DERIVATION OF THE BLACK SCHOLES MODEL AND ITS IMPLICATIONS UCSI UNIVERSITY

BY JEEVAN KUMAR & JACKSON TAM KONG WAI

2013

B.SC. (HONS) ACTUARIAL SCIENCE

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ACKNOWLEDGEMENTS

First and foremost, we would like to express our sincerest acknowledgements and gratitude to

our supervisor, Mr. Lim Fang Ching, for his guidance and support throughout the process of

completing this research study. Mr. Lim showed his interest in the subject and his

professionalism and also his patience while supervising us in this research paper. Without his

supervision, this research would not have completed in such a timely and professional

manner. We sincerely appreciate the guidance, encouragement and counsel that he gave to us.

Next, we would like to thanks to our course mates who gave their hand to us when we meet

some difficulties. Lastly, we would like to express our deepest acknowledgement to our

parents who have been very supportive during our study in this course.

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CONTENTS

1.0 Summary 1

1.1 General Introduction 2

1.2 Problem Statement 4

1.3 Research Objectives 5

1.4 Definition of Terms 6

2.0 Brief History of Options 8

2.1 Options Trading in Malaysia 8

3.0 Literature Review 10

3.1 Empirical Testing on Different Market 12

3.2 The Important Assumptions 13

3.3 The Advantages of Black Scholes Model 14

3.3.1 Implied Volatility 14

3.3.2 Volatility Smile 15

3.3.3 Volatility Skew 16

3.3.4 Implication of Volatility Skew for Option Pricing 17

3.4 Errors Implying towards the Black Scholes Model 18

3.5 Summary of Chapter 29

4.0 Methodology 31

4.1 Stochastic Differential Equation 31

4.1.1 Geometric Brownian Motion 32

4.2 Arbitrage 32

4.3 Self-Financing 33

4.4 Risk Neutral Valuation 33

4.5 Ito‟s Lemma 34

4.6 Log normal Dynamics 36

4.7 Factors Affecting the Option Value 38

4.8 Black Scholes Model 39

4.8.1 American and European Options 42

4.8.2 American and European Calls 43

4.8.3 Solution of the Black Scholes Equation 44

4.8.4 Volatility 48

4.8.5 Historical/Empirical Method 49

4.8.6 Implied Volatility 50

4.9 Dividend Paying Stocks 50

4.9.1 Continuous Dividend Yield Model 50

4.10 Options on Futures 51

4.10.1 American Options on Futures 51

4.10.2 European Options on Futures 52

4.10.3 Put-Call Parity 52

4.10.4 Black‟s Model 53

4.11 Binomial Model 54

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5.0 Findings and Analysis 58

6.0 Conclusion 60

6.1 Summary of Black Scholes Model 60

6.2 Volatility; Implied or Historical? 61

6.3 Risk Neutral Valuation 62

6.4 Advantages and Limitations 63

6.4.1 Limitation of Black Scholes Model 63

6.4.2 Other Similar Models to Black Scholes Model 65

7.0 References 66

8.0 Appendix 68

8.1 VBA Code of CRR Binomial Model in Excel 68

8.2 VBA Code of Black Scholes Model in Excel 70

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1.0 Summary

This present thesis “The Derivation of Black Scholes Model” aims at exploring and

explaining the fundamental of Black Scholes Model and the various equations surrounding it.

Throughout these derivations, core concepts in financial mathematics, such as Ito‟s Lemma,

arbitrage pricing, risk-neutral pricing and put-call parity are also presented. We will also

introduce the immense effects and contributions of the model towards the economy, and the

limitation of the Black Scholes Model that has brought many financial experts to question

about its reliability.

First and foremost, we will introduce the option background and point out the emergence and

discussion of the research problem. In the end of this discussion a disposition of the entire

process will be given. Later we will illustrate the general methodology applied in this work.

The methodological design as well as methods of data collection will be discussed. Then

finally the theoretical framework is presented later in the chapter where we provide more

details about options and option trading, explain the origin of the Black-Scholes model, its

input variables, restrictions and how it is applied. Furthermore, a summary covering the

limitations of the Black-Scholes Model that results in financial crisis.

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1.1 General Introduction

Finance has become one of the most rapidly changing and fastest growing areas in the

corporate business world. Because of this revolution, modern financial instruments have

become extremely complex. Many new mathematical models have been derived but

dismissed by many financial experts due to its poor applicability in the market. Some are

essential to implement and price these new financial instruments. Basically, the world of

corporate finance that was once controlled by the business students is now governed by

mathematicians and computer scientists.

Published in the Journal of Political Economy 1972, Black-Scholes Model is considered one

of the biggest achiever in terms of approach and applicability. The model was developed

during the early 1970‟s where economist Fisher Black, Myron Scholes, and Robert Merton

derived the Black-Scholes option-pricing formula, which helps investors and speculators to

deduce a value for call or put option. Scholes and Merton were later awarded the 1997 Nobel

Prize in Economics for their contribution. Unfortunately Fisher Black died in 1995, or he

would have also received the award. The Black-Scholes Model displayed the importance of

Mathematics in the field of Finance and also let to the development and success of the new

field of mathematical finance or financial engineering. Its biggest strength is the possibility of

estimating market volatility of and underlying asset generally as a function of price and time

without direct reference to specific investor characteristics like expected yield, risk aversion

measures or utility functions, Besides, the model is capable of replicating itself or hedging

i.e. an explicit trading strategy in underlying assets and risk-less bonds whose terminal payoff

is equal to payoff of a derivative security at maturity. This means that basically an investor

can continuously buy and sell derivatives by the strategy and never to incur any losses. This

kind of strategy plays a part as insurance for loss in the sense that if loss is incurred in one

part of the portfolio at payoff, the loss will be compensated from the gain on the other side of

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payoff. Normally referred to as dynamic hedging as it involves continuous hedging. These

kind of trading strategies are normally automated. Their existence is assumed and challenged

by arbitrage. An arbitrage assumes that as time goes on, money is only moved around and

within the portfolio. No money is taken out or added to the portfolio almost behaves like self-

financing. Also Black-Scholes Model is built on Brownian motion which can be seen as

continuous time limit of random walks.

In order to answer the applicability and reliability of Black-Scholes Model, this paper focuses

more on examining the use of Black-Scholes Model in the U.S. Stock Exchange Markets and

some limitations of the model that has made investors question about its reliability. The

empirical results shall serve as evidence to what extent the Black-Scholes Model has

performed and contribute in the world of corporate finance.

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1.2 Problem Statement

It is assumed that the Black-Scholes Model made it possible for option traders to calculate

their delta hedge and to price option. This statement is highly debatable as stocks are actively

traded in the 17th

century stated by Joseph De La Vega himself. What‟s unique about the

model is that the model claims to remove the necessity of risk-based drift from the underlying

security, eventually to make the trade “risk-neutral”. But one does not require dynamic

hedging as simple put-call parity can suffice. Basically options have deeper history than

shown in the conventional literature. Forward contracts can be seen dating all the way back to

Mesopotamian clay tablets dating around 1750 B.C. Gelderblom and Jonker showed that

Amsterdam grain dealers had used options and forwards in the early 1550‟s. Early records

also showed that in the late 1800 and early 1900, options were actively traded in London and

New York, as well as Paris several other European exchanges. Option market it seems, were

active and extremely sophisticated.

Recently, the campaign against The Black-Scholes Model option pricing has raised many to

question: Do we actually need the Black-Scholes Model in the first place? This is basically

brief questions which critically claimed by many bankers and traders that actually cause the

sudden evaporation of billions in investment funds. Black-Scholes Model was partially

blamed for the subprime crisis. Essentially, the argument was raised blaming investors

blindly fetch too much risk too eagerly, by having so much faith on the model. But many still

oppose the argument and concluded that the model has nothing to do with today‟s subprime

mortgage crisis.

Basically, the model simply does not withstand or resists any wear or criticism. Financial

prices tend to jump and bring about a volatility that is not constant. The key strategy behind

the model is basically known as dynamic hedging, a mathematical formula for constructing a

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portfolio made up of underlying assets and cash that will theoretically mirror the option‟s

value at any given time. Unfortunately, as much convincing the model has been, it‟s

application simply not feasible in the real world. Veteran traders and best-selling authors

Nassim Taleb and Espen Haug criticize the model saying that: It was not used (even by those

who think they have used it), it wasn’t needed, and it wasn’t original in the first place.

1.3 Research Objectives

Main objective of this study is to derive and explain the birth of the model alongside with its

contributions towards the world of corporate finance. The problems are addressed by

performing empirical testing on the performance of Black-Scholes Model that commonly

used for pricing options. The empirical study basically conducted to perform the usage of the

model in real life stock exchange under the U.S. stock markets. Stocks under the U.S.

markets will be randomly selected and calculated to derive an option price using the Black-

Scholes Model. Should the model perform poorly, this study will analyze all the side effects

of the model and look into other alternatives if possible. A probable pricing bias includes

stock strike price, time to maturity, volatility of respective underlying share, and risk free

interest rate. Basically the model may over or under perform based on respective probable

pricing biases.

