very important to get

the accurate option price. There are different kinds of volatility; historical and implied

volatility. Historical volatility is the volatility of the underlying during the past. If you

know the actual option price you can calculate the implied volatility. With past data you

can get al the input of the Black Scholes formula and keep the volatility unknown while

the actual option price is already known. With solving this equation you get the implied

volatility. Basically implied volatility will give you the price of an option while historical

volatility will give you an indication of its value. Before you can get an option price you

must have an estimate of the future volatility. The future volatility can be better estimated

with the implied volatility rather than the historical volatility. The implied volatility can

be a function of the stock price. There is some empirical evidence that there is a negative

correlation between the stock price and the volatility of stocks for small firms (Stulz,

2003). An explanation for this can be as follow; if firm value drops (stock price drops)

then leverage will rise, and if leverage rise then equity becomes riskier so that the

volatility will increase. The Black and Scholes model is modified to take to in account the

negative relationship between stock prices and volatility. If the volatility can change

randomly and it not perfectly correlated with the stock price then the binomial model will

not longer work. There are a lot reasons why volatility could change randomly, a good

example of one of these reasons is the announcement of earnings.

The Black Scholes model assumes that the volatility is constant during the life of the

option. If the time to maturity is long for a specific option, than there is some bias in

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calculating the option price with the Black Scholes model. The chance is great that the

volatility will change during the life of the option; this means that the constant volatility

input can create bias. A better option pricing technique for an option with a long time to

maturity is the Binomial model because there is the opportunity to change the volatility

during the life of the option.

3.2 Geometric Brownian motion

The Black Scholes model also assumes that the price of the underlying security follows a

geometric Brownian motion with constant volatility. This means that all options on the

same underlying should produce the same implied volatility. However many empirical

research, like research by Rubinstein (1994), Dumas (1998), Tompkins (2001), have

shown that there is an existence of a systematic relationship between the implied

volatility and the strike price. This is known as the volatility smile. With the volatility

smile there is a long-observed pattern in which at-the-money options tend to have lower

implied volatilities than other options.

A related concept of the researchers is that of the term structure of volatility, this refers to

that implied volatility differs on options with the same underlying security and the same

strike price but with different times to maturity. This means that the log-normality

assumption (geometric Brownian motion) of the Black Scholes model does not hold.

Empirical study of Jackwerth and Rubinstein (1996) has also shown this. They show that

the implied risk-neutral probability densities are heavily skewed to the left and are highly

leptokurtic. A distribution with a positive kurtosis is called leptokurtic, this means that

the distribution has more a peak around the mean (a higher probability of returns near the

mean than that’s the case with a normally distributed return) and has more fat tails (a

higher probability of extreme returns than that’s the case with a normally distributed

return). So this means that the log-normality assumption in the Black Scholes model does

not hold (Tobias Herwiq, 2006).

3.3 Constant risk-free interest rate

Another assumption of the Black Scholes model that could not be hold is that there is a

known risk-free interest rate that remains constant during the life of the option. You can

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use a stochastic interest rate instead of a constant risk-free rate to price options. A model

that uses a stochastic interest rate is that of Amin and Jarrow (1992). Their has been done

empirical research by Krister Rindell (1994) to check whether the Black Scholes model

price stock index options better than of the model of Amin and Jarrow. He has used data

on European style stock index options from the Swedish option market. Krister Rindell

concluded that the Amin and Jarrow model outperforms the Black Scholes model. He has

found pricing errors (difference between the observed options price and the price found

with the model) with the Black Scholes model, if the model is correct these errors should

be random. But these errors are correlated with the time-to-maturity of the options so

there is time-to-maturity bias. This time-to-maturity bias found in the test of the Black

Scholes model disappears when the options are priced with the Amin and Jarrow model.

3.4 No transaction costs

The Black Scholes model also assumes that there are no transaction costs, this is not true

in the real world. The Black Scholes model is based on the theory of no arbitrage, but

arbitrage relationships in theory are always affected by the transaction costs in practice.

