thesis draft (2)

Effectiveness of the Black-Scholes model for option pricing

Bachelor Thesis Paper

Submitted to the “Department of Finance”

At the Faculty of Management Technology

German University in Cairo

Done by:

Sarah Adel Fahmy Hamed

ID no.: 28-4254

Supervised by:

Dr. Maria Do Rosario Correia

Submitted on:

Tuesday 3rd of May, 2016.

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Declaration

I herewith declare that this report is in full accordance with the Plagiarism Guidelines of

the Faculty of Management & Technology at the GUC.

Signature

Sarah Adel Fahmy

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Abstract

This paper aims to explore, test, and evaluate the effectiveness of the Black-Scholes

model for option pricing as a primarily concern. The further objective is to test whether

there will be a smile effect or a skewed effect in volatility.

Keywords: Hedging, lognormality, stochastic process, Brownian motion, option

valuation, volatility, implied volatility, volatility smile, volatility skewness.

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Table of ContentsDeclaration...........................................................................................................................2

1. Introduction:.............................................................................................................52. Literature Review.....................................................................................................6

2.1 BRIEF ILLUSTRATION OF THE BLACK- SCHOLES OPTION PRICING MODEL:......62.2 OPTIONS..............................................................................................................132.3 LIMITATIONS OF THE BLACK-SCHOLES OPTION PRICING MODEL:...................162.3.1 Application of the Black-Scholes-Merton model in reality:.............................18

3. Methodology:..........................................................................................................193.1 An introduction to research design:.....................................................................193.2 Testing the Effectiveness of the Black-Scholes model for option pricing:.........21

3.3 Results & findings:......................................................................................................234. Conclusion..............................................................................................................32BIBLOGRAPHY...........................................................................................................34

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1. Introduction:

In their classical paper proposed in 1973 on the hypothesis of option pricing,

Black and Scholes present a method of investigation that has altered the hypothesis of

corporate liability pricing (Merton, 1975). Factually, inequities between parties in an

options environment as well as, uncertainty of potential returns on hedging investments,

were needed to be addressed in finance. In 1973, finance professor, Myron Scholes and

finance analyst, Fischer Black constructed a remarkable model with Robert Merton which

was concerned about mitigating risk of potential returns on hedging investments and

diminishing inequities between the long position and the short position in an options

environment. Founders of the model formulated a distinct field in Calculus referred to

the “stochastic” process.

By convention, one of the influential forces used in the Black-Scholes option pricing

formula is the concept of stock volatility (Black & Scholes, 1973). In fact, the hypothesis

which assumes a volatility-free set of conditions has made adaptability for its application

in a field where we had interminable components that could affect the business sector

(Kou, 2002). For instance, the Black-Scholes option pricing model is a classical study

based on the theory of option pricing and is mainly concerned about the call options

gains/losses (Merton, 1975).

This paper regards the effectiveness of the Black-Scholes model for option pricing as

a primarily concern. To explore the importance of such a model, first, a brief illustration

about the Black-Scholes-Merton model and its equation are clarified as well as, the

required assumptions for applying the model in reality. Second, we explain the

relationship between the degree of moneyness and how it correlates with pricing errors.

Moreover, this paper relates the Black-Scholes model to option pricing and how

effectively it acts as a grant to this special type of derivative. Last but not least,

limitations of the model are explained, followed by a real-life application. In fact, using

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the Black-Scholes model could not be the best model for pricing Employee Stock

Options (ESOs) as well as pricing warrants due to the assumption of constant variance

which consequently, has lead to biases in model pricing for nearly all warrants

(Kreps,1982).

2. Literature Review

2.1 BRIEF ILLUSTRATION OF THE BLACK- SCHOLES OPTION PRICING MODEL:

The Black-Scholes model was named after the financial analyst, Fischer Black

and finance professor, Myron Scholes in the year 1973. It was particularly created to

mitigate the uncertainty of potential returns on hedging investments (Black-Scholes

method for pricing).

Understanding the effectiveness of this model requires sufficient knowledge in the

practice of hedging. In fact, derivatives are considered the central to the practice of

hedging, such contracts are made between two or more parties and their value

depends on the underlying asset such as: stocks, bonds, and currencies. There are two

forms of derivatives, first is the futures contract which is used to ensure and guarantee

price stability. While the second form, is the option derivative, which is an alliance

that gives the holder of the option the right to either buy (put) or sell (call) a stock in a

company and is mainly characterized by a short period agreement having a strike

price (Tucker, 2007).

In 1973, Fischer Black and Myron Scholes constructed a theoretical study which

created an updated foundation for finance and business. This study was primarily

concerned about inequities between both parties in an options environment. To

highlight the importance of this theoretical study, Black, Scholes, and Merton,

constructed a booming fresh field in Calculus which is known as “stochastic” process

which is sometimes referred as a “random” process. This process assumes that the

relevant variables follow standard stochastic processes models which could reflect the

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behavior of these variables in reality. For instance, those who aim to predict the short-

term and long-term state of their investments would be ecstatic from this model of

non-linear thinking as it would aid financial specialists in constructing theoretical

estimates of the price of European-style options.

The major focus of the Black-Scholes model is on the call options gains/losses,

which are the difference between the pay-off of the option and the purchasing cost of

the call. When calculating the option price, there are six aspects of the stock to

address: The price of the underlying stock, the price to exercise the option, the

volatility of the stock price, time to expiration, Interest rate, and finally, the dividend

rate (Tucker, 2007). Also, it is crucial not to neglect the concept of stock volatility

which is one of the dynamic forces used in this formula. Where-by, any unforeseen

adversities in the marketplace, would consequently lead to dramatic losses for

investors. In fact, options ensure price stability through protecting investors from

such significant losses. Thus, when the stock volatility rises, the value of the option

also rises; implying a direct relationship between stock volatility and the value of the

option (Black & Scholes, 1973).