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1.4 Definition of Terms

For the purpose of this study, the following terms need some clarification

Option – The right but not an obligation, to buy or sell something at some time in the future

at a given price. Options are trading based on some underlying assets such as stocks, bonds

and index funds. Currently, options are not available in Malaysia but they are available in the

U.S. markets.

Call Option – The right but not an obligation to buy a particular good at an exercise price.

Put Option – The right but not an obligation to sell a particular good at an exercise price.

Warrants – A long term call options written on a specified number of ordinary shares by

Issuer Company and traded on an organized exchange.

Call Warrants – A relatively new type of financial instrument introduced to the KLSE. A

short-term call options (Not more than 2 years) that can be issued by a third party entity

specified in the Security Commission guidelines based on existing shares.

Futures – Dealing in futures means having the right to buy or sell a financial instrument at an

agreed price at a future date. In Malaysia, the popular futures product traded in the market

are: KLCI Futures (FKLI) and Crude Palm Oil Futures (FCPO). In other countries, the future

commodity market includes wheat, corn, soybeans, gold, silver, and so on.

Exercise Price or Strike Price – The fixed price at which the good may be bought or sold.

American Option – An option that can be exercised on any day up until the expiry date.

Pseudo-American Option – An option that can be exercised on a future specified date up to

its expiry date.

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European Option – An option that can be exercised on the last day of the option.

Premium – The cost of an option

The General Market

The Dow Jones Industrial Average (DJIA) – It consists of 30 widely traded stocks such as

Microsoft and Wal-Mart in the US.

The Nasdaq Composite – It is the house of most but not all the technology stocks in America.

It has over 4000 innovative and fast growing high-tech companies.

The Hang Seng Index (HSI) – It consists of 33 blue chips stocks in Hong Kong stock market.

The Straight Times Index (STI) – STI also consists 30 widely traded stocks in Singapore.

The Nikkei 225 Index (NI) – This index consists of 225 top-tier Japanese companies in the

Tokyo stock exchange.

The FBM Kuala Lumpur Composite Index (KLCI) – 30 blue chips stocks that are traded in

Malaysia.

The Future Kuala Lumpur Index (FKLI) – The KL futures can act as a guide to where the

KLCI is heading.

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2.0 Brief History of option

Options trading can be traced back to 332BC where during that time, a man known as Thales

bought the rights to buy olive prior to a harvest, reaping a fortune. Options then turned up

again during the tulip mania of 1636 where options on tulips were widely purchased in order

to speculate on the rising price of tulips. At the end of the seventeenth century in London, a

market was formed to trade in both call and put options. That was the first instance of trading

both call and put options over an exchange. By 1872, Russel Sage introduced “Over the

Counter” call and put options trading to the United States which was unstandardized and

illiquid. The emergence of options trading today comes with the setting up of the Chicago

Board of Options Exchange (CBOE) and the Options Clearing Corporation (OCC) in 1973

where standardized exchange traded call options were introduced. By 1977, put options were

also introduced by the CBOE and since then, options trading took on the standardized

exchange traded form that we are familiar with today.

2.1 Option Trading in Malaysia

Options trading are relatively new in Malaysia. Bursa Malaysia Derivatives currently offers

the trading of options contracts in Malaysia.

An option is a financial derivative that derives its value from its underlying instrument. This

could be in the form of a stock, stock indices, commodities, and debt instruments or futures

contracts. For example, the options being offered by Bursa Malaysia Derivatives is an index

option with the FBM KLCI as its underlying instrument. In Malaysia, FBM KLCI options is

the only option being traded on Bursa Malaysia Derivatives and it is an option on FBM KLCI

index, which can be thought of as the trading of a portfolio of shares that tracks our local

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share market. However, as index options usually exclude dividend payments and voting

rights, they are not considered an exact substitute for actual ownership of the shares.

The FBM KLCI, the underlying asset for the FBM KLCI options, is the weighted average

market capitalisation of 30 largest Malaysian companies listed on the Bursa Malaysia by full

market capitalisation. To trade the option, bids and offers are entered into the automated

system by exchange members and the transaction is executed when a match is made.

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3.0 Literature Review

A review of related literature is compiled to compare the knowledge of financial economists,

compensation professionals, option traders, academics and other experts in the area of Black

Scholes Model as to option pricing. The review will indeed serve as a path to compare and

contrast the views of experts and eventually provide an abstract framework for the study.

The Model with hedging errors

Throughout the research, Black Scholes Model is associated with hedging errors. The

formula relates for a “lag” resulting from the non-simultaneous trading in the option and its

underlying asset when implementing the hedging portfolio. The lags may largely affect

problems of liquidity in the market place – “The extended Black Scholes model with lags and

hedging errors”, International Journal of Banking and Finance

Self-Financing?

The Black Scholes Model is considered to be a self-full filling prophecy quoted by Cherian

and Jarrow (1994). The formula basically relies on few unrealistic assumptions, which the

primary assumption was the transaction cost that was thought to be zero. In reality,

transaction cost cannot be avoided alongside with the volatility in the market. But yet option

prices in the real world are remarkably close to those predicted by the model. One main

possibility could be the mass use of this model by traders as a working assumption for their

option pricing - Black, Merton and Scholes: Their work and its consequences

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Black Scholes Model revealed that, under certain assumption on the random sequence {Rt}

and for a certain value of Xo, it is possible to find a policy such that Xt = ΦSt. The value Xo

then regarded as a fair price for Po for option at time 0. Under these conditions, the contract

is self-financing and risk free to the buyer and seller – “On the Structure of Proper Black-

Scholes Formulae”, Journal of Applied Probability, Vol. 38, Probability, Statistics and

Seismology (2001), pp. 243-248

The beginning of the model

Black and Scholes model claimed that in many cases their famous model could be used as an

approximation to give an estimate of the option value. The Chicago Board Options Exchange

(CBOE), the first public options exchange, started trading in April 1973, and by 1975, traders

on the CBOE were using the model to price and hedge their option positions. Ever since the

model was widely introduced, “thousands of traders and investors use the formula everyday”,

noted by the Nobel committee (Marsh & Kobayashi, 2000). Making it more simple and

conventional, it was widely used in those personal computer days that Texas Instruments sold

a handheld calculator specially programmed to produce Black-Scholes options prices and

hedge ratios – “Application of Option Pricing Theory: Twenty Five Years Later”

In the Malaysian context, Malaysia Derivatives Exchange Bhd (MDEX) and KLSE-RIIAM

Information System (KLSE-RIS) has inserted a Black-Scholes Option Pricing Model

Calculator in their respective website for investor‟s convenience in checking out BS price.

KLOFFE (before MDEX) has become the first exchange to provide Black-Scholes Model

calculator over internet in Southeast Asia, beginning 16th April 2001. Table below shows a

list of local institutions that often apply the Black-Scholes Model to value Malaysian

warrants and advise investors. The model brings forward a great impact to Malaysian equity

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warrant investors since it is widely used as equity warrant evaluator by local research

institutions and securities firms.

List of Local Institutions/ Publications that Apply Black-Scholes Model

TA Securities

RHB Research Institute

Kenanga Research

Affin Securities

AmSecurities

iCapital.biz

GK Goh Research

Malaysian Business

AMS Research

Dynaquest's Monthly Digest

3.1 Empirical Testing on Different Markets

There have been an enormous number of empirical tests on the Black-Scholes model. The

size of the option market and especially the difference in size between the U.S., Japan as well

as emerging markets in Asia, and Malaysia in particular is interesting from the viewpoint of

researchers. Empirical researches on call options traded on developed market in the U.S.

include Macbeth and Merville (1980), Rubinstein (1985), Lauterbach and Schultz (1990),

Leonard and Solt (1990), and Hauser and Lauterbach (1997). Macbeth and Merville (1980)

and Rubinstein (1985) reported actual call option prices on the U.S. market tend to trade

above their Black-Scholes values when the options are out-of-the money and below their

Black-Scholes values when the options are in-the-money.

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3.2 The Important Assumptions

The Black Scholes Model has a huge breakthrough in the pricing of stock options. It had a

huge influence on the way that trader‟s price and hedge options (Hull, 2006). The Black

Scholes Model is based on few simplified assumptions (Blacks & Scholes, 1973)

First and foremost, stock price follows a random walk in continuous time with a variance rate

proportional to the square of the stock price. Basically today‟s price will reflect

corresponding to today‟s news and information in the stock market. Thus the distribution of

possible stock price at the end of any finite interval is log-normal. The variance rate of return

on the stock is constant.

The short term interest rate is known and constant through time.

The option is “European”, means it can only be exercised at maturity.

The stocks pay no dividend or other distributions (The model was later adjusted)

There are no transaction costs in buying or selling the stock or the option.

It is possible to borrow any fraction of the price of a security to buy it or hold it, at the short-

term interest rate.