This means that transaction costs create bounds around the theoretical price within which

the market price may fall without giving rise to a profitable arbitrage opportunity large

enough to cover the cost of exploiting it. (Stephen Figlewski, 1989) This effect of the

transaction cost that lead to market efficiency is also stated by Black and Scholes, see

page 17.

3.5 Elasticity of variance and other properties

Important empirical research about the pricing of options with the Black Scholes model

has been done by C.Cox and Mark Rubinstein. They have created alternative option

pricing models and compared it with the Black Scholes model. They have created these

alternative option pricing models by assuming that some of the following important

properties that led to the Black Scholes formula will not be true:

• The possible percentage changes in the stock price over any period will not depend on

the level of the stock price at the beginning of the period. So it is assumed that there is a

random walk.

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• Over a very small interval of time, the size of the change in stock prices is also small.

This means that although we were certain a change in the price of the underlying would

occur, not much could happen before we could do something about it.

• Over a single period, only two stock price outcomes were possible. This feature is also

used by the Binomial model.

By considering one of the properties not true, like the second of the above properties, you

consider the probability that the stock price can have a jump movement. There has been

concluded that the alternative option pricing models has less bias then that of the Black

Scholes model.

The model of Cox uses the constant elasticity of variance to price options. The constant

elasticity of variance functions of C.Cox calculates the sensitivities, theoretical price and

implied volatility of options. Thereby they use a valuation technique based on the

constant elasticity of variance option pricing model. This model considers the possibility

that the volatility of the underlying asset is dependent upon the price of the underlying

asset while the Black Scholes model assumes an elasticity of zero. Also empirical study

of Black (1976) and Beckers (1980) have found that the variance of the rate of return is

inversely related to stock prices.

Other researchers, like Thorp and Gelbaum (1980) say that the model of Cox is a more

realistic version of the Black-Scholes model because studies have shown that price

variances do change as the asset price changes. Although the Black-Scholes model

assumes a constant asset price volatility regardless of the level of the security price.

3.6 Validity of a pricing model

You have seen that there is a lot of research that stated that some assumptions made by

the Black Scholes model do not hold. There cannot directly said that the Black Scholes

model should be rejected due to these deviations from the basic assumptions of the

model. A study of Bhattacharya (1981) shows that even if the Black Scholes model can

give a wrong price of the option if some of the underlying assumptions do not hold, the

deviations are usually small and on average insignificant. He state that the deviations

usually decrease as time to maturity increases and as the degree of the option that is in the

money increases. According to Milton Friedman (American economist and noble price

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winner), the validity of a model should not be tested by testing its assumptions

approximation of reality. He state that the validity of a model should be tested by its

ability to predict future events.

There are four approaches of validating option pricing models (Katz, 2005):

(1) Testing pricing models by means of simulations and quasi-simulations of deviations

from the basic assumptions of the model.

(2 )Comparing the prices calculated by the model with the actual prices.

(3) Creating a neutral hedge positions and testing the behavior of the returns from the

investment. This approach is suggested by Black and Scholes in 1972. The idea of this

approach is to create with options and their underlying security a position. This position

should be riskless if the option pricing model is correct.

(4) Imputing the standard deviations (volatility) from actual option prices by using an

option pricing model.

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4 How large are the pricing errors with use of the Black Scholes model?

There has been a lot of research to investigate the performance of option pricing models.

The first few researchers have used historical volatility. The problem with this is that

such an estimate of historical volatility reflects the pas history of the stock, while the

relevant volatility is the one that market expects to hold over the life of the option. So a

better input is implied volatility, the most recent papers have used this implied volatility.

4.1 Whaley

A comprehensive empirical study of the pricing of stock options has been done by

Whaley (1982). He has used the Black and Scholes model to price 15,582 call prices.