Relying on the Black-Scholes model requires a number of assumptions. Firstly,

we assume that the option is exercised only at the time of expiration. Secondly, the

risk-free interest rate and underlying stock volatility won’t be altered and will remain

constant within a given period. In addition to, no sudden fluctuations in stock prices

will occur. Last but not least, this method is based on the fact that the underlying

stock itself does not pay dividends. Such a theory might be regarded as “irrelevant” in

real world applications. In fact, this model was constructed to relate the constants in

this environment (such as: risk-free interest rates or current underlying stock prices)

while examining the factors that lead to volatility (Black & Scholes, 1973).

By convention, Fischer Black and Myron Scholes have created a model that

was truly needed for a financial environment. For instance, the assumption that states

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a volatility-free set of conditions has created flexibility for its application in a field

where we had infinite factors that could have an impact on the market (Kou, 2002).

In the upcoming phase, the Black-Scholes option pricing formula is clarified. Also,

we highlight the specific type of motion and distribution which the model follows.

According to implied volatility, we explain the reasons behind its downward sloping

curve and biases that occur from volatility smiles.

2.1.1 The Black-Scholes-Merton Option Pricing Formula:

The following, illustrates the Black-Scholes Option Pricing Formula:

The Black-Scholes Option Pricing Formula:

(Voniatis, 2013).

To illustrate briefly, the Black-Scholes model is based on Brownian motion as well

as Normal distribution. Brownian motion might lead to arbitrage opportunities. In other

words, risk-less profits are likely to occur (Kou, 2002). Brownian motion produces a log-

normal distribution for stock price between any two points in time (Merton, 1976). To

introduce this type of lognormal distribution, there are two forms of Brownian motion;

first is the geometric Brownian motion which implies that the distribution of the terminal

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underlying asset is lognormal. Second is the arithmetic Brownian motion, which implies

that the distribution of the terminal underlying asset is normally distributed. Thus, we can

predict that the underlying instrument is following an arithmetic Brownian motion and

the underlying stock has a position in the instrument with a long position in a zero-strike-

put-option. Such a mathematical formula is applied while taking into consideration the

above mentioned conditions discussed earlier and the theory of Brownian motion in this

model implies the arbitrage or the presence of risk-less profits (Brooks et Chance,

2014).

To understand lognormal distribution, figure 1 illustrates:

(Figure extracted from Hull 2005)

Fischer Black and Myron Scholes have both created this parametric model to price

the call options through using an assumption that states that the underlying asset is

following a random diffusion process. Also, the model assumes continuous trading that

has no dividends and taxes applied to the stocks. Furthermore, Geometric Brownian

motion of stock prices is frictionless. In fact, this model proves that the option price of

call options and put options is a function of of asset price (S), exercise price (X), time to

maturity (T), risk free interest rate (r) and volatility of asset price (s).

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The formulas for call and put option, respectively, (Hajizadeha et Seifi, 2011).

C = S0N(d1) - Xe-rT N(d2)

P = Xe-rTN(-d2 ) - S0N(-d1)

Where

d (ln( S / X ) (r s 2 / 2)T ) /s T

d = S X + r -s T s T = d1 -s T

In order to derive the option pricing formula, “ideal conditions” in the market must

be assumed for the stock as well as the option. These conditions are:

(1) “Frictionless” markets which implies zero transaction costs or differential taxes.

Trading is continuous, borrowing, and short selling are allowed with no restriction and

with complete available proceeds. Moreover, the rate of borrowing and lending is exactly

the same.

(2) The short term interest rate is known and remains constant throughout the lifespan of

the option.

(3) No dividends or other distributions are paid for the stock during the lifespan of the

option.

(4) We assume that the option is “European” which implies that it can only be exercised

at time of expiration.

(5) The stock price follows a “geometric” Brownian motion which leads to a log-normal

type of distribution for the the stock price between any two points in time. (Merton,

1976).

It is also worth mentioning Merton’s demonstration which states that the interest

rate is stochastic, and assumes that the stock pays no dividends (Merton, 1976). The risk

neutral approach which was proposed by Black and Scholes (1973) is considered in

finance as a standard paradigm which is extensively used by specialists. Volatility smiles

and skewness are considered two of the major biases for this model (Rubinstein, 1994).

First, implied volatility has a tendency to rise for in-the-money options as well as, out-of-

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the-money options. Basically, in the degree of moneyness, put options are underpriced by

the Black-Scholes formula. Thus, implied volatility has a downward sloping curve.

Second, volatility smile is characterized with a risk neutral density having a kurtosis

higher than that of a normal density. However, the existence of skeweness premia implies

that the return distribution has a left tail which is fatter than that of the right tail (Bates,

1995).

According to the standard option pricing theory, implied volatilities for all option

contracts having different strike prices, should be exactly the same in a declining market.

Although, implied volatilities are known to have a left skew, market prices of options are

usually higher (lower) than the Black-Scholes-Merton prices for in the money (out-of the

money) call options. Rubinstein (1994) named this type of market imperfection as

“volatility skewness”. Actually, the presence of transaction costs does contribute to such

imperfection (Maclean, Zhao, et. al., 2011). The augmentation of the option premium

lured by an augmented volatility, might offset the necessary transaction costs. Despite

that, the hedging portfolio does not delineate the option pay off (Leland, 1985).

Furthermore, the strike price is considered as a supplementary parameter in the

formulation of the Black-Scholes Model. Basically, according to Leland’s approach,

deviation in prices occurs due to the fact that augmented volatility is not dependent on

strike prices of options. Thus, the assumption of Brownian motion was considered in the

settlement of an alternated adjusted volatility model (Maclean, Zhao, et. al., 2011). Such

a settlement indicates a risk-neutral condition whereby, in the Black-Scholes option

pricing model, the risk neutral valuation in consistent with arbitrage-free pricing

(Maclean, Zhao, et. al., 2011).

2.1.2 Pricing Errors:

As for pricing errors, by convention, the degree of moneyness has a relation to deal

with pricing errors where-by, in order to incur a higher volatility, both out-of-the-money

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puts and calls’ degree of moneyness and its square are employed as explanatory variables

(Madan, Carr, et. al, 1998). Option maturity and implied volatilities are directly related

where-as, approaching the maturity of an option, leads to higher implied volatility. Thus,

option maturity is employed as an explanatory variable. Furthermore, the level of interest

rates is used as a supplementary regressor. In fact, Black-Scholes implied volatility

proposes that the coefficient of the degree of moneyness ought to be negative, while the

the coefficients for the square of moneyness and the maturity of the option should be

negative (Madan, Carr, et. al, 1998).