There no penalties to short selling. A seller who does not own a security will simply accept

the price of the security from the buyer, and will agree to settle with the buyer on some future

date by paying him/her an amount equal to the price of the security on that date.

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3.3 The Advantages of Black Scholes Model

Based on the these underlying assumptions, the BS model says that the price of an option is a

function of the current stock price S, exercise price K, risk-free interest rate r, time to

expiration t, stock price volatility σ. Therefore, the BS model requires little information to

price options, and the computational demands of the model are quite modest. Derman and

Kani (1994) assert that the main appeals of the BS model are its simplicity and ease

of implementation.

3.3.1 Implied Volatility

All parameters in the Black-Scholes model, except the volatility of stock price, can be

directly observed from the market data and the terms specified by option contracts.

Therefore, if market price for a particular option is available, the market-based

estimate of the stock price volatility can be obtained by numerically inverting the Black-

Scholes formula, yielding the implied volatility (Huang and Chen, 2002). It is the implied

volatility that makes the option value derived from the BS model equal to its market value

(Chriss, 1996). If the BS model is correct, the implied volatility should provide a measure of

the expected volatility of the underlying asset over the remaining life of the option contract

(Chriss, 1996). Nowadays many traders quote options‟ market prices in terms of the implied

volatility (Derman and Kani, 1994).

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3.3.2 Volatility Smile

“The volatility smile is the way in which implied volatility varies with strike price for options

of a fixed expiration”. (Chriss,1996:341) Today the volatility smile has become a remedy of

testing the validity of the Black Scholes model by seeing if the implied volatility is actually

independent of strike price as assumed by the BS model (Chance, 2004b)

Implied Volatility Smile

As mentioned earlier, the fundamental assumption going into the Black Scholes Model is that

the stock prices move with the constant volatility, consequently the implied volatilities

obtained by numerically inverting the Black Scholes formula should be exactly the same for

options on the same underlying with the same maturities but with a different strike prices

(Rubinstein, 1994). Therefore, if the Black Scholes model is correct, the implied volatility

smile should be flat and basically this statement is supported by some empirical evidence. For

example, Rubinstein in 1994 reported that, whole not strictly true, S&P 500 index options on

the same underlying with the same underlying with the same maturity tended to have the

same implied volatilities across different exercise price before 1987.

Figure 1: Implied Volatility Smile

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3.3.3 Volatility Skew

Rubinstein (1994) later then discovers that in practice the flat volatility smile has been rarely

observed since the stock market crash in October 1987. The graph of implied volatility as a

function of strike price for the identical options has tended to depart from the flat volatility

smile since the 1987 crash. As Chriss (1996: 328) states, “implied volatilities change from

strike to strike and from expiration to expiration, and there is a recognizable pattern for these

changes.” When first observed, the implied volatilities were u-shaped in equity market,

giving appearance of a smile as exhibited in Figure 1. Hence this pattern was named volatility

smile. The volatility is relatively low for at-the-money options and becomes progressively

higher as an option moves either in-the-money or out-of-the-money (McDonald, 2006).

Figure 2: Volatility Skew

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3.3.4 Implications of volatility skew for option pricing

Volatility skew implies that the Black Scholes formula frequently misprices the in-the-money

and out-of-the-money options (Derman, 2004). Option values calculated by the Black

Scholes model increases as volatilities of the underlying assets rises (Dupire, 1994). From

other perspective point of view, options that are traded at a higher implied volatilities are

more expensive, assuming all other things being equal (Huang and Chen, 2002). Due to the

negative relationship between exercise price and implied volatility, the implied volatility rises

as strike prices fall, and accordingly options values go up due to the increased implied

volatility. When the level of the implied volatility is higher than the flat smile assumed by

the Black Scholes model, the out-of-the-money puts are priced more expensive than the

Black Scholes model predicts (Chriss, 1996).

Conversely, implied volatilities drop as strike prices rise, consequently the option values goes

down due to the decrease in volatility. Once the implied volatility fall below the flat smile

assumed by the Black Scholes model, the out-of-the-money puts (and the in-the-money

calls) are less expensive than the values arising from the Black Scholes model (Chriss,

1996).

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3.4 Errors implying towards the Black Scholes Model

The model assume constant volatility

The most significant assumption is that volatility. Volatility is the measure of how much a stock

can be expected to move in the near-term or short term, is a constant over time. While volatility

can be relatively constant in very short term, it is never constant in longer term. Some advanced

option valuation models substitute Black-Scholes constant volatility with stochastic-process

generated estimates. The Black-Scholes model assumes that the underlying prices follow the

geometric Brownian motion. The probability distribution of stock return is normally

distributed and the stock price is independent of all the previous trading. In this case, each

trade contributes to the overall volatility in the same way.

McDonald (2006) argues that the Black Scholes model fails to consider all of the factors that

enter into the pricing of an option. This model only accounts for the stock price, the exercise

price, time to expiration, the dividends, and the risk-free rate. Therefore, the implied volatility

is more or less a catch, capturing whatever variables are missing and thus causing the model

is improperly specified or blatantly wrong.

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According to Rubinstein (1994), the constant volatility Black-Scholes model will fail under

any of the following five violations:

1. The local volatility of the underlying asset is a function of the concurrent underlying asset

price or time.

2. T he local volatility of the underlying asset depends upon the prior path of the underlying

asset price.

3. The local volatility of the underlying asset is a function of a state-variable which is not the

concurrent underlying asset price or the prior path of the underlying asset price.

4. The underlying asset price changes with large jumps.

5. The market has imperfections such as significant transactions costs, restrictions on short

selling, taxes, non-competitive pricing and etc.

Interest rates constant and known

Same as volatility, interest rates are also assumed to be constant in the Black-Scholes model.

The Black-Scholes model uses the risk-free rate to represent this constant and known rate. In the

real world, there is no such thing as a risk-free rate, but it is possible to use the U.S. Government

Treasury Bills 30-day rate since the U. S. government is deemed to be credible enough.

However, these treasury rates can change in times of increased volatility.

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Wrong assumption on return distribution

The Black-Scholes model assumes that the percentage changes in assets prices are log

normally distributed (McDonald, 2006). However, the volatility smile (or skew) indicates that

this assumption is not true. As Lekvin and Tiwari (2001) points out, logarithmic percentage

changes in underlying prices follows non-normal distribution, leading to the existence of the

volatility smile. Jackwerth and Rubinstein (1996) also believe that the Black-Scholes model

fails to capture the true nature of stock return distribution which is characterized by excess

kurtosis and negative skew.

There has been much literature documenting excess kurtosis and negative skew in stock

return distribution. On the one hand, empirical evidence shows that, in equity markets, center

peak of the stock return distribution is much higher, that is, the real return distribution has

greater kurtosis than the lognormal distribution (Cox and Ross, 1976; Merton, 1976; Jarrow et

al., 1977). On the other hand, large negative stock returns are much more common than

large positive ones, so stock return distribution should be negatively skewed. This

phenomenon is also called „contemporaneous asymmetry‟. This negative skew, or

contemporaneous asymmetry, reveals up clearly in the pattern of extreme moves in stock prices

in the postwar period (Campbell and Hentschel, 1992). For example, of the five largest one-

day movements in the S&P 500 index since World War II, four were declines in the index and

only one was increase (Cutler et al., 1989). Ignoring asymmetries in return distributions, such

us negative skew, can lead to underestimation of risk and thus the incorrect pricing of

financial instruments such standardized option contracts.

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Several explanations for the excess kurtosis and asymmetry in stock returns have been provided.

The first and perhaps most well-known explanation is the leverage effect (Christie, 1982).

According to the leverage effect hypothesis, the drop in the market value of the firm following

bad news increases the operating and financial leverage and thus increases the volatility of

the subsequent returns. Financial leverage is measured by the ratio of debt to equity, referring

to the extent that a firm is financed by borrowed funds (Brealey et al., 2006). It is used to

measure the risk to shareholders arising from the firm‟s capital structure. Firms with higher

financial leverage may be at higher risk of bankruptcy due to the high level of borrowed funds

in the firm‟s capital structure, which increases the volatility of the stock return. As Christie

(1982) points out, the volatility of stock return is an increasing function of financial

leverage. Likewise, operating leverage refers to the division between fixed and variable costs.

Firms with greater amount of fixed as opposed to variable costs are said to have high

operating leverage (Brealey et al., 2006). It is not easy for these firms to reduce costs in

response to falling sales. Hence, compared with firms with low fixed costs, these firms are

facing relatively higher risk and accordingly the firms‟ stock return is subject to higher

volatility. Christie (1982) has found that stock volatility has strong positive relationship with

operating leverage. All in all, the higher financial or operating leverage leads to the larger

volatility of stock returns and thus higher risk, resulting in greater probability to earn negative

returns and thereby making the return distribution negatively skewed (Lev, 1974). However, it

has been contended that the leverage effect cannot fully explain the observed asymmetry

in the stock return distribution. In particular, the leverage effect seems to be the case for

daily or more frequent return data (Schwert, 1989).