These call prices are of dividend paying call options. If you look at dividend paying call

options there is need for dividend information to calculate the options prices. This can be

accurately estimated by inspection of the underlying historical dividend payments. This

has also been done by Whaley. Whaley found an average mistake of 3.16 cent for call

options with an average call price of 4.1388. This is 0.76 % of the average call price. He

has concluded that the Black Scholes model performed well but has some bias. The Black

Scholes model has overpriced options on high volatility stocks and underpriced options

on low volatility stocks. Whaley stated that this phenomenon, a lot of researchers come to

these same result, could be attributed to the following resources:

• non-stationarity of the stock return standard deviation parameter

• assumption of a known dividend

• assumption of a zero tax rate differential between the dividend and capital gain income

Black and Scholes have also found the above mentioned result on their own model.

Thereby they used past data to estimate the variance and state that this is the reason why

there is an underpricing of low volatility stocks, and overpricing of high volatility stocks.

With a better measurement of the volatility this problem can be solved. This means that if

someone buys and sells options for his own account where he uses the Black Scholes

valuation model to price options, then he can expect to lose money if other traders have

information about the volatility that is not contained in the past data. If it is assumed that

all market traders uses all available information efficiently, then the buying of an

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undervalued option at the Black Scholes model price and selling an overvalued option at

the Black Scholes model price would result in significantly negative excess portfolio

returns. But buying an undervalued option at the market price and selling an overvalued

option at the market price will result in significantly positive excess portfolio returns. So

while the model overestimates the value of an option on a high variance security, market

traders will underestimate and vice versa. However if you include the transaction cost of

trading in options than the implied profits of buying options on a low volatility securities

and selling options on a high volatility securities will disappear. This means that there is

market efficiency (Black and Scholes, 1972).

But Whaley (1995) has not found a significant relation between the underpricing of the

options with the Black Scholes model and the degree of moneyness of the options (how

much an option is in the money; underlying price minus the exercise price minus the

option premium), so the underpricing cannot be explained by the degree of the

moneyness. This result is contradicting with other research results. Whaley also has

found a significant relationship between the pricing errors and the dividend yield, and

also with the time to maturity. He found less pricing errors if the dividend yield increased

(negative correlation). If the time to maturity increased, then also the pricing errors

increased (positive correlation). Hence, the Black Scholes model underprices options on

high-dividend stocks and overprices options with a long time to maturity. (Whaley, 1995)

4.2 Rubinstein

Another import research that has been done to look at the Black Scholes pricing errors is

a study of Rubinstein (1985). He has considered the prices of two options of the same

stock that differ along some dimension. For example it could be that one option has a

higher price than the other option. With the Black Scholes formula both options should

have the same implied volatility. If this is not true, for example, if the option with the

higher exercise price has a higher implied volatility then there could be implied means

that the Black Scholes formula has some bias. Rubinstein found that there were

statistically significant differences between the implied volatilities. He found that the

implied volatility of out-of-the-money options increases as time to maturity becomes

shorter. Thus, short-maturity options are valued more than implied by the Black Scholes

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model. The biases that Rubinstein documented changed over time. The important

changes in the biases of the Black Scholes model took place after the stock market crash

of 1987. After 1987 there has also been a lot of analysis of stock index options like the

S&P500. Also the stock index option can be good priced with the Black Scholes model

where you can treat the index as the underlying. There also has been found that deep-in-

the-money and deep-out-of-the-money of the stock index options have higher implied

volatility (Stulz, 2003).

4.3 Dan Galai, Macbeth and Merville

Dan Galai also has looked at the validity of the Black Scholes model in pricing of

options. He has concluded the following:

• The Black Scholes model approximates market prices well for at-the-money options

with medium to long time-to-maturity, although unexplained deviations from model

predictions are still observed for these options.

• There are significant relative deviations between market prices and model prices for

deep-in and deep-out-the-money options. This result is also found by the study of

Whaley.

• If the option is close to maturity, then there are more observed violations.

• No other alternative model offers a consistently better explanation for the actual

behavior of option prices over time than the Black Scholes model does.

• Despite the deviations from model predictions, the markets for options seem to be quite

efficient in the sense of at allowing a trader to make consistently above-normal profits on

an after-commissions and after-tax basis.

