بسم الله الرحمن الرحیم

بسم الله الرحمن الرحیم

Myron S. Scholes Robert C. Merton Fischer Black

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997

Prize motivation: “For a new method to determine the value of derivatives"

Presented By Abdolmohammad Kashian

P.H.D Student of Economics, ISU, Tehran

Myron S. Scholes Born: 1 July 1941, Timmins, ON, CanadaAffiliation at the time of the award: Long Term Capital Management, Greenwich, CT, USAPrize motivation: "for a new method to determine the value of derivatives"Field: financial economics

Education:

McMaster University (1961) (liberal arts,

majoring in Economics)

University of Chicago (1964) finance and

economics

University of Chicago (1969) Financial

Economics

Contribution: Developed a method of determining the value of derivatives, the Black-Scholes formula (together with Fischer Black, who died two years before the Prize award). This methodology paved the way for economic valuations in many areas. It also generated new financial instruments and facilitated more effective risk management in society. The work generated new financial instruments and has facilitated more effective risk management in society

Robert C. Merton Born: 31 July 1944, New York, NY, USAAffiliation at the time of the award: Harvard University, Cambridge, MA, USAPrize motivation: "for a new method to determine the value of derivatives"Field: financial economics

EducationB.S., Columbia University (Engineering Mathematics), 1966M.S., California Institute of Technology (Applied Mathematics), 1967Ph.D., Massachusetts Institute of Technology (Economics), 1970

Contribution: Had a direct influence on the development of the Black-Scholes formula and generalized it in important ways. By devising another way of deriving the formula, he applied it to other financial instruments, such as mortgages and student loans. The work generated new financial instruments and has facilitated more effective risk management in society.

Fischer BlackBorn: January 11, 1938 – August 30, 1995Education: Harvard UniversityDied: August 30, 1995 (aged 57)New York, U.S.Field: financial economics

Risk management:

Risk management is essential in a modern

market economy. Financial markets enable

firms and households to select an appropriate

level of risk in their transactions, by

redistributing risks towards other agents who

are willing and able to assume them

The Instrument of Risk Management and

its Valuation:

Markets for options, futures and other so-called

derivative securities - derivatives, for short -

have a particular status.

The valuation of these instrument is very

important. Effective risk management requires

that such instruments be correctly priced.

Fischer Black, Robert Merton and Myron Scholes made

a pioneering contribution to economic sciences by

developing a new method of determining the value of

derivatives. Their innovative work in the early 1970s,

which solved a longstanding problem in financial

economics, has provided us with completely new ways

of dealing with financial risk, both in theory and in

practice. Their method has contributed substantially to

the rapid growth of markets for derivatives in the last

two decades. Fischer Black died in his early fifties in

August 1995.

Black, Merton and Scholes´ contribution

extends far beyond the pricing of derivatives,

however. Whereas most existing options are

financial, a number of economic contracts and

decisions can also be viewed as options: an

investment in buildings and machinery may

provide opportunities (options) to expand into

new markets in the future.

The history of option valuation

Attempts to value options and other derivatives

have a long history. One of the earliest endeavors

to determine the value of stock options was made

by Louis Bachelier in his Ph.D. thesis at the

Sorbonne in 1900. The formula that he derived,

however, was based on unrealistic assumptions, a

zero interest rate, and a process that allowed for a

negative share price.

Case Sprenkle, James Boness and Paul Samuelson

improved on Bachelierís formula. They assumed that

stock prices are log-normally distributed (which

guarantees that share prices are positive) and allowed

for a non-zero interest rate. They also assumed that

investors are risk averse and demand a risk premium in

addition to the risk-free interest rate.

In 1964, Boness suggested a formula that came close to

the Black-Scholes formula, but still relied on an

unknown interest rate , …

The attempts at valuation before 1973 basically

determined the expected value of a stock

option at expiration and then discounted its

value back to the time of evaluation. Such an

approach requires taking a stance on which risk

premium to use in the discounting.. But

assigning a risk premium is not

straightforward.