To focus on the attractive features of the Black-Scholes-Merton option pricing

model, finance practitioners highlight in their studies the possibility of obtaining closed,

exact formulas for pricing options and other financial derivative securities. The model is

of high quality and its features are regarded as vital. In fact, this model is very useful

when fast calculations are required in finance and mathematics (such as: calibration

purposes and hedging risky assets) (Ortisi et Zuccolo, 2013). Bernstein (1992) refers to

the Black-Scholes model as “The Universe Financial device”.

As for hedging, the Black-Scholes no-arbitrage argument provides us with a

rational option pricing formula, as well as, a hedging portfolio. This hedging portfolio

requires continuous trading, leading to a prohibitively expensive process due to the fact

that the market is characterized by proportional transaction costs (Barles et Soner, 1998).

Basically, such a market lacks the portfolio which replicates European call options. Thus,

it is mandatory to relax the hedging condition, causing domination in the portfolio rather

than a replication in the value of the option. Such a relaxation implies a trivial dominating

hedging portfolio of holding a single share of the stock where the call is written (Barles

et Soner, 1998). The straight-forward arbitrage argument states that all viable option

prices should be less that the smallest initial capital which can support a dominating

portfolio. Despite the interesting results which this approach has proposed for markets

with no transaction costs, an alternate relaxation of perfect hedging in markets with

transaction costs was heavily needed. So, an adjustment in volatility was proposed by

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Leland who derived an option price which is equal to the Black-Scholes price but with

an adjusted volatility (Barles et Soner, 1998).

Equation extracted from (Barles et Soner, 1998).

where _ is the original volatility, _ is the proportional transaction cost and _t is the

transaction frequency. In this formula, both _ and _t are assumed to be small while

keeping the ratio µ/√ ∆t order one (Barles et Soner, 1998).

2.2 OPTIONS

An option is one type of derivatives which gives the holder the right to buy or sell an

asset. Just like all derivatives, options are subject to certain conditions, having a specified

time period. “American” options are ones that can be exercised at any point of time until

the option expires. On the other hand, “European” options can only be exercised at time

of expiration. When an option is exercised, the price paid for exercising the option is

referred as the “exercise price” or “strike price”. Lastly, the “maturity date” or the

“expiration date” is the time when the option is expired (Black et Scholes, 1973).

Options and stocks are directly related where-as, the higher the price of the stock, the

greater the value of the option. When the price of a stock exceeds the exercise price, the

option is likely to be exercised. Consequently, the current value of an option will be

almost equal to the stock price minus the pure discount bond price which matures on the

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same date of the option, having a face value that is exactly equal to the strike price of the

option (Black et Scholes,1973). However, when the stock price is lower than the

exercise price, the option is likely to get expired without being exercised, implying that

the value of option will be close to zero (Black et Scholes,1973).

Concerning the maturity of the option, when the expiration date is far in the future,

the price of the bond that pays the exercise price on the expiration date would be very

low, leading to an equilibrium in the value of the option and the price of the stock (Black

et Scholes, 1973). However, when the expiration date gets very close, the value of the

option and the stock price would be approximately the same (while deducting the

exercise price). Lastly, when the stock price is lower that the exercise price, the option

would have zero value. In fact, the value of an option diminishes through maturity if the

stock value is not altered (Black et Scholes, 1973).

figure 2,(Black et Scholes, 1973) illustrates the relationship between the option value

and the stock price.

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When an underlying asset’s price satisfies a stochastic differential equation which is

an impact of random motions developed from standard distributions, Option pricing

theory is applied. An options’ value depends on the way the underlying asset is

distributed (Tsibirdi et Atkinson, 2003).

In 1990, Bachelier’s used the “Central Limit Theorem” in order to restrict the option

pricing model to the normal distribution and this has dominated the whole theory of

option pricing. Then, it was modified onto a log-normal for keeping the underlying asset

value positive. The Central Limit Theorem declares that the only limiting distribution,

identically distributed (iid) random variables is the normal distribution. Such a normal

distribution assumes that the unforeseen alterations in an asset’s price aggregates

throughout the life of an option are the addition of the (iid) random shocks which happen

daily and occur minute by minute. Such random shocks are regarded as the collective

outcome of millions of agent transactions. Black and Scholes (1973) is the most well-

known model which illustrates the option pricing formula. Although, in reality, asset

prices have a long fat tail due to the normality assumption that neglects market crashes.

In a normal distribution, a vast alteration will occur only as an outcome of a vast number

of small changes. Also, the delta hedging strategy of the Black and Scholes model,

eradicates the uncertainty (risk) through perpetual rebalacement of the portfolio. This can

no longer be applied in other models which have infinite number of discontinuities. In

actual situations, the Black-Scholes world cannot hedge uncertainty completely due to the

transaction costs which are entangled when the portfolio is frequently rebalanced. For

instance, delta hedging is not considered as a sufficient hedging strategy in more general

models which have huge fluctuations in prices of options (Tsibirdi et Atkinson, 2003).

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2.3 LIMITATIONS OF THE BLACK-SCHOLES OPTION PRICING MODEL:

Although the Black-Scholes option pricing model has created a boom in finance and

economics, two empirical phenomena were discovered. Firstly, asymmetric leptokurtic

features which imply that the return distribution has a left skew, and has a higher peak

with two heavier tails than the normal distribution. Secondly, the volatility smile. In fact,

if this model was completely accurate, the implied volatility would have been constant. In

actual situations, the implied volatility curve represents a “smile” which implies a convex

curve of the strike price (Kou,2002). The lognormal probability distribution of the

underlying asset assumption is considered very controversial (Hajizadeh et Seifi,2011).

To overcome the smile bias which the Black-Scholes option pricing model presents,

several variant models were proposed for option pricing. However, the results proposed

by such models were not as significant as the Black-Scholes due to their complexity in

computations and analytical intractability. Basically, all the option pricing models were

originally generated from the Black-Scholes; thus, they are just considered as extensions

of the classical Black-Scholes option pricing model which was constructed by Fischer

Black and Myron Scholes in (1973).