Also the study of MacBeth and Merville in 1979 found that the Black Scholes model

underestimates (overestimates) market prices for in-the-money (out-of-the-money)

options. Further they found that the extent of the mispricing decreases as the time to

maturity decreases.

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5 How accurate is the Black Scholes model in pricing AEX index call options?

The most AEX- index options have short time to maturity, I will use the Black Scholes

model to price these call options. There will also be an analyse of call option with a long

time maturity, so that there can be done a comparison. Further there will be only data

analyse of the call options. To check the validity of the Black Scholes model in this case I

will compare the prices calculated with the Black Scholes model with the actual prices

(method 2, see page 16). The call options start at different days in May (16, 17 and 24

May) and the most of this end in a couple months. But there are also options that just

mature in years like the year 2010. These AEX-index call options are of European style.

The AEX index is not adjusted for the dividend payments of the firms that are in the

AEX index. The stock market is working on a new AEX index that is adjusted for the

dividend payments.

To calculate the option price I have collected the five inputs and the actual option price

on the website of Euronext Amsterdam and converted it to excel. The risk free interest

rate has been found on the website of “de nederlandse bank”, this interest rate is about

3.3%. This interest rate is also for the month May. The most important input, the

volatility, can be found in different ways. First I will use historical volatility to price the

call options.

5.1 In the case of historical volatility

There has been used historical data of the AEX-index prices over the last five years, from

28 May 2002 until 28 May 2007. To get the volatility I have calculated the returns with

the following formula:

Return (St) = (St – St-1) / St-1

These are daily returns because I have used the daily AEX-index prices. With the use of

the function STDEV in excel there has been found the daily volatility, this has been

converted to the yearly volatility. To calculate the yearly volatility I have used 250 days

in the year, because that is the number of trading days in a year. The calculated historical

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volatility is 22.52 %. With the use of this volatility there has been calculated the Black

Scholes option prices. The average pricing errors is €10.78 with an average call option

price of €34.78. The standard deviation is also large, it’s 10.64. The average pricing error

is about 30.98% of the average actual call price. There can be concluded that the Black

Scholes model is not accurate with the use of historical volatility. I also have looked at

the difference in pricing errors between call options with a short maturity and a long

maturity. I have divided the call options in 2 sides; call options with a short maturity (less

than a half year) and with a long maturity (longer than a half year). The average pricing

errors of the call options with a short maturity is €5.22 with a standard deviation of 4.44

while that of options with a long maturity is €20.72 with a standard deviation of 11.27.

Thus, the pricing errors with the use of historical volatility are less with options with a

short maturity but are still great. Empirical research of Whaley, MacBeth and Merville

has stated that there is a positive correlation between time to maturity and pricing error.

This means that the pricing error rises if also the time maturity rises. The next graph

shows the relationship between the pricing error of the AEX-index call options and the

time maturities of these call options.

Figure 5.1 Relationship between pricing error and time to maturity

Historical volatility

-20

0

20

40

60

0,0

9

0,1

0,1

0,1

7

0,1

9

0,2

6

0,3

4

0,4

4

0,5

8

0,8

3

1,0

9

1,6

4,6

Time to maturity (years)

Pri

cin

g e

rro

r (e

uro

's)

On this graph can be seen that the pricing error rises if the time-to-maturity increase. The

correlation coefficient has been calculated with the use of excel function CORREL, the

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correlation coefficient is 0.94. This is a very high correlation coefficient, so there is a

highly positive correlation between the pricing error and time to maturity. Thus, AEX

index call options with a short time to maturity contain less error. This result is in line

with the empirical study of Whaley, Macbeth and Merville. Another result of the study of

Whaley is that there is no significant relationship between the pricing error of call options

and the degree of moneyness. But research of Bhattacharya has stated that the pricing

error decrease as the degree of moneyness increase. The degree of moneyness can be

calculated as follow:

Underlying price / strike price

If this ratio is more (less) than 1, than the call option is in the money (out the money).