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Pindyck (1984) claims that the volatility feedback theory can explain the negative skewness

and excess kurtosis in stock return distribution as well. This theory relies first of all on the

well-documented fact that the stock volatility is persistent (Pindyck, 1984). It seems plausible

that changes in volatility have important effects on the stock returns (French el al., 1987). The

increase in volatility makes the stock less attractive, and therefore pushes its price down. This

hypothesis states that the significant news, good or bad, increases the volatility of a stock and

hence its risk premium, which in turn magnifies negative returns and reduces positive returns

(Campbell and Hentschell,1992).

To sum up, the arrivals of large piece of good or bad news increase the volatility of stocks

and hence their risk premium. As a result, negative returns are magnified while positive returns

are shrunk, leading to stock return distribution with excess kurtosis and negative skewness.

However, Poterba and Summers (1986) question the plausibility of this volatility feedback

hypothesis as volatility changes in the market generally are too short-lived to affect stock

return distribution.

In addition, negative skew can also be explained by „stochastic bubbles hypothesis‟

(Blanchard and Watson, 1982). When stock market bubble takes place, the stocks are traded

in high volume and prices are considerably overvalued. Consequently, the market bubbles

are usually followed by a sudden drop in prices, known as market crash or bubble burst. The

volatility is typically higher after the stock market falls than after it rises, resulting in higher

risk in the market crash and hence more negative return. In one word, the stock return

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distribution with negative skewness might be generated by the popping of stochastic bubbles

which creates substantial loss in the market.

Assumption on price movement

The Black-Sholes model assumes that the prices of the underlying assets change smoothly with

no jumps, but extreme stock movements are more common than would be expected from the log

normally-distributed stock return as assumed by the Black Scholes model. Therefore, the Black

Scholes model fails to capture the true nature of stock price movements which are observed to

have discrete jumps. Price jumps always arise from the arrival of new information, such as

takeover attempt, verdict in an important lawsuit, earnings announcement, and so on. The

existence of large price movements leads to the excess kurtosis in the stock return distribution

Suppose a piece of important news due in a few days and the stock price is expected either to rise

or to fall. Thus a single jump in stock price can be anticipated. According to Hull (2006: 384), in

this case, “The probability distribution of the stock price might consist of a mixture of two

lognormal distributions, the first corresponding to favorable news, the second to unfavorable

news. The true probability distribution is bimodal (certainly not lognormal)”. The situation is

illustrated in figure 3. The solid line shows the mixtures of lognormal distribution for the stock

price, which is the true return distribution. The dashed line demonstrates the lognormal

distribution with the same mean and standard deviation as the true distribution. As can be seen,

the true probability distribution has higher kurtosis than the lognormal distribution. Bollerslev

24

(1987) asserts that large price movement is always followed by greater amount of gain or loss

than usual, which can explain the excess kurtosis in the bimodal distribution.

Wrong valuation of risk-neutral

The primary feature of Black-Scholes model is that the values of contingent claims do not

depend upon investors‟ risk preference. As Bodie et al. (2006) point out; the Black-Scholes

model is based on the risk-neutral valuation and assumes that all the investors in such risk-

neutral world are indifferent to risk.

Figure 3: Effect of a single large jump on return distribution

Source: Hull, J. (2006) Options, Futures and Other Derivatives. 6th ed. Upper

Saddle River: Pearson/Prentice Hall. Pp.384

25

Figure 4 illustrates that the observed volatility smile for equity options is downward sloping

and thus deviates from the flat volatility smile. The negative volatility skew indicates that

investors pay higher price for the options with low strike prices because they have higher

implied volatility than suggested by the Black-Scholes model. As shown in the Figure 4, it is

interesting that the pattern of negative volatility skew has existed only since the stock market

crash of October 1987. Implied volatilities were much less dependent upon strike price prior

to October 1987. This phenomenon led Rubinstein (1994) to argue that one reason for the

volatility skew may be so-called “crashophobia”. That is, traders are concerned about the

possibility of another market crash similar to the October 1987, and they price options like

risk-averse investors. “There is some empirical support for this explanation. Declines in the

S&P 500 tend to be accompanied by a steepening of the volatility skew. When the S&P

increases, the skew tends to become less steep” (Hull, 2006: 381). In summary, traders are

risk-averse, rather than risk-neutral.

Figure 4 Volatility skew for put option before and after 1987 crash

Strike Price

Imp

lied

Vo

lati

lity

%

Downward-slopping volatility skew

Data from put price after 1987 crash

Flat volatility smile

Source: The Sardonic Smirk: The Volatility Smile and the Asymmetry of Risk.

26

For example, after 1987 crash, the out-of-the-money puts with lower strike prices have higher

implied volatilities and thus higher option value than the prediction made by the Black Scholes

model which assumes the flat volatility smile.

Existence of arbitrage opportunity

The most popular explanation for the volatility smile to date is the erroneous assumption in

the Black Scholes model regarding the log normally distributed return of underlying assets.

Hence it is said that, if the implied volatilities were calculated by the „correct‟ option pricing

model based on the right distributional assumptions, volatility smile would disappear and the

implied volatility would become flat (Bates,1996; Jackwerth and Rubinstein, 1996). In fact, it

is extremely difficult to derive the „correct‟ model because the arbitrage opportunities do exist

in the real world and are very hard to be incorporated into the option pricing model (Das and

Sundaram, 1999; Ederington and Guan, 2002). So the ideal assumption concerning no

arbitrage activities in markets threatens the reliability of the Black Scholes model.

However, Ederington and Guan (2002) find that such trading strategy based on the Black

Scholes volatility smile yields substantial returns. On the one hand, profits generated by such

trading strategy should vary randomly around zero on the presumption that the true volatility

smile is flat and the observed volatility skew arises from the erroneous distributional

assumptions, but the actual profits vary roughly in line with the Black Scholes model‟s

27

predictions (Ederington and Guan, 2002). Profits are high when the Black Scholes formula

predicts they are high and low when Black Scholes model predicts they are low. This finding

suggests that there are arbitrage activities in the markets. On the other hand, the trading

profits are not as great as the Black Scholes formula predicts, suggesting that the true smile is

indeed flatter than the observed volatility smile but not flat. “Such divergence between the

reality and theory may be due to deficiencies in the Black Scholes model, such as the

erroneous assumptions that volatility is constant and that returns are log-normally distributed”

(Ederington and Guan, 2002: 10).

In addition, based on the Black Scholes model, an important relationship between the call

value and put value can be derived as follows:

Where C and P denote the call and put option value respectively, K refers to the strike price, r

is the risk-free interest rate, T represents the time to maturity and S0 stands for the current

stock price.

The relationship is known as put-call parity. It shows that the value of European call with a

certain strike price and expiration date can be deducted from the value of a European put

with the same strike price and expiration date, and vice versa (Hull,2006). If the equation

does not hold, there are arbitrage opportunities in the market. Unfortunately, numerical studies

have reported that there are frequent and substantial violations of put-call parity involving

28

both overpricing and underpricing of calls or puts (Wilson and Fung, 1991; Kamara and

Miller, 1995; Klemkosky and Resnick, 1979). Because the actual put or call prices do deviate

from the parity price in the real world, opportunities exist for investors to set up a riskless

arbitrage position and earn more than the risk-free rate of return (Klemkosky and Resnick,

1979). The worse is that, almost two thirds of these arbitrage activities result in loss (Kamara

and Miller, 1995). Therefore, it can be said that the assumption of no-arbitrage opportunity

is unrealistic and weakens to the accuracy of the option value calculated by the Black

Scholes model.

The model only assumes for European Option

The model only assumes European-style options which can only be exercised on the expiration

date. Where else, American style options can be exercised at any time during the life of the

option, making the American options more valuable due to their greater flexibility in exercising

their maturity.

No commissions and transaction costs and liquidity of the market

The Black-Scholes model assumes that there are no required fees for buying and selling options

and stocks and no barriers for trading. This seems ridiculous because parameters such as this

could not be avoided in the real world of market trading. Besides the model also assumes that the

markets are perfectly liquid and it is possible to purchase or sell any amount of stock option or

their fractions at any time given.

29

3.5 Summary of chapter

The Black-Scholes formula, the most commonly used option pricing model today, follows

some ideal assumptions. First, asset prices follow geometric Brownian motion with constant

volatility σ and the asset returns are log normally distributed. Second, the Black-Scholes

model is based on risk-neutral valuation, assuming that all the traders are indifferent to the

risk. Last but not least, the Black-Scholes model assumes that the arbitrage opportunities do not

exist in the real world.

Implied volatility can be obtained by numerically inverting the Black-Scholes formula. Since

assumption of constant volatility goes into the Black Scholes model, the implied volatility

should be the same for all options on the same underlying assets. In fact, implied volatilities

for the equity options with the same expiration date but different strike prices consistently

differ, often displaying a persistent negative skew pattern on a graph. The existence of

volatility skew suggests that the Black-Scholes model is not a perfect measure to price option

due to its unrealistic assumptions.