In this model, option prices are given as a function of volatility meaning that if the

price of an option is known in the market, we can invert this relationship to know the

implied volatility. Thus, If the model was free from limitations, implied volatility would

be exactly the same for all option market prices, but this case is not proved in reality. The

stock price of European option under scrutiny and maturity are correlated with the

implied Black-Scholes volatilities (Dupire, 2004).

The method which was proposed by the Black-Scholes-Merton model was

constructed to value traded options (risk-neutral valuation) and with no doubt, has

revolutionized finance. On the other hand, the suggested instrument was designed for

much simpler investment vehicles rather than the stock options that firms promote to

their employees. Over the past 40 years, sophisticated models were developed than the

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Merton-Black-Scholes (MBS) and reflected the substantive characteristics of traditional

(“ at-the money” and time-vested) ESOs more properly. In fact, more complex

instruments are now being used by firms for compensating their key employees (such as:

premium, capped, and market-based instruments) (Rudkin et Bosco, 2014).

The basic assumptions in the Black-Scholes determination are that exchanging

happens continuously in time. Furthermore, the value motion of the stock has a persistent

way with the probability one. It is pompous to assert that the Black-Scholes investigation

is invalid on the grounds that persistent exchanging is unrealistic and in light of the fact

that no exact time arrangement has a ceaseless example way. In another connection,

Merton demonstrates that consistent exchanging arrangement will be a substantial

asymptotic estimate to the discrete-exchanging arrangement. Under these same

conditions, the profits on the Black-Scholes "no-risk" arbitrage portfolio will be facing

some risk. However, the level of this risk will be a limited, continuous function of the

trading interval length, and the risk will go to zero as the trading interval goes to its

continuous limit. So, the interval length is not "too extensive," the contrast between the

Black-Scholes constant exchanging alternative cost and the "right," discrete-exchanging

cost cannot contrast by much without making a "virtual" arbitrage plausibility. In fact, the

Black-Scholes arrangement is not legitimate, even in the persistent breaking point, when

the stock value progress can't be spoken to by a stochastic procedure with a ceaseless

example way. Basically, the legitimacy of the Black-Scholes formula relies on upon

whether the stock value changes to fulfill a sort of "nearby" Markov property. I.e., in a

short interval of time, the stock price can only change by a small amount (Merton, 1975).

Another limitation was proposed by (Geske & Roll, 1984) whereby, the Black-

Scholes-Merton model exhibits systematic empirical biases in valuating American call

options. Thus, an ad hoc adjustment of the European call option formula was further

developed in order to obtain several call values. The maximum of these values is referred

to “pseudo” American- call price whereas, it is closest to the actual American value

(Geske & Roll, 1984). Furthermore, the assumption of a constant default free interest

rate of the Black-Scholes model could be regarded as further limitation. In fact,

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generalizing the model to allow for stochastic interest rates could be an another possible

deficiency due to the concept of variance. If the variance of the interest rate correlated

with the stock price or the interest rate level, α2 might differ from zero (Lauterbach et

Schultz, 1990).

2.3.1 Application of the Black-Scholes-Merton model in reality:

2.3.1.1 Employee Stock Options:

According to the application of Black-Scholes model in reality, Rudkin and Bosco

(2014) have a refutation towards the Black-Scholes model claiming that it is not the best

way for valuing employee stock options (ESOs) specifically because they differ from

traded options. In fact, ESOs are more complicated than the simple traded European

options which the Black-Scholes model and its progeny (Modified Black-Scholes) are

created to value. So, newer models were developed for valuing ESOs and are considered

more accurate and propose typically lower fair value estimates that the Merton-Black-

Scholes model (Rudkin et Bosco, 2014).

2.3.1.2 Pricing Warrants:

As for pricing warrants, biases in model pricing for approximately all warrants have

occurred as a result of the constant variance assumption of the dilution of the Black-

Scholes model (Lauterbach et Schultz, 1990). Furthermore, the model does not

highlight the importance of frequently trading securities whereas, trading securities

frequently, can enable a "few" multiperiod securities to span in "many" states of nature.

In the Black-Scholes model there are two securities and uncountably many states of

nature, but because there are infinitely many trading opportunities and, what is worth

mentioning and is of great importance, uncertainty resolves "nicely", markets are

effectively complete. Even though there are far fewer securities than states of nature,

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nonetheless there is a complete (or nearly complete) set of contingent claims markets.

Therefore, risk is allocated efficiently (Kreps,1982).

3. Methodology:

3.1 An introduction to research design:

The approach of this research is deductive in light of the fact that hypotheses are

obtained and tested in order to validate or not the underlying theory. Deduction implies

the inspection of the hypothesis and it is considered as the most used research approach

as a part of common sciences. It is worth mentioning that all the arguments, assumptions,

and expectations for the occurrence of a specific phenomenon, which permit them to be

overseen and controlled, depend on certain laws and regulations (Saunders, Lewis, et

al., 2009).

A deductive examination ought to follow five consecutive steps: the initial step is to get a

speculation from the hypothesis to test connection between two or more variables.

Moreover, theory is presented in operational terms to elucidate how the variables will be

tested. Furthermore, the operational hypothesis should be measured. Then, determining

whether the hypothesis was proven or not. Last but not least, is to redesign the hypothesis

taking into account the outcomes; in any case, this step is not fundamental. However, it is

mandatory for the research to experience all those steps effectively and in the right

sequence.

As for the attributes, the deduction approach has a few attributes and characteristics

which are: it shows the relationship between two or more variables, gives controls or

permits different variables to be held consistent keeping in mind the end goal to help

hypothesis testing, takes after an all around acceptable and replicable system, infers that

the scientist must be absolutely free of what is being considered, tests perceptions

quantitatively through the operational theory, and the last attribute is speculation, which

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refers to the addition of results on an example that can speak to an entire population

(Saunders, Lewis, et al., 2009).

