Chapter 1

Introduction and history of derivatives

Derivatives definition:

A derivative instrument is a financial contract whose payoff structure is

determined by the value of an underlying commodity, interest rate, share
price index, exchange rate, oil price.

Derivative is a product whose value is derived from the value of one or more
basic variables, called bases (underlying asset, index, or reference rate), in a
contractual manner.

The underlying asset can be equity, forex, commodity or any other asset.

Example, wheat farmers may wish to sell their harvest at a future date to
eliminate the risk of a change in prices by that date. The price of this
derivative is driven by the spot price of wheat which is the "underlying".

Derivatives are securities under the SC(R)A and hence the trading of
derivatives is governed by the regulatory framework under the SC(R)A.

Emergence of financial derivative products:

Derivative products initially emerged as hedging devices against fluctuations

in commodity prices, and commodity-linked derivatives remained the sole
form of such products for almost three hundred years.
Financial derivatives came into spotlight in the post-1970 period due to
growing instability in the financial markets. However, since their emergence,
these products have become very popular and by 1990s, they accounted for
about two-thirds of total transactions in derivative products.

In recent years, the market for financial derivatives has grown tremendously
in terms of variety of instruments available, their complexity and also
turnover. In the class of equity derivatives the world over, futures and
options on stock indices have gained more popularity than on individual
stocks, especially among institutional investors, who are major users of
index-linked derivatives. Even small investors find these useful due to high
correlation of the popular indexes with various portfolios and ease of use.

Factors driving the growth of Derivatives:

Over the last three decades, the derivatives market has seen a phenomenal
growth. A large variety of derivative contracts have been launched at
exchanges across the world.

Some of the factors driving the growth of financial derivatives are:

1. Increased volatility in asset prices in financial markets

2. Increased integration of national financial markets with the international

markets

3. Marked improvement in communication facilities and sharp decline in their

costs

4. Development of more sophisticated risk management tools, providing

economic agents a wider choice of risk management strategies
5. Innovations in the derivatives markets, which optimally combine the risks
and returns over a large number of financial assets leading to higher returns,
reduced risk as well as transactions costs as compared to individual financial
assets.

Participants in the Derivatives Markets:

There are basically three broad categories…

1) Hedgers
2) Speculators
3) Arbitrageurs

1) Hedgers:
Hedgers face risk associated