The most popular explanation for the existence of the volatility skew is that it is due to

erroneous assumptions in the Black Scholes model regarding the constant volatility and the

log normally distributed return. On the one hand, the local volatility of the underlying assets

may be a function of state variables such as asset price, time, etc. And asset price do

experience discrete jumps. So the assumption of constant volatility is unrealistic. On the other

hand, empirical evidence shows that the true stock return distribution is characterized by

excess kurtosis and negative skew, which can be explained by leverage effect, volatility

30

feedback hypothesis and stochastic bubbles hypothesis.

Moreover, traders are willing to invest in stock and simultaneously purchase put option on

that stock as insurance against the stock price declines, which bids up the value of put

options with lower strike price. In other words, investors have great fear of market crash

stating they are risk-averse, rather than risk-neutral as assumed by the Black-Scholes model.

A complementary explanation for the volatility smile is that it is due to the failure of the no-

arbitrage assumption of the Black Scholes model. The trading strategies based on the

volatility smile, namely, buying options at or near the bottom of the B-S smile and selling

options at or near the top of the smile, do make profits. In addition, empirical evidence shows

that there are substantial violations of the put-call parity in the real world, indicating that

arbitrage opportunities exist for investors to earn riskless return.

31

4.0 Methodology

4.1 Stochastic Differential Equation

A stochastic differential equation or SDE is a differential equation which one or more of terms is

a stochastic process, hence resulting in a solution which itself a stochastic process. The SDE‟s

capable of modeling the randomness of the underlying asset in financial derivatives. They are

utilized throughout pricing derivative assets because they produce rather a conventional model

on how an underlying asset‟s price changes in respect to time. Basically, the randomness of the

underlying instrument is essential, when traders have the desire to eliminate or take the risk,

finally leading to the existence of derivative assets.

Let be the price of a security. We would be interested to know , the next instant‟s

incremental change in the security price. The dynamic behavior of the asset price in time interval

can then be represented by the SDE, which is

( ) ( ) , )

Where is an innovation term in representing unpredictable events that occur during the

infinitesimal interval , ( ) is the drift parameter and ( ) is the diffusion parameter

which depend on the level of observed asset price and on the time . SDE‟s are vital in

stochastic environment as the evolution of the asset price at time contains uncertainty.

32

4.1.1 Geometric Brownian motion

If ( ) is a Brownian motion with drift rate and variance rate , the process * ( )

( ) + represents the geometric Brownian motion, or also known as the exponential

Brownian motion, or the log normal diffusion. The mean is given by , ( )- ( ) ,

while the variance is , ( )- ( ) ,

-

4.2 Arbitrage

Arbitrage is a trading strategy that involves two or more securities being misplaced relative to

each other to realize a profit without taking risk. In practice, arbitrage opportunity are normally

rare, short-lived and therefore immaterial with respect to the volume of transactions. Therefore,

the market seemingly does not allow risk free profits. The primary tools used to determine the

fair price of a security or a derivative asset relies on no arbitrage principle. It is a fundamental

assumption about the market.

The no arbitrage principle is that a portfolio yielding a zero return in every possible

scenario must have a zero present value. Any other value would imply arbitrage

opportunities, which one can realize by shorting the portfolio if its value is positive and

buying it if its value is negative.

If one makes risk free profit in the market, then arbitrage opportunities do really exist and

it implies that the economy is in state of disequilibrium. An economic disequilibrium is a

situation in which there is a mispricing in the market and investors trade. Their trading

33

causes prices to change, moving them to new economic equilibrium. The mispricing is

corrected by trading and arbitrage opportunities no longer exist.

4.3 Self-financing strategy

This is a trading strategy in which the change in value of a portfolio is a result of a change in the

value of the underlying asset and not due to the change in the portfolio structure. Let‟s just say

that we have units of a stock and units of a bond , therefore the portfolio‟s value is

This strategy is self-financing only if and . We therefore have

readjusted the portfolio while the prices have remained the same, and the total value has not

changed.

4.4 Risk Neutral Valuation

Basically is an assumption that the world is risk neutral while having the valuation of derivatives.

A risk neutral world is where assets are valued solely in terms of their expected return. The

return on all securities is the risk free interest rate all individuals are indifferent to risk. Thus the

risk neutral valuation principle is rather vital in option pricing. Indeed it implies that all the

expected return must be zero. Therefore, derivative prices are determined by the expected

present value payoff. We make an assumption that the world is risk neutral state and the price

obtained is correct, not just in a risk neutral world but also in the real world.

34

4.5 Ito’s Lemma

In stochastic calculus, Ito‟s lemma or Ito‟s formula, is the extension of the chain rule of ordinary

calculus and contributes a very vital role in mathematics. Speaking to the fact, the importance of

Ito‟s lemma in finance cannot be exaggerated widely. There are several versions of Ito‟s lemma,

but this paper focuses more on the Ito‟s lemma on Brownian motion.

Ito‟s Lemma for Brownian motion

Let ( ) be a function of t, which satisfies the stochastic differential equation

( ) ( )

Where ( ) and ( ) are deterministic functions of x and t, and z represents a standard

Brownian motion.

Let ( ) be a twice continuously differentiable function of x and t.

( ) 0 ( )

( ) 1 ( ) ( )

Here means

, means

, and means

Thus, the above formula actually referred to as Ito‟s Lemma or Ito‟s formula

The non-rational derivation of the Ito‟s lemma derived using the Taylor‟s expansion to expand

in the second order and replace with ( ) ( ) ,

( )

( )

35

( ( ) ( ) )

( )

( ( )

( ) ) ( ( ) ( ) ) ( )

Throughout the calculation, we just ignore the ( ) and terms and later replace ( )

with , then equation will be

( ) ( )

( )

0 ( )

( ) 1 ( )

If a variable ( ) follows a geometric Brownian motion, it follows the stochastic differential

equation of the form

( )

And the Ito‟s lemma for function ( ) as

0

1

Where and are constants.

If a stock price, S follows a random process in (4.3). Let ( ) , then

,

( )

36

Substituting (4.4) into (4.3), we eventually get

( ) .

/

Judging from above, is a Brownian motion with drift parameters .

/ and variance .

Integrating the expression above from to , we extract an explicit formulation for the evolution

of the stock price.

∫ ( )

.

/ ∫

∫

( ) ( ) .

/ , ( ) ( )-

And ,(

) √ -

, where ( )

Therefore, the stock dynamics follows a lognormal distribution.

4.6 Lognormal Dynamics

The rate of return of a stock can be presented as

√ , where ( )

As time passes by an amount of , the rate of return changes by , and also jumps up and

down by a random amount of √ . By making the time interval smaller and smaller, as

, the random process becomes a continuous process. Taking √ , the above equation

can be expressed as

37

( )

We then denote the right hand side of (4.5) by

( )

The variable is known as the drift rate.

( ) , write as

Logarithm of is normally distributed. Therefore, the distribution of is lognormal. The

lognormal distribution implies the following advantages

A lognormal distribution variable only takes on positive values , ) compared to

normal distribution which has both positive and negative values.

It is mathematically amenable as we can derive solutions for values of options.

It exhibits a skew with its mean and median. The stock dynamics will be treated as log

normally distributed with specified mean and variance.

4.7 Factors affecting the option value

The primary determinants of an option are the current stock price , interest rate . Strike

price , expiration date , stock price volatility , and the dividend expected during the life of

the option. It is also crucial to identify whether the option is an American or European style

option.

38

Increase in , means a higher intrinsic value if the call was in the money and hence

higher the premium, and vice versa for a put option.

Increase in raises the intrinsic value of a put while lowering the intrinsic value of a call.

The longer the time spans of an option, the greater the chances that it will be possible to

exercise the option profitably.

The higher the volatility, the greater the movement in the price of the underlying

instrument, the more valuable the option is.

Dividend payment reduces the , thus increasing the probability for a call option to be

out of money, making the option less valuable. The reverse applies to put option having

to be in the money.

The increase in interest rate reduces the present value of the exercise price for the option.

A call option gives someone the right but not an obligation to buy a particular good at an

exercise price and higher degree of discount means more valuable the right is. Similarly,

for put option the right becomes less valuable due to the low interest rate.

Factor Call Option Put Option

Strike Price, Decrease Increase

Spot Price, Increase Decrease

Interest Rate, Increase Decrease

Time to maturity, Increase Increase

Volatility, Increase Increase

Dividend, Decrease Increase

Table 4.1: A summary of the general effect of each of the six variables

39

4.8 Black Scholes Model

In 1973, Fisher Black and Myron Scholes formulated and solved the partial differential equation

governing the behavior of the contingent claims and this changed the general view of pricing

derivatives as financial instruments

Under the assumptions of lognormal dynamics of derivatives, Fisher Black and Myron Scholes

together with Merton developed their European option pricing model. Under their model, they

made a few assumptions

Stock prices exhibit the following properties:

- Continuously compounded returns are normally distributed

- Volatility is constant

- Future dividends are known

No transaction costs of taxes.

No risk free arbitrage opportunities exist.