According to the philosophy of this study, it is positivism which is known as the

customary theory of most researchers. The positivism reasoning uses a great deal of

observational techniques and quantitative examination. This logic demonstrates that the

consequences of the study will be founded on genuine social perceptions; accordingly, it

ought to be reliable and dependable. The final result ought to resemble a general law

extracted from genuine records of information and can be utilized for further looks into.

An advocate of the positivism philosophy is called a “resources” researcher which is the

exact contrast of a “feelings” researcher. The independence of the “resources” researcher

has no limit which implies that he/she has no impact on the subject of the study and also,

is not affected by the subject of the study. The fact that the researcher has a separate

identity from the study and is considered as a third party, implies consistency in the

results obtained. Thus, the research conducted by a “resources” researcher is considered

free from personal value judgments due to the fact that the “resources” researcher is

regarded as a value free researcher. To conclude, the positivism philosophy targets

reliability through the independency of social factors and externality (Saunders et al.,

2009).

Concerning information and data collection techniques, data can be gathered

through utilizing a number of different strategies or techniques. The technique that was

utilized to gather information and test the hypotheses for this study is the

document/records technique. This technique is additionally called the archival research

strategy whereby, it gathers information from repository archives. Those archived

documents can either be recent or historical like on account of this study. Recorded

information speaks to a part of reality; subsequently, they ought to be broke down to

mirror the genuine every day exercises. The historical technique can be utilized for

exploratory, graphic, and logical investigates (Saunders et al., 2009).

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3.2 Testing the Effectiveness of the Black-Scholes model for option pricing:

In order to test the effectiveness of the Black-Scholes model against it’s alternatives,

a number of studies relied on regressions such as (Lauterbach & Schults, 1990) the

following equation was used for each warrant for each quarter over 1971-1980:

Other studies, obtained the option data from the Chicago Mercantile Exchange (CME)

where the stocks were actually being traded in the market (Madan, Carr, et. al, 1998). After

reviewing a number of studies on the Black-Scholes model, it was observed that further

studies were originated from the original Black-Scholes due to its effectiveness in the

valuating of options. Models like the Trinomial tree, Monte-Carlo, and the Variance

Gamma process were developed for the purpose of pricing financial derivatives such as:

Option contracts. Recent studies constructed simpler formulas for option pricing such as

(Dupire, 2004). However, all the proposed studies are considered branches of the

original Black-Scholes model.

The primary objective of this paper is to test whether there will be a volatility smile

effect or a skewed volatility effect. Volatility Skewness is a type of market imperfection

whereby, the term “skew” demonstrates a trader’s expectation of future volatility (Sircar

& Papanicolaou, 1998). The concept of implied volatility was proposed in empirical

studies by (Latane & Redelman, 1976) and also by Beckers (1981), Rubinstein

(1985), and Canina & Figlewski (1993) and has been used to reveal the denoting

discrepancy between both, market prices and Black-Scholes model prices; whereby, the

strike price and the time to maturity of the options contract are two variables which lead

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to alterations in implied volatilities of market prices. For instance, the alteration of

implied volatility with strike price for options of equivalent time-to-maturity is usually a

U-shaped curve with minimum at or near the money. Thus, implying a smile curve which

is approximate to the current asset price (Sircar & Papanicolaou, 1998). According to

the concept of volatility smile, studies which revealed the limitations of the Black-

Scholes model, acknowledged the smile effect: that implied volatilities of market prices

are not constant across the range of options, but differ with strike price and time to

maturity of the contract (Sircar & Papanicolaou, 1998).

In this chapter, the model’s accuracy will be tested through assessing whether

theoretical prices provided by the Black-Scholes (using the historical standard deviation

of returns for the underlying asset) deviate or not from the correspondent market prices.

A number of studies show that discrepancies in prices are more significant for in the

money and out of the money options. Robert Merton demonstrated that, for deep-in-the-

money options, there is comparatively slight probability that the stock would decelerate

below the exercise price (Merton, 1975).

Information about the underlying equity stock symbol, option details (delivery

date, strike price, option class, and option symbol) were obtained from

http://finance.yahoo.com/options/ . In order to obtain a clear and a fair assessment, two

call options as well as two put options that are actively traded in the market were chosen.

The call options chosen are: UUP and EEM while the put options chosen are: GDX and

SPY. Four firms were chosen: Power Shares DB US Dollar Bullish, iShares MSCI

Emerging Markets, Market Vectors Gold Miners (ETF) GDX, and SPDR S&P 500 ETF

(SPY). The most recent available information of the past three months prices for each

stock was collected and from this information, the daily price returns for each was

calculated as well as the standard deviation of returns (daily STDEV and annual

STDEV). The Straddle for each call option and put option was obtained from

22

http://finance.yahoo.com/options/

http://finance.yahoo.com/options/ as well in order to get the strike price. Lastly, the

difference between the Black & Scholes price and the actual price was calculated.