The Black-Scholes formulaThis years laureates resolved these problems by recognizing that it is not necessary to use any risk premium when valuing an option. This does not mean that the risk premium disappears, but that it is already incorporated in the stock price. In 1973 Fischer Black and Myron S. Scholes published the famous option pricing formula that now bears their name (Black and Scholes (1973)). They worked in close cooperation with Robert C. Merton, who, that same year, published an article which also included the formula and various extensions (Merton (1973))

The idea behind the new method developed by Black, Merton and Scholes can be explained in the following simplified way:European call option (gives the right to buy a certain share at a strike price of $100 in three months).The value of this call option depends on the current share price; the higher the share price today the greater the probability that it will exceed $100 in three months, in which case it will pay to exercise the option.

A formula for option valuation should thus determine exactly how the value of the option depends on the current share price. How much the value of the option is altered by a change in the current share price is called the "delta" of the option.

As the share price is altered over time and as the time to maturity draws nearer, the delta of the option changes. In order to maintain a risk-free stock-option portfolio, the investor has to change its composition. Black, Merton and Scholes assumed that such trading can take place continuously without any transaction costs (transaction costs were later introduced by others). The condition that the return on a risk-free stock-option portfolio yields the risk-free rate, at each point in time, implies a partial differential equation, the solution of which is the Black-Scholes formula for a call option:

where the variable d is defined by:

𝐶=𝑆𝑁 (𝑑 )−𝐿𝑒− 𝑟𝑡𝑁 (𝑑−𝜎 √𝑡)

𝑑=𝑙𝑛 𝑆𝐿+(𝑟+

𝜎2

2 )𝑡𝜎 √𝑡

According to this formula, the value of the call

option C , is given by the difference between

the expected share price - the first term on the

right-hand side - and the expected cost - the

second term - if the option is exercised

The option pricing formula is named after

Black and Scholes because they were the first

to derive it. Black and Scholes originally based

their result on the capital asset pricing model

(CAPM, for which Sharpe was awarded the

1990 Prize). While working on their 1973

paper, they were strongly influenced by

Merton. Black describes this in an article

(Black (1989))

Scientific importanceThe option-pricing formula was the solution of a more than seventy-year old problem. As such, this is, of course, an important scientific achievement. The main importance of Black, Merton and Scholes´ contribution, however, refers to the theoretical and practical significance of their method of analysis. It has been highly influential in solving many economic problems. The scientific importance extends to both the pricing of derivative securities and to valuation in other areas.

Pricing of derivativesThe laureates initiated the rapid evolution of option pricing that has taken place during the past two decades.

Corporate liabilitiesBlack, Merton and Scholes realized already in 1973 that a share can be interpreted as an option on the whole firm

و اخر دعوانا ان الحمد للّه رب العالمین

31

The Options Are Two Kinds:

Vanilla optionsExotic options

32

Vanilla

options

Vanilla are the basic options also known as European

and American. European options can be exercised only

at maturity date where American options can be

exercised at anytime up to the maturity date. Although

European options are easier to analyze, mostly American

options are traded on the real market

33

Exotic

options

We sort options in two main groups, the vanilla options

described above and exotic options which are more complex

derivatives constructed on standard vanilla options. Barrier

option are part of exotic options but they are not the only

ones. We can invent every kind of exotic options and there

exists plenty of them, usually traded over-the-counter.

However the most common ones are:

34

Exotic

Options Barrier are normal puts and calls except that they

disappear or appear if the underlying asset price cross a given level.

Asian have a pay out not determined by the underlying price at maturity but by the average underlying price over some pre-set period of time.

Lookback is a path dependent option where the option owner has the right to buy (sell) the underlying instrument at its lowest (highest) price over some preceding period.

Forward start option is an option whose strike price is determined in the future.

35

Exotic

Options Basket option is an option on the

weighted average of several underlying's. Digital/Binary option pays a fixed amount,

or nothing at all, depending on the price of the underlying instrument at maturity.

Bermudan options is an option where the buyer has the right to exercise at a set (always discretely spaced) number of times. This is intermediate between a European option which allows exercise at a single time, namely expiry date, and an American option which allows exercise at any time (the name is a pun: Bermuda is between Americand Europe).