Risk free interest rate, is known and constant.

Short-selling is costless, and the interest rate on borrowing is the risk free rate.

There are no restrictions on the amount of the underlying asset investors can hold.

40

Let a stock price follows

( )

where is the trend, is the volatility and follows a Wiener process. Now, suppose that is

the price of a call option and other derivative contingent on . The variable must be some

function of and . Hence, by Ito‟s Lemma

0

1

( )

The discrete versions of ( ) and ( ) are

( )

0

1

( )

The Wiener process of underlying‟s and is the same and can be carried off by choosing an

appropriate portfolio of the stock and the derivatives. We therefore choose a portfolio of

We short one derivative and long an amount

of shares. We then define as the value of the

portfolio and we have

( )

The change in in the value of the portfolio in the interval is given by

41

( )

Substituting ( ) into( ), we eventually retrieve

,

- ( )

The portfolio is now risk less due to the elimination of the term. The portfolio must then

earn a return similar to the other short term risk free securities. Therefore

( )

Where is the risk free interest rate. Substituting ( ) and ( ) into( ), we obtain

0

1 ,

- ( )

Finally we have

( )

This is the famous Black-Scholes-Merton differential equation.

Solving the partial differential equation in ( ) gives an analytical formula for pricing the

European style options. European options can only be exercised at the maturity date. The

American style options can be exercised anytime up to the maturity date. Thus the analytical

formula we will derive is not appropriate for pricing them due to this early exercise privilege.

42

4.8.1 American and European Options

American options are almost similar to European options except that American option allows an

early exercise privilege. Considering if we know the price of a European option, we can price the

parallel American option by determining the impact of early exercise privilege. Therefore, the

value of the right to exercise before expiration is also known as early exercise premium. Thus the

American option must be worth at least as much as the European option.

( ) ( )

( ) ( ) ( )

Where the and

American and European Puts

The respective American and European lower boundary conditions that are determined by the

free arbitrage free option prices are

( )

( ) ( ) ( )

The upper boundary conditions are

( )

( ) ( ) ( )

43

The price difference between American and European options hugely depends on the extent to

which the option is in-the-money, the interest rate and the time remaining. The early exercise of

an American put casts aside the value of waiting to see how stock prices evolve. For an

American put on a dividend paying stock, the optimal time to exercise is basically immediately

after the dividend payment. Dividend payment reduces the value of the stock and this forces the

put further into-the-money. A put option when held in conjunction with the stock insures the

holder against the stock price falling below the certain level. However, it may be optimal for

investors to forgo this insurance and exercise early in order to realize the strike price

immediately. But, it is also optimal to exercise a put before maturity date on a non-dividend

paying stock.

4.8.2 American and European Calls

For a non-dividend paying stock, early exercises is never optimal and the price of an American

call holds the exact values as its European counterpart. The respective lower boundary and upper

boundary conditions are given by

( ) ( ) ( )

and ( ) ( ) ( )

If the underlying stock pays a dividend, it is rather wise to actually exercise early, and basically

an American call can worth more than the European call. The early exercise should occur

immediately before a dividend payment as a dividend payout lowers the current stock price and

this in turn lowers the call intrinsic value.

44

The American call on a non-dividend paying stock should not be exercised early as the call

option behaves like insurance to the holder against the stock price falling below the exercise

price. This insurance fells when the option is exercised. The latter the strike price is paid out, the

better for the option holders

4.8.3 Solution of the Black Scholes Equation

The restriction here given here is ( ) ( ). The lower and upper boundary

is given by 4.20. These are the conditions needed to be satisfied by the PDE.

Now let , where represents the expiration time and as the present time. The

equation ( ) can be written as

( )

Let

( )

We now induct a new notation ( ) ( ). Using the above equation of (4.22), the

Black Scholes PDE appears to be a diffusion equation

,

-

( )

45

Apparently has a fundamental solution as a normal function

( )

√ *

[ (

) ]

+ ( )

The solution to (4.23) is

( ) ∫ ( )

( ) ( )

Using the payoff conditions and the fundamental solutions of equation (4.24), we derive

( )

√ ∫ (

) *

[ (

) ]

+

√ ∫ (

) *

[ (

) ]

+ ( )

We then denote the distribution function for a normal variable by ( ),

( )

√ ∫

( )

We later express equation (4.26) as

( )

√ ∫

,

( )

-

√ ∫ ,

( )

-

( )

where .

/ .

/ . We now consider the second term in the right

hand side of (4.28). Let

( ) √ ( )

46

Then using equation (4.29), the becomes

√ ( )

given that the limits of (4.28) using (4.29) are

√ (

)

√ ( )

By changing the variable from to , the second term of equation (4.28) becomes

√ ∫

√ ∫

( ) ( )

The integrand of the first term in (4.28) is expressed as

0 ( )

1

, ( )

-

, ( ) ( )

( )

-

, , ( ) -

-

,

, ( ) -

- ( )

47

We therefore use the definition of A to have

( )

Introducing (4.33) and (4.34) into (4.28), the first term then becomes

√ ∫ ,

, ( ) -

-

( )

By changing the variables, we derive

√ ∫

( ) ( )

The equation (4.26) can be written as

( ) [ ( ⁄ ) (

)

√ ] [

( ⁄ ) (

)

√ ] ( )

and finally it implies that

( ) ( ) ( )

Where

( ⁄ ) (

)

√ and √ ( )

We now have the Black Scholes formula for the price at time zero of a European call option on a

non-dividend paying stock. We then can derive the European put option by using put call parity

by . The European put therefore is given as

( ) ( ) ( )

48

The European put and call option mainly has grown its popularity throughout the world as its

importance towards pricing an option has brought ease to many investors. Calculating the option

prices requires parameters that can be observed from the market, but the volatility especially has

been the talk of the town of many financial mathematicians as this parameter cannot be observed

directly from the market. Therefore it becomes rather necessary to find appropriate methods in

estimating the volatility.

4.8.4 Volatility

Volatility is defined as the standard deviation measure of an asset‟s potential deviating from its

current price. Basically the definition is also close related to what we know as risk. The price

volatility creates greater value for a given option, for the greater the volatility of the underlying,

the greater the value of the option. Volatility is good for option holders but bad for financial

assets holders due the fact that financial assets have both risks but option holders only enjoy the

upside potential and not the downside risk.

The fluctuation of price volatility in assets market is mainly caused by the release of information,

the process of trading, and market making for financial instruments. Explaining further,

information release can be divided into anticipated and unanticipated information.

The volatility estimate is the measure of the uncertainty about the returns on the asset. When

pricing an option, the volatility is assumed to be time homogeneous, where it is the same over the

life of an asset and constant between the pricing date and expiry of the option. Basically, the two

major approaches for the estimation of the volatility are the historical and implied volatility

methods.

49

4.8.5 Historical/Empirical Method

The method works on estimating the volatility by calculating the standard deviation if the logs of

the price changes of a sample time series of historical data for the asset price. The daily return is

given as ( )⁄

The variance is estimated by the sample variance, which is normalized by to make it an

unbiased statistic

,∑ (

)]

The standard deviation computed equals to the daily volatility if daily data is used. Annual

volatility is computed as

√

refers to the annual volatility, is the daily volatility and the is the mean value of the

daily return. We take 252 days as the number of days traded in a year. If the asset pays

dividends, the asset price sequence must later be adjusted to reflect the non-homogenous nature

of the data series. Transition from cum dividend to ex dividend will affect the price of the assets.

A dividend payment increases the return to be paid to the buyer. If the buyer has an asset that

pays dividend, then the daily price return is restated as ,( ) -⁄

50

4.8.6 Implied volatility

The volatility of an underlying asset which when substituted into the Black Scholes model, gives

a theoretical price close to the market price. The implied volatility is considered deficient as the

options currently trading volatility is treated as being the true constant assets price volatility

parameter. Besides, options with different strike prices and same maturity often demonstrate

different implied volatilities, also known as volatility smile. Newton-Raphson method or any

suitable numerical method is used to derive volatility with respect to the price of the option.

4.9 Dividend Paying Stock

By modifying the Black Scholes model, we can derive a dividend paying model and study the

effects of dividend on the value of European options.

4.9.1 Continuous Dividend Yield Model

We let as the constant continuous dividend yield. The holder receives a dividend within

the time interval of . The value of share later decrease after the payout of the dividend and so

the expected rate of return of a share becomes ( ). The geometric Brownian motion

model in (4.7) becomes

( ) ( )

51

The modified PDE in (4.16) becomes

( )

( )

Therefore,

( ) ( ) ( )

( ) ( ) ( )

Where

( ⁄ ) (

)

√ and √ ( )

The dividend payment reduces the stock price from and the risk free interest rate

from ( ).

4.10 Options on Futures

Futures option is a contract that grants the holder the right, but not obligation, to buy or sell a

futures contract at a fixed price or strike price, upon the specified expiration date.

4.10.1 American Options on Futures

The minimum value of an American call and put futures is its intrinsic value. The respective call

and put intrinsic values are

52

( )

P ( ) ( )

where is the futures price and is the strike price.