With a ratio of 1 then the call option is at the money, if the ratio differs far from 1 than

the option is deeply out (in) the money. The next scatter diagram shows the relationship

between the pricing error and degree of moneyness. Remember that the pricing error

contains the difference between the calculated Black Scholes model price and the actual

call price.

Figure 5.2 Relationship between pricing error and degree of moneyness

Historical volatility

-10

0

10

20

30

40

50

60

0 0,5 1 1,5 2 2,5 3 3,5

Degree of moneyness

Pri

cin

g e

rro

r

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On this diagram can not be clearly seen a pattern. The correlation coefficient between

these two variables is -0.17. So there is a slightly negative correlation between the pricing

errors and degree of moneyness. This has also been concluded by the study of

Bhattacharya, but in this case it’s also almost the same as the result of Whaley because

the correlation coefficient of -0.17 is not far from 0 (no correlation). In other words,

AEX-index call options that are in the money contain a little bit less pricing error.

This information tells us not much because I have concluded that the pricing of the call

options with the use of historical volatility is not so accurate. Other researcher also stated

that historical volatility is a poor input to calculate the option prices, they stated that

historical volatility is a poor estimator of the future volatility.

5.2 In the case of implied volatility

Implied volatility can maybe be better used to price call options. But how can the implied

volatility of the AEX-index be found? The volatility of the American market is almost the

same as that of the Netherlands. In other words, the volatility of the SP500 is the same as

that of the AEX-index. There exists a volatility index on the SP500, the VIX. The value

of VIX is the same as the implied volatility of the AEX-index. So I have used the value

of the VIX on the date when I started to price the options, namely on 24 May. The

implied volatility is less than the historical volatility, it’s 14.02%. This means that we can

expect that the pricing errors with the use of implied volatility is less, because option

price is low with a low volatility. Vice versa, option price is high with a high volatility.

We have seen that the average option pricing error with the use of historical volatility is

€10.78 above the actual option price. This means that the pricing error is less than €10.78

with a lower volatility; there is also a greater change that the pricing error is negative. In

other words that the Black Scholes model price is less than the real price (underpricing). I

have calculated the call prices with the use of the implied volatility. The average pricing

error is € 0.29 with a standard deviation of 4.08. This is about 0.82 % of the average

actual call price. Whaley has found an average pricing error of 0.76% of the average call

prices and concluded that the Black Scholes model is accurate enough. In this case there

also can be concluded that the pricing of the AEX index call options with the Black

Scholes model is accurate enough. The pricing errors, like expected, are less than in the

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case that there has been used historical volatility. Thus, implied volatility can be better

used to price AEX-index call options. The average pricing error of the call options with a

short maturity is € -0.95 with a standard deviation of 2.15, this negative sign means that

the Black Scholes model price is less than the actual option price (underpricing). So we

can conclude that the pricing of the call options with a short maturity is accurate. The

average pricing error of the call options with a long maturity is € 2.50 with a standard

deviation of 5.55. With the next table you can compare the results of the pricing errors

with the use of historical and implied volatility.

Table 5.1 Pricing error

Pricing error Average

(historical

volatility)

Average

(implied

volatility)

Standard

deviation

(historical vol)

Standard

deviation

(implied vol.)

Total 10.7755

0,2863

10.6444

4,0834

Short 5.2190

-0,9512

4.4432

2,1451

Long 20.7188

2,5008

11.2677

5,5526

With the use of implied volatility I also can conclude that the pricing errors are less with

call options with a short maturity. Pricing AEX index call options with a long maturity is

also better with the use of implied volatility rather than with the historical volatility. The

following graph shows us the relationship between time to maturity and pricing errors.

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Figure 5.3 Relationship between pricing error and time to maturity

Implied volatility

-20

0

20

40

0,0

9

0,1

0,1

0,1

7

0,1

9

0,2

6

0,3

4

0,4

4

0,5

8

0,8

3

1,0

9

1,6

4,6

Time to maturity (years)

Pri

cin

g e

rro

r (e

uro

's)

There can be seen that the pricing error rise with time to maturity. To see how high the

correlation is between these two variables there has been calculated the correlation

coefficient. There is also a high correlation between the time to maturity and pricing error

because the correlation coefficient is 0.72. The correlation between the time to maturity

and pricing error is much higher if there has been used historical volatility (0.94) rather

than implied volatility (0.72). This means that with the use of implied volatility the

pricing error increase less with time to maturity.