3.3 Results & findings:

3.3.1 Call Options:

3.3.1.1 UUP Power Shares DB US Dollar Bullish (In the money)

Date Adj Close Daily Returns Strike Price Difference

1/4/2016 25.690001 23 2.690001

1/5/2016 25.84 0.005821829 23.5 2.34

1/6/2016 25.77 -0.002712654 24 1.77

1/7/2016 25.51 -0.010140493 24.5 1.01

1/8/2016 25.559999 0.001958058 25 0.559999

1/11/2016 25.68 0.004683888 25.5 0.18

1/12/2016 25.719999 0.001556382 26 -0.280001

1/13/2016 25.68 -0.001556382 26.5 -0.82

1/14/2016 25.75 0.002722148 27 -1.25

1/15/2016 25.690001 -0.002332777 27.5 -1.809999

1/19/2016 25.74 0.001944352 28 -2.26

1/20/2016 25.75 0.000388425 28.5 -2.75

1/21/2016 25.719999 -0.001165767 29 -3.280001

1/22/2016 25.860001 0.005428552 29.5 -3.639999

1/25/2016 25.780001 -0.003098376 30 -4.219999

1/26/2016 25.73 -0.00194141 30.5 -4.77

1/27/2016 25.709999 -0.000777644 31 -5.290001

1/28/2016 25.59 -0.004678332 31.5 -5.91

1/29/2016 25.860001 0.010495762 32 -6.139999

23


2/1/2016 25.690001 -0.006595562 32.5 -6.809999

2/2/2016 25.67 -0.000778855 33 -7.33

2/3/2016 25.23 -0.017289232 33.5 -8.27

2/4/2016 25.059999 -0.006760853 34 -8.940001

2/5/2016 25.17 0.0043799 34.5 -9.33

2/8/2016 25.110001 -0.002386596 35 -9.889999

2/9/2016 24.93 -0.007194315 35.5 -10.57

2/10/2016 24.879999 -0.00200767 36 -11.120001

2/11/2016 24.809999 -0.00281747 36.5 -11.690001

2/12/2016 24.92 0.004423937 37 -12.08

2/16/2016 25.139999 0.008789469 37.5 -12.360001

2/17/2016 25.129999 -0.000397852 38 -12.870001

2/18/2016 25.139999 0.000397852 38.5 -13.360001

2/19/2016 25.07 -0.002788251 39 -13.93

2/22/2016 25.26 0.007550205 39.5 -14.24

2/23/2016 25.27 0.000395804 40 -14.73

2/24/2016 25.290001 0.000791179 40.5 -15.209999

2/25/2016 25.27 -0.000791179 41 -15.73

2/26/2016 25.440001 0.006704856 41.5 -16.059999

2/29/2016 25.49 0.001963441 42 -16.51

3/1/2016 25.5 0.000392234 42.5 -17

3/2/2016 25.48 -0.000784621 43 -17.52

3/3/2016 25.32 -0.006299233 43.5 -18.18

3/4/2016 25.25 -0.002768442 44 -18.75

3/7/2016 25.200001 -0.001982122 44.5 -19.299999

3/8/2016 25.209999 0.000396667 45 -19.790001

3/9/2016 25.219999 0.000396589 45.5 -20.280001

3/10/2016 24.969999 -0.009962227 46 -21.030001

3/11/2016 24.950001 -0.000801202 46.5 -21.549999

3/14/2016 25.049999 0.003999925 47 -21.950001

3/15/2016 25.07 0.000798125 47.5 -22.43

3/16/2016 24.799999 -0.010828299 48 -23.200001

24

3/17/2016 24.58 -0.008910509 48.5 -23.92

3/18/2016 24.639999 0.002437994 49 -24.360001

3/21/2016 24.719999 0.003241494 49.5 -24.780001

3/22/2016 24.790001 0.002827794 50 -25.209999

3/23/2016 24.92 0.005230308 50.5 -25.58

3/24/2016 24.93 0.000401204 51 -26.07

Daily STDEV 0.005150828

Annual STDEV 0.000324472

Comment: The strike price increases by 0.5 for every running period and as it

increases, the difference between the Black & Scholes price and the actual price,

decreases due to the fact that discrepancies in prices are more significant for in the money

options. The results obtained validate the effectiveness of the Black & Scholes option

pricing model.

3.3.1.2 EEM iShares MSCI Emerging Markets (In the money):

Date

Adj

Close Daily Returns K (Strike Price) Difference

3/1/2016 31.4 29.5 1.9

3/2/2016 31.82 0.01328713 30 1.82

3/3/2016 32.18 0.011250119 30.5 1.68

3/4/2016 32.82 0.019692944 31 1.82

3/7/2016 32.77 -0.001524623 31.5 1.27

3/8/2016 32.21 -0.01723653 32 0.209999

3/9/2016 32.41 0.006190085 32.5 -0.09

3/10/2016 32.46 0.001541514 33.5 -1.040001

25

3/11/2016 33.14 0.020732451 35 -1.860001

3/14/2016 32.94 -0.006053287 35.5 -2.560001

3/15/2016 32.47 -0.014371077 40 -7.529999

3/16/2016 33.12 0.019820682 40.5 -7.380001

3/17/2016 33.85 0.021801641 41 -7.150002

3/18/2016 34.03 0.005303519 41.5 -7.470001

3/21/2016 34.1 0.002054867 42 -7.900002

3/22/2016 34.02 -0.002348739 42.5 -8.48

3/23/2016 33.44 -0.017195829 43 -9.560001

3/24/2016 33.36 -0.002395151 43.5 -10.139999

3/28/2016 33.47 0.003291938 44 -10.529999

3/29/2016 33.93 0.013650034 44.5 -10.57

3/30/2016 34.28 0.010262486 45 -10.720001

3/31/2016 34.25 -0.0008755 45.5 -11.25

4/1/2016 34.15 -0.00292392 46 -11.849998

4/4/2016 33.74 -0.012078508 46.5 -12.759998

4/5/2016 33.08 -0.019755206 47 -13.919998

4/6/2016 33.48 0.012019315 47.5 -14.02

4/7/2016 32.81 -0.020214868 48 -15.189999

4/8/2016 33.38 0.017223571 48.5 -15.119999

4/11/2016 33.81 0.012799698 49 -15.189999

4/12/2016 34.33 0.015263019 49.5 -15.169998

4/13/2016 34.94 0.01761261 50 -15.060001

4/14/2016 34.77 -0.00487733 50.5 -15.73

4/15/2016 34.57 -0.005768692 51 -16.43

4/18/2016 34.73 0.004617613 51.5 -16.77

4/19/2016 35.26 0.015145253 52 -16.740002

4/20/2016 35.1 -0.004548047 52.5 -17.400002

4/21/2016 34.75 -0.010021501 53 -18.25

4/22/2016 34.54 -0.00606147 53.5 -18.959999

4/25/2016 34.32 -0.006389827 54 -19.68

26

4/26/2016 34.69 0.010723158 54.5 -19.810001

4/27/2016 34.9 0.006035454 55 -20.099998

4/28/2016 34.54 -0.010368785 55.5 -20.959999

4/29/2016 34.39 -0.004352306 56 -21.610001

5/2/2016 34.3 -0.00262047 56.5 -22.200001







pricing model.