4.10.2 European Options on Futures

The lower boundary of a European call and put are respectively given by

,( ) ( ) -

,( ) ( ) - ( )

4.10.3 Put-Call Parity

We construct two portfolios, A and B. Portfolio A will consist of a long futures and a long put on

the futures. This can be referred to as a protective put, where it involves an investment strategy

of the use of long position in a put and a stock to provide a minimum selling price for the stock.

Portfolio B consists of a long call and long bond with the face value equal to the exercise price of

the futures contract minus the futures price. The current value of portfolio A is which is a

protective put and that of portfolio B is ( ) ( ). Since the portfolio B acts like a

protective put too, then its current value will equal to the current value of portfolio A. We

therefore say agree that

( ) ( ) ( ) ( ) ( )

53

4.10.4 Black’s Model

For futures and forward prices to be equal, we have to assume that the interest rates are non-

stochastic. The modification of the Black Scholes model on a spot instrument gives us the

European call option on futures as

( ), ( ) ( )- ( )

The future price takes the place of the stock price . For the European put options on futures,

we have

( ), ( ) ( )- ( )

Where

( ⁄ ) (

)

√ and √ ( )

Risk free rate is ignored due to the fact that it captures the opportunity cost of funds tied up in

stock. If the option is on futures contract, no funds are invested, thus no opportunity cost as the

cost to carry is zero. The cost to carry refers to the cost in holding or storing an asset that

consists of storage costs and interest lost on funds tied up. The dividend in the other does not

show up because the futures itself captures all the effects of the dividend, so including the

dividend would rather be unnecessary.

54

4.11 Binomial Model

The model was first introduced by Cox, Ross and Rubinstein (CRR) in 1979 and assumes that the

stock price movements were composed of large number of smaller binomial movements.

Binomial models come in handy particularly when the holder decides to exercise early or prior to

maturity as the model is able to accommodate complex options pricing difficulties. The binomial

model reduces the possibilities of price changes, removes the possibility for arbitrage by

assuming a perfectly efficient market, and also shortens the duration of the option. Under these

simplifications, it is able to provide a mathematical valuation of the option at each point in time

specified. Basically the model takes the risk neutral probabilities into approach. It only assumes

the underlying price for stocks can either increase or decrease according to time until the option

expires.

First we divide the life time , - of the option into time subintervals of length ,

where . Suppose that is the stock price of the beginning of a given period of time.

Then the binomial model of price movements assumes that at the end of each period, the price

will either go up by with probability or go down by with probability ( ). The and

are the up and down factors with .

Using the log normal distribution with the parameters , and all satisfying the risk neutral

valuation, we have

( )

( ) ( )

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The variance therefore is

, - , -

, - , - , -

[ ] ( ) , ( ) -

This is then simplified into

( ) ( )

We assume , then

√

√

( )

Basically is called the risk neutral probability. It ensures all bets are fair and no arbitrage is

available. The expectation of share price can be written as , - ( ) , where

is the price after one period.

The Cox, Ross and Rubinstein model contains Black Scholes analytical formula as the limiting

case as the number of steps tends to infinity. The value of the call option at the first movement

is given by

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( )

( )

As for the risk neutral call option price at the present time is

, ( ) - ( )

Now we extend the model to 2 periods. At time for two consecutive upward and downward

stock movements will be represented by and . For one upward and one downward

movement will be represented by .

( )

( )

( ) ( )

The values of call option at time are

, ( ) -

, ( ) - ( )

For a dividend paying stock, the risk neutral probability will be adjusted as

√

√

( )

( )

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Where the probability of stock prices increases varies with the level of the continuous dividend

rate, .

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5.0 Finding and Analysis

The results for American and European options using the multi-step binomial model are

compared to those prices obtained using the Black Scholes model.

Type of Option N

BS Value

100 500 1000 5000

European Call 267.8332 267.619 267.6261 267.6156 267.6184

American Call 267.8332 267.619 267.6261 267.6156

European Put 85.574 85.3598 85.3669 85.3564 85.3593

American Put 123.1602 123.1959 123.1985 123.2018

Based on the results obtained, we can conclude that the Black Scholes model can only be used to

value American call options and put option given that early exercise is does not take place. The

value of American put option is higher than the corresponding European put option due to the

fact that we have early exercise premium. Sometimes the early exercise of the American put

option can be optimal. Besides the American option on the dividend paying stock is always

worth more than its European counterparts. A very deep-in-the money American option has a

high early exercise premium. The premium of both the put and call option decreases as the

option goes out-of-the money.

The American and European call options are not worth as the same as it is optimal to exercise

the American call early on a dividend paying stock. A deep-out-of-the money American and

Table 5.1 Comparison of the Multi Step Binomial and CRR Analytical formula to Black Scholes’s value

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European call and put options are worth the same. This is due to the fact that they might not be

exercised early as they are worthless.

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6.0 Conclusion

6.1 Summary of the Black Scholes Model

Determining the value of derivatives had been a longstanding problem in finance for 70 years. In

the early 1970‟s, Black and Scholes made a pioneering contribution to finance through the

development of the Black Scholes model. The Black-Scholes formula is generally one of the

most commonly used option pricing model today. It has a magnificent influence on the way that

a trader price and hedge options today. This model basically relates the option value to five

parameters, which are the risk-free interest rate, strike price, time to maturity, underlying asset

price and the volatility of the underlying asset return. The Black-Scholes model is based on the

following ideal assumptions:

The underlying asset prices move with constant volatility.

The underlying asset returns are normally distributed.

All the traders are indifferent to the risk, that is, they are risk-neutral.

There are no riskless arbitrage opportunities.

Because one fundamental assumption underlying the Black Scholes model is that the asset prices

move with constant volatility, the implied volatilities obtained by numerically inverting the

Black-Scholes formula should be exactly the same for options on the same underlying asset

with different strike prices. Therefore, one way to test the validity of the Black Scholes model is

to see whether the implied volatility for given underlying assets is constant across different strike

prices.

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6.2 Volatility; Implied or Historical?

Volatility is generally the most critical parameter for option pricing because option prices are

very sensitive to price changes and this parameter cannot be directly observed but can be

estimated by various mathematical models. Whilst implied volatility is the volatility of the option

implied by current market prices and is commonly used the argument that this is the "best"

estimate. Skilled or professional options traders will not rely solely on implied volatility but will

look behind the estimates to see the fluctuation that you would expect from historical and current

volatilities, and hence whether the price of the options are supposed to be more expensive or

cheaper than perhaps they should be.

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

an indication of its value. It's important to understand both parameters. Basically if your forecast

of volatility based on historical prices that is greater than current the implied volatility (options

under valued) you might consider calling a straddle, and if your historical forecast is less than

the implied volatility you might want to long a straddle.

Implied Volatility – Calculation of American and European options with and without

dividends.

Equally Weighted Historical Volatility – Estimated using the historical prices, one or

more high low, open and closing prices.

Exponentially Weighted Historical Volatility – Estimated using the EWMA

(Exponentially Weighted Moving Average) or the GARCH (Generalized Autoregressive

Conditional Heteroscedasticity) model.

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Volatility Forecasting – By applying the GRACH model, it helps you to predict how the

volatility moves in the future.

6.3 Risk Neutral Valuation – Does the Expected Stock Return Matters?

Significantly, the expected rate of return of the stock is not one of the variables in the Black-

Scholes model or in any other model to evaluate an option price. The important implication here

is that the value of an option is completely independent of the expected growth of the underlying

asset and is therefore risk neutral. The fact that a prediction of the future price of the underlying

asset is not necessary to value an option may appear to be counter intuitive, but it is proven

correct over the years. Dynamically hedging a call using underlying asset prices generated from

Monte Carlo simulation perhaps is the ideal way to stimulate this situation. Irrespective of the

assumptions regarding stock price growth built into the Monte Carlo simulation the cost of

hedging a call (i.e. dynamically maintaining a delta neutral position by buying & selling the

underlying asset) will always be the same, and will be very close to the Black-Scholes value.

Putting it another way, whether the stock price rises or falls after. For example, writing a call, it

will always cost the same (providing volatility remains constant) to dynamically hedge the call

and this cost, when discounted back to present value at the risk free rate, is very close to the

Black-Scholes value. (http://www.hoadley.net/options/bs.htm)

Therefore the price evaluated by Black Scholes model is nothing more than the amount an option

writer would require as compensation for writing a call and hedging the risk. The main concept

here actually states that the price of an option is rather independent of the risk preferences of

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investors which are called the Risk Neutral Valuation. It states that all derivatives can be valued

by assuming the return from their underlying assets is the risk free rate.

6.4 Advantages and Limitations

The main advantage of the Black-Scholes model is generally its speed of evaluating option

prices. The model it lets you evaluate a very large number of option prices in a very short time

period, making it easy for options traders to price options instantaneously. The biggest strength is

the possibility of estimating market volatility of an underlying asset generally as a function of

price and time without direct reference to specific investor characteristics like expected yield,

risk aversion measure or utility functions. Secondly, the model also acts as a self-replicating

strategy or hedging, meaning investors can continuously buy and sell derivatives by the strategy

and never incur any losses.