With the use of a scatter diagram can also be seen the relationship between the pricing

error and the degree of moneyness.

Figure 5.4 Relationship between pricing error and degree of moneyness

Implied volatility

-10

-5

0

5

10

15

20

25

30

35

0 0,5 1 1,5 2 2,5 3 3,5

Degree of moneyness

Pri

cin

g e

rro

r

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There can be seen that the AEX index call options that are out the money contain less

pricing error than in the money options. Vice versa, call options that are in the money

contain more pricing error. To check how strong this correlation is I have calculated the

correlation coefficient, this is 0.20. So there is not a strong positive correlation between

the degree of moneyness and the pricing error.

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6 What are the causes in the eventually pricing errors, and how can they be

reduced?

In the previous chapter there has been described the pricing error that exist in pricing the

AEX index call options with the Black Scholes model. There has been concluded that

implied volatility as input is a better way to price the call options. But also in this case

there remains pricing error. Can the pricing error further been reduced?

6.1 Dividend payments

In calculating the AEX index options there has been used the AEX index as underlying

price. But this index has not been adjusted for the dividend payments of the 25 firms that

are in the AEX. So there must been developed a new index that takes the dividend

payments in account. With the adjusted AEX index the pricing error can be less with the

use of the Black Scholes model.

6.2 Risk-free interest rate

Another input that can be discussed is the risk-free interest rate. There has been used a

risk free interest rate of 3.3 %, maybe this risk free rate is to low or to high. To check of

another interest rate provides less error, I have used different interest rates to check

whether the pricing errors reduce. With increasing the interest rate the pricing error also

increase, so the interest rate should not be higher. But if there has been reduced the

interest rate then the pricing error first decrease and then increase. The interest rate at

which the decrease change in an increase is 3%. Thus, a risk free interest rate of 3 %,

3.1% and 3.2% provides less pricing error. The optimal interest rate of 3.1 % provides an

average pricing error of € -0.05598 with a standard deviation of 3.78. So both the average

and standard deviation are less with the use of a risk free interest rate of 3.1 %. Because

the interest rate is not far from 3.3%, it can be that the actual risk free interest rate in the

period that I started pricing the options is 3.1%. Another way in which pricing error can

be reduced is by using stochastic interest rate. The Black Scholes model assumes a

constant risk free rate interest rate; with using a stochastic interest rate in the model of

Amin and Jarrow pricing error can be reduced. The AEX index call options that I have

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priced are that of in the days 16, 17 and 24 May. Because there has been little time

between these days there can be assumed that the risk free interest rate is constant, so

using stochastic interest rate shall not reduce the pricing error a lot. This also the case for

stochastic volatility.

6.3 Lognormal distribution

Another assumption of the Black Scholes model is that the returns of the underlying are

lognormal distributed. The causes of the pricing error can maybe explained by the fact

that the assumption of a lognormal distributed returns does not hold. To test this

assumption I have collected the AEX index prices from 28 May 2002 until 28 May 2007.

The lognormal returns are calculated with excel, the formula used to calculate these log

returns is:

Log returns (St) = LOG (St) –LOG (St-1).

The calculated log returns are converted to SPSS to make a histogram with a normal

curve.

Figure 6.1 Lognormal distribution of the AEX index

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The above histogram shows that the distribution of the AEX index seems like a

lognormal distribution. The coefficients of the skewness and kurtosis are calculated with

SPSS to have more information how accurate the lognormal distribution is. With a

lognormal distribution the coefficient of the skewness is 0 and coefficient of the kurtosis

is 3.