3.3.2 Put Options:

3.3.2.1 GDX : Market Vectors Gold Miners ETF (GDX) (In the money)


1/4/2016 14.09 11 3.09

1/5/2016 14.02 -0.004980444 13.5 0.52

1/6/2016 14.25 0.016272025 14 0.25

1/7/2016 14.88 0.043261123 14.5 0.38

1/8/2016 14.52 -0.02449102 15 -0.48

27

1/11/2016 13.92 -0.042200354 15.5 -1.58

1/12/2016 13.61 -0.022521838 16 -2.39

1/13/2016 13.61 0 16.5 -2.89

1/14/2016 13.13 -0.035905128 17 -3.87

1/15/2016 13.09 -0.003051108 17.5 -4.41

1/19/2016 12.47 -0.04852282 18 -5.53

1/20/2016 12.85 0.030018052 18.5 -5.65

1/21/2016 12.91 0.004658394 19 -6.09

1/22/2016 13.03 0.009252186 19.5 -6.47

1/25/2016 13.38 0.026506664 20 -6.62

1/26/2016 13.97 0.043151119 20.5 -6.53

1/27/2016 14.2 0.016329791 21 -6.8

1/28/2016 13.86 -0.024234971 21.5 -7.64

1/29/2016 14.21 0.024938948 22 -7.79

2/1/2016 14.65 0.030494393 22.5 -7.85

2/2/2016 14.3 -0.024180798 23 -8.7

2/3/2016 15.35 0.070855937 23.5 -8.15

2/4/2016 16.15 0.050804576 24 -7.85

2/5/2016 17.049999 0.054230095 24.5 -7.450001

2/8/2016 17.469999 0.024334922 25 -7.530001

2/9/2016 16.75 -0.042086809 25.5 -8.75

2/10/2016 17.139999 0.023016597 26 -8.860001

2/11/2016 18.370001 0.069304099 26.5 -8.129999

2/12/2016 18.84 0.025263315 27 -8.16

2/16/2016 17.209999 -0.090491717 27.5 -10.290001

2/17/2016 17.82 0.03483087 28 -10.18

2/18/2016 18.9 0.0588405 28.5 -9.6

2/19/2016 18.379999 -0.02789886 29 -10.620001

2/22/2016 18.530001 0.008128032 29.5 -10.969999

2/23/2016 18.92 0.020828469 30 -11.08

2/24/2016 19.110001 0.009992247 30.5 -11.389999

2/25/2016 19.389999 0.014545607 31 -11.610001

28

2/26/2016 18.690001 -0.036768743 31.5 -12.809999

2/29/2016 19.379999 0.03625288 32 -12.620001

3/1/2016 18.57 -0.042694179 32.5 -13.93

3/2/2016 19.049999 0.025519674 33 -13.950001

3/3/2016 19.82 0.03962448 33.5 -13.68

3/4/2016 19.709999 -0.005565458 34 -14.290001

3/7/2016 20.4 0.03440883 34.5 -14.1

3/8/2016 19.42 -0.049231438 35 -15.58

3/9/2016 19.51 0.004623692 35.5 -15.99

3/10/2016 20.379999 0.043626824 36 -15.620001

3/11/2016 19.98 -0.019822206 36.5 -16.52

3/14/2016 19.120001 -0.043996813 37 -17.879999

3/15/2016 19.530001 0.021216836 37.5 -17.969999

3/16/2016 20.860001 0.065881701 38 -17.139999

3/17/2016 20.43 -0.020829089 38.5 -18.07

3/18/2016 20.610001 0.008772035 39 -18.389999

3/21/2016 20.57 -0.00194274 39.5 -18.93

3/22/2016 20.58 0.000486027 40 -19.42

3/23/2016 19.030001 -0.078302997 40.5 -21.469999

3/24/2016 19.459999 0.022344292 41 -21.540001

3/28/2016 19.42 -0.002057563 41.5 -22.08

3/29/2016 20.540001 0.05607079 42 -21.459999






29


pricing model.