6.4.1 Limitation of Black Scholes Model

Despite its popularity and the wide spread use, the model was built under some non-real

assumptions, and those assumptions are challenged by experts about its reliability throughout

these years.

Constant volatility – Volatility can be constant over short term period but it is never

constant in a longer term as large price changes tend to be followed large price changes,

and vice versa. This trend leads to a property called the volatility clustering. While

measures of volatilities are negatively correlated with the assets price returns (Leverage

Effect), trading volumes in the other hand is positively correlated, thus making

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volatility to be unstable over time. Therefore, some advance option valuation models

substitute the constant volatility under Black Scholes model with stochastic process

generated estimates.

The movement of stocks in the market – Stocks moves randomly in the market, also

known as random walk. The stock at time is independent to time (Martingale

Property of Brownian Motion). Stock prices are determined by many economic factors

that cannot be assigned the same probability of price fluctuation in each period of time.

Returns of log normally distributed underlying stock prices are normally

distributed – This assumptions is reasonable in the real world, though not fitting

observed financial data accurately. Assets returns have a finite variance and semi heavy

tails contrary to stable distributions like log normal with infinite variance and heavy

tails (Clark, 1973). As time scale over which return of assets are calculated increases,

the distribution of asset prices look more like the normal distribution with heavy tails

despite the fact that autocorrelation of asset prices are often insignificant.

Constant interest rates – The interest rates are represented by the risk free rate, while

there is no such thing as risk free rates but it is possible to use the Treasury Bills 30-

Day rate if the government is deemed to be credible. Yet so the treasury rates can

fluctuate based on the increase of volatility.

No commission and transaction costs – The Black Scholes model assumes that there

are no fees for buying and selling options, and also there are no barriers for trading.

Basically this is not true as stock brokers charge rates based on spreads and other

criteria.

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Only can be applied for European style options which can only be exercised on the

expiration date – American style options can be exercised any time during the life

span of the option, making it more valuable than European style options due to their

greater flexibility.

Markets are perfectly liquid – The Black Scholes model allows the possibility to

purchase and sell any amount of stock or options at any time. This is not possible as

investors are limited by the amount of money they can invest, policies of the company

and the permission or interest of the sellers to sell.

The huge advantage the Binomial model has over the Black Scholes model is that it can be used

to price American options accurately. This is due to the fact that the Binomial model allows early

exercises. The only disadvantage the model possesses is that the model is relatively slow; as the

time expands the slower the calculation leads to a price for an option.

6.4.2 Other Similar Models to the Black Scholes Model

The pricing of American options using analytical models are more unmanageable compared to

European options. Throughout the years, many models have been developed to handle American

options pricing in an efficient manner. The three most widely used models are:

Roll, Geske and Whaley analytic solution - The RGW formula can be used for pricing

an American call on a stock paying discrete dividends.

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Black's approximation for American calls - Although the RGW formula is an analytic

solution it involves solving equations iteratively and thus it is slower than Black-Scholes

model. Black's approximation basically involves using the Black-Scholes model after

making adjustments to the stock price and expiration date to take account of early

exercise.

Barone-Adesi and Whaley quadratic approximation - An analytic solution for

American puts and calls paying a continuous dividend. Like the RGW formula it involves

solving equations iteratively so whilst it is much faster than the binomial model it is still

much slower than Black-Scholes.

Many models have been proposed over the years, all attempting to actually mimic the

characteristics of market fully. Generally saying, every aspect of the market cannot be

considered in any fraction of time, as every factor capturing the price of a financial security

cannot be captured mathematically. Finally, we can only attempt to capture most of the factors,

as what is proposed by Levy models and other advance models.

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7.0 References

1. Lyuu Y. (2002). Financial Engineering and Computation: Principles, Mathematics,

Algorithms. Cambridge University Press, UK.

2. Neftci S. (2000). An Introduction to the Mathematics of Financial Derivatives. Academic

Press, Second Edition, New York.

3. Baz J. and Chacko G. (2004). Financial Derivatives: Pricing, Applications and

Mathematics. Cambridge University Press, Cambridge

4. Black F. and Scholes M. (1973). The Pricing of Options and Corporate Liabilities.

Journal of Political Economy, Vol.81 (3), 637-654.

5. Chance D. (1998). An Introduction to Derivatives. The Dryen Press, Fourth Edition,

USA.

6. Hull J. (2003). Options, Futures and other Derivatives. Pearson Education Inc., fifth

Edition, New Jersey

7. Das S. (1997). Risk Management and Financial Derivatives, A guide to the Mathematics.

McGraw Hill, New York.

8. Wilmott P., Howison S. and Dewyne J. (1995). The Mathematics of Financial

Derivatives. A Student Introduction. Cambridge University Press, Cambridge.

9. Kolb R. (1999). Futures, Options and Swaps. Blackwell Publisher Ltd, Third Edition, UK

10. Peter Carr, Liuren Wu, 2002. “Time Change Levy Process and Option Pricing” August

2002, Accessed 20.11.2010

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11. Bryman, A. (2004) Social Research Methods. 2nd

ed. Oxford: Oxford University

Press.

12. Campbell, J. and Hentschel, L. (1992) “No News Is Good News: An Asymmetric

Model Of Changing Volatility In Stock Returns”. Journal of Financial Economics. 31

(3): 281-318.

13. Chance, D (2004b). An Introduction to Derivatives & Risk Management. 6th

ed. Ohio:

Thomson/South-Western.

14. Cox, J. C., and Ross, S. A. (1976) “The Valuation of Options for Alternative

Stochastic Processes”. Journal of Financial Economics. 4 (2): 145-166.

15. French, K., Schwert, G. and Stambaugh, R. (1987) “Expected Stock Returns And

Volatility”. Journal of Financial Economics. 19 (1): 3-29.

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8.0 Appendix

8.1 VBA code of CRR Binomial Model in Excel

'CallPutFlag - use "c" for call and "p" for put option

'S - spot price

'K - option strike

'T - option maturity

'r - risk free rate

'v - volatility

'ExerciseType - use "a" for American and "e" for european

'N - no of time steps for the binomial tree

Public Function CRR_Price(CallPutFlag, S, K, T, r, v, ExerciseType, N) As Double

S0 = S

If CallPutFlag = "c" Then

CallPutFlag = 1

Else

CallPutFlag = -1

End If

dt = T / N

u = Exp(v * dt ^ 0.5) 'size of up jump

d = Exp(-v * dt ^ 0.5) 'size of down jump

p1 = (u - Exp(r * dt)) / (u - d) 'probability of up jump

p2 = 1 - p1 'probability of down jump

ReDim Smat(1 To N + 1, 1 To N + 1) 'hlods stock prices

Smat(1, 1) = S0

For i = 1 To UBound(Smat, 1) - 1

Smat(1, i + 1) = Smat(1, i) * Exp(v * dt ^ 0.5)

For j = 2 To i + 1

70

Smat(j, i + 1) = Smat(j - 1, i) * Exp(-v * dt ^ 0.5)

Next j

Next i

ReDim Cmat(1 To N + 1, 1 To N + 1)

For i = 1 To N + 1

Cmat(i, N + 1) = Application.Max(CallPutFlag * (Smat(i, N + 1) - K), 0)

Next i

For i = UBound(Smat, 2) - 1 To 1 Step -1

For j = 1 To i

present_value = Exp(-r * dt) * (p2 * Cmat(j, i + 1) + p1 * Cmat(j + 1, i + 1))

immediate_val = CallPutFlag * (Smat(j, i) - K)

If ExerciseType = "a" Then

Cmat(j, i) = Application.Max(present_value, immediate_val)

Else

Cmat(j, i) = Application.Max(present_value, 0)

End If

Next j

Next i

CRR_Price = Cmat(1, 1)

End Function

8.2 VBA code of The Black Scholes Model in Excel

Function Black_Scholes(ByVal S As Double, ByVal K As Double, ByVal r As

Double, ByVal q As Double, ByVal t As Double, ByVal sigma As Double) As

Double

Dim d1 As Double

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Dim d2 As Double

Dim a As Double

Dim b_call As Double

Dim b_put As Double

Dim c As Double

Dim call_price As Double

Dim put_price As Double

a = Log(S / K)

b_call = (r - q + 0.5 * sigma ^ 2) * t

b_put = (r - q - 0.5 * sigma ^ 2) * t

c = sigma * Sqr(t)

d1 = (a + b_call) / c

d2 = (a + b_put) / c

call_price = S * WorksheetFunction.NormsDist(d1) - K * Exp(-r * t) *

WorksheetFunction.NormsDist(d2)

put_price = K * Exp(-r * t) * WorksheetFunction.NormsDist(-d2) - S *

WorksheetFunction.NormsDist(-d1)

Black_Scholes = Array(call_price, put_price)

End Function