Table 6.1 log returns of the AEX index

Valid 1925 N

Missing 0

Skewness -,106

Std. Error of Skewness ,056

Kurtosis 3,888

Std. Error of Kurtosis ,112

The calculated skewness coefficient is -0.106 and that of kurtosis is 3.888. So there can

be concluded that the lognormal distribution is not perfect. This also can be one of the

causes of the calculated pricing error in chapter 5.

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Conclusion

In the literature review there has been stated that the Black Scholes model is accurate in

the pricing of call options. This is also the case for AEX index options if there will be

used implied volatility. Implied volatility is a much better way to price options rather

than with historical volatility. With the use of historical volatility there is no accuracy in

pricing the AEX index options. These pricing errors are larger for call options with a long

time to maturity and if the degree of moneyness decrease (out the money options). The

pricing error for call options with a long time to maturity is also larger in the case there

has been used implied volatility, but this positive correlation is not so large is in the case

of historical volatility. This time to maturity bias has also been found by other empirical

researchers like Whaley. Investors can better use other pricing techniques like the

Binomial model to price AEX index call options with a long time to maturity. The time to

maturity bias can not be reduced with the model of Amin and Jarrow because there has

only be observed call options on three days in the same week. This means that stochastic

interest rate that is used in the model of Amin and Jarrow shall not reduce the pricing

error that I have calculated.

The relationship between the pricing error and degree of moneyness in the case that there

has been used implied volatility is the opposite as in the case of historical volatile. So

pricing error increase as the AEX index options are more in the money, but this positive

correlation is not strong. This result is not in line with empirical research of Whaley,

MacBeth and Merville because they found no relationship between the two variables.

If the risk-free interest rate decrease with 0.1% than the pricing error is minimal given

that other inputs remain unchanged. This means that the observed risk-free interest rate

may not be correct. Their are also other reasons that can explain the pricing error. An

important one is that the AEX index is not been adjusted for the dividend payments of the

25 firms that are in the AEX index. Further can be stated that the distribution of the AEX

index is not perfect lognormal distributed. There exist some skewness and kurtosis that

create the pricing error. The pricing error can maybe decrease if these reasons will be

taken into account. There can be done research in the future to check how much the

pricing decrease if these reasons are taken into account.

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Appendix

Black scholes formula:

S = underlying price

X = strike price

t = time remaining until expiration, expressed as a percent of a year

r = current continuously compounded risk-free interest rate

v = annual volatility of the underlying price (the standard deviation of the short-term

returns over one year).

ln = natural logarithm

N(x) = standard normal cumulative distribution function

e = the exponential function

Example of the Binomial model

Consider a call option that lasts only one period. Assume that at the end of the

period the underlying stock can only have 2 possible values. The current stock price is S0

= 50, and at the end of the period the stock price will either grow (with 30%) to S1, upp =

(1.3)*50 = 65, or fall (with 20%) to S1, down = (0.8)*50 = 40.

The call strike price X is 48, and the risk-free rate is 10% per period.

S0 = 50

Su = (1.3)50 = 65

Sd = (0.8)50 = 40

C0 = ?

Cu =Max[0,65-48] = 17

Cd = Max[0,40-48] = 0

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Before you can determine the cal price (C0) you must determine a replicating portfolio.

The value of the replicating portfolio must be the same as the call price. Consider the

following replicating portfolio: ∆ number of shares and R dollars invested in the risk-free

bonds with a risk free rate of 10 %.

So the value of the replicating portfolio should be the same as the option values.

∆65 + 1,1 R = 17

∆40 + 1,1 R = 0

With solving this you get ∆= 0.68 and R = -24.73

C0 = ∆S =R = 50*0.68 – 24.73 = 9.27

Risk-neutral valuation formula

Call option value = [ p × Option up + (1-p)× Option down] ÷ (1+r)

= [ p × (S up - strike) + (1-p)× (S down - strike) ] ÷ (1+r)

Su= 65

S0=50

Sd= 40

C0=

Max(0, 65-48)= 17

Max(0, 40-48)= 0

Stock price tree Call price tree

RP0= ∆50 + R

RPu= ∆65 + 1,1 R

RPd= ∆40 + 1,1 R

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References

Articles

Avramidis Athanassios N and L'Ecuyer Pierr, 2006, Efficient Monte Carlo and Quasi-

Monte Carlo Option Pricing Under the Variance Gamma Model, Management science;

vol.52 (2006) nr.12 p.1930-1944 (15 refs.)

Black Scholes, 1972, The valuation of option contracts and a test of market efficiency,

Journal of Finance

Chriss, Neil A, 1997, “Black-Scholes and Beyond” pages 190-192. Chicago, IL: Irwin

Professional Publishing.

Duan J.-C and Yu M.-T, 1999, Capital standard, forbearance and deposit insurance

pricing under GARCH., Journal of banking and finance; vol.23 (1999) nr.11 (November)

p.1691-1706

Figlewski,Stephen, 1989, “Options arbitrage in imperfect markets,” Journal of finance

Kimmel R, 2007, Maximum likelihood estimation of stochastic volatility models, Journal

of financial economics; vol.83 (2007) nr.2 (February) p.413-452

Koch, I. , Schepper, A.D, 2007, Distribution-free option pricing, Insurance;

vol.40 (2007) nr.2 (March) p.179-199

Krister Rindell, 1994, pricing of index stock options when interest rates are stochastic,

journal of banking and finance;1994) vol.19 (1995) nr.5 (08) p.785-802

Minton,Bernadette A,, Schrand,Catherine M, Walther,Beverly R, 2002, The Role of

Volatility in Forecasting; Review of accounting studies, vol.7 (2002) nr.2-3 (June) p.195-

215

Szakmary,A, Ors,E, Kyoung Kim J,Davidson, W.N, 2003, The predictive power of

implied volatility: Evidence from 35 futures markets, Journal of banking and finance;

vol.27 (2003) nr.11 (November) p.2151-2175

Books

Whaley, Robert, 1995, “Valuation of American call options on dividend paying stocks;

Emperical test,” Journal of Financial Economics

Bachelor thesis

34

Zimmermann H and Hafner W and Hafner W, 2007, Amazing discovery; Vincenz

Bronzin's option pricing models, Journal of banking and finance;vol.31 (2007) nr.2

(February) p.531-546

Brenner and Menachem, 1983,Option pricing : theory and applications

Herwiq and Tobias, 2006, Market-conform valuation of options

Huisman and Juno J.M, 1999, Effects of strategic interactions on the option value of

waiting

Hull and John C, 2006, Options, futures, and other derivatives

Jarrow Robert A and Rudd, 1983, Option pricing

Kabir and Rezaul, 1997, New evidence on price and volatility effects of stock option

introductions

Kallianpur, Gopinath, 2000, Introduction to option pricing theory, p32

Katz Jeffrey Owen, Advanced Option pricing models, page 91

Shaffer and Sherrill, 1989, Immunizing options against changes in volatility

Stulz, 2003, risk management and derivatives

Willmott and Paul, 1995, Option pricing : mathematical models and computation, page

41

Internet

http://hccbeleggenforum.nl/index.php?option=com_content&task=view&id=158&Itemid

=93 ).

http://www.investopedia.com/articles/optioninvestor/02/031102.asp

h t t p : / / e n . wi k i p ed i a . o r g/ w i ki /R at i on a l_p r i c in g# A rb i t r a ge _f r ee _p r i c in

g).

http://www.hoadley.net/options/bs.htm)

h t t p : / / e n . wi k i p ed i a . o r g/ w i ki /V ol a t i l i t y_ sm i l e

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http://www.fintools.com/docs/risk_neutral_valuation.pdf

h t t p : / / w w w. b eh r . n l /B eu rs / aex . h t m lT

h t t p : / / w w w. sb t io n l in e . co m /n i s t / ed a / s ec t io n3 / e d a3 66 1 . h t m

http://en.wikipedia.org/wiki/Option_time_value

(http://www.fintools.com/doc/options/optionsConstant_Elasticity_of_Variance_.html