30

3.3.2.2 SPY SPDR S&P 500 ETF (In the money):


1/4/2016 199.988533 155 44.988533

1/5/2016 200.326785 0.001689928 160 40.326785

1/6/2016 197.799825 -0.012694424 165 32.799825

1/7/2016 193.054297 -0.024284054 170 23.054297

1/8/2016 190.935221 -0.011037267 175 15.935221

1/11/2016 191.124249 0.000989521 180 11.124249

1/12/2016 192.666298 0.008035932 185 7.666298

1/13/2016 187.86108 -0.025256913 190 -2.13892

1/14/2016 190.945164 0.016283534 195 -4.054836

1/15/2016 186.84631 -0.02169988 200 -13.15369

1/19/2016 187.095027 0.001330246 205 -17.904973

1/20/2016 184.697389 -0.012897903 210 -25.302611

1/21/2016 185.732061 0.005586352 215 -29.267939

1/22/2016 189.542411 0.020307701 220 -30.457589

1/25/2016 186.677184 -0.015231968 225 -38.322816

1/26/2016 189.224046 0.013550905 230 -40.775954

1/27/2016 187.164675 -0.010942897 235 -47.835325

1/28/2016 188.139642 0.005195619 240 -51.860358

1/29/2016 192.725988 0.024084965 245 -52.274012

2/1/2016 192.65634 -0.000361449 250 -57.34366

2/2/2016 189.184257 -0.018186535 255 -65.815743

2/3/2016 190.318407 0.005977051 260 -69.681593

2/4/2016 190.616871 0.001567007 265 -74.383129

2/5/2016 186.985591 -0.019233942 270 -83.014409

2/8/2016 184.468574 -0.013552442 275 -90.531426

2/9/2016 184.478517 5.38993E-05 280 -95.521483

2/10/2016 184.31935 -0.000863167 285 -100.68065

2/11/2016 181.921712 -0.01309341 290 -108.078288

31

2/12/2016 185.672372 0.020407239 295 -109.327628

2/16/2016 188.806202 0.016737423 300 -111.193798

2/17/2016 191.890302 0.016202762 305 -113.109698

2/18/2016 191.104347 -0.004104267 310 -118.895653

2/19/2016 191.014812 -0.000468623 315 -123.985188

2/22/2016 193.780547 0.014375343 320 -126.219453

2/23/2016 191.333178 -0.012710022 325 -133.666822

2/24/2016 192.208652 0.004565215 330 -137.791348

2/25/2016 194.536641 0.01203902 335 -140.463359

2/26/2016 194.088953 -0.002303956 340 -145.911047

2/29/2016 192.566805 -0.007873442 345 -152.433195

3/1/2016 197.093462 0.02323491 350 -152.906538

3/2/2016 197.978894 0.004482386 355 -157.021106

3/3/2016 198.754891 0.003911933 360 -161.245109

3/4/2016 199.401549 0.003248264 365 -165.598451

3/7/2016 199.560732 0.000797985 370 -170.439268

3/8/2016 197.381967 -0.010977841 375 -177.618033

3/9/2016 198.356949 0.00492741 380 -181.643051

3/10/2016 198.516117 0.00080211 385 -186.483883

3/11/2016 201.719595 0.016008299 390 -188.280405

3/14/2016 201.460935 -0.001283098 395 -193.539065

3/15/2016 201.132626 -0.00163097 400 -198.867374

3/16/2016 202.296621 0.00577052 405 -202.703379

3/17/2016 203.58001 0.006324056 410 -206.41999

3/18/2016 204.380005 0.003921934 415 -210.619995

3/21/2016 204.669998 0.001417886 420 -215.330002

3/22/2016 204.559998 -0.000537595 425 -220.440002

3/23/2016 203.210007 -0.00662136 430 -226.789993

3/24/2016 203.119995 -0.000443049 435 -231.880005

3/28/2016 203.240005 0.000590659 440 -236.759995

3/29/2016 205.119995 0.009207578 445 -239.880005

32


Annual

STDEV 0.000743631





pricing model.

4. Conclusion

In the last 30 years, financial derivatives have developed from a minimal action to

possess an inside stage position in money related monetary hypothesis and budgetary

practice. At the same time, mathematical finance has become one of the fundamental

branches of applied mathematics. The single biggest credit for these wonderful

advancements is because of Fisher Black, Myron Scholes, and Robert Merton, whose

exemplary 1973 papers gave a hypothesis of how to value options. Without this

prescription, valuating options would have remained a greater amount of a craftsmanship

rather than a science, and exchanging options would have been less liquid and less

important, as traders would have had less knowledge than nowadays on how to fairly

value and hedge the options. Then again, this appreciated accomplishment lays on the

assumption of no arbitrage opportunities, lognormality for spot price dynamics, and

frictionless exchanging. In reality, despite the fact that the state of arbitrage free and the

assumption of lognormality are contended to be attractive more often than not, exchange

expenses undenibly exist.

33

In order to explore the effectiveness of the Black & Scholes option pricing model, the

model was briefly explained through historical evidence where Fischer Black and Myron

Scholes first constructed this parametric model in 1973. This model was primarily

created to mitigate risk whereby, the no arbitrage opportunity assumption reflects this.

The major focus of this model is on the call options’ gains/losses which are basically the

differences between the pay-off of the option and the purchasing cost of the call.

Assumptions like lognormality and no-arbitrage opportunities are the principles of this

model. Furthermore, the Black, Scholes, Merton model formula is briefly explained as

well as the required conditions to ensure a fair application of the model. Moreover, the

relationship between the option value and stock price is summarized in a simple graph.

According to the limitations and criticism of this unique option pricing model, they

are as follows: First, the lognormal probability distribution of the underlying asset

assumption is considered very controversial. Second, the Black Scholes model exhibits

systematic empirical biases in valuating American call options. Third, generalizing the

model to allow for stochastic interest rates could be considered as an another deficiency

of the model due to the concept of variance. Fourth, the Black Scholes option pricing

model is not the most preferable model for valuating ESOs (Employee Stock Options).

Fifth, biases in model pricing for approximately all warrants have occurred due to the

constant variance assumption of the dilution of the Black Scholes model.

The primarily concern of this study is to test whether there will be a volatility smile

or a skewed-volatility effect. Volatility skewness is a type of market imperfection.

After selecting the methodology and collecting the data, consistent results were obtained

which support literature on the Black-Scholes option pricing model. Minor differences in

results between call options and put options were observed and the fact that discrepancies

in prices for in the money options was validated. As for the strike price, we observed the

same increase of $0.5.

To conclude, this model is of high quality and very beneficial despite the fact that

some of its assumptions are unrealistic. However, if Fischer Black & Myron Scholes

have not constructed such a hypothesis, valuating financial derivatives (specifically,

35

options) could have almost been impossible and the existence of newer models and

studies which already exist in today’s financial environment, would not have emerged

like nowadays.

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Abbas Seifia, Ehsan Hajizadeha. 'A Hybrid Modeling Approach For Option Pricing'.

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Web.

Black, Fischer, and Myron Scholes. 'The Pricing Of Options And Corporate

Liabilities'. Journal of Political Economy 81.3 (1973): 637. Web.

Company, R. et al. 'Explicit Solution Of Black–Scholes Option Pricing Mathematical

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Modelling 45.1-2 (2007): 80-92. Web.

Don Chance, Robert Brooks. 'Some Subtle Relationships And Results In Option Pricing'.

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El-Khatib, Youssef, and Abdulnasser Hatemi-J. 'On Option Pricing In Illiquid Markets

With Jumps'.ISRN Mathematical Analysis 2013 (2013): 1-5. Web.

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Journal n. pag. Web.

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LAUTERBACH, BENI, and PAUL SCHULTZ. 'Pricing Warrants: An Empirical Study

Of The Black-Scholes Model And Its Alternatives'. The Journal of Finance 45.4

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MacLean, Leonard, Yonggan Zhao, and William T. Ziemba. 'An Endogenous Volatility

Approach To Pricing And Hedging Call Options With Transaction Costs'. SSRN

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Madan, D. B., P. P. Carr, and E. C. Chang. 'The Variance Gamma Process And Option

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'Pricing With A Smile'. (2015): n. pag. Print.

Rodney J. Bosco, Ron D. Rudkin. 'Valuation Of Employee Stock Options'. (2014): n.

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Results and findings were obtained from: http://finance.yahoo.com/options/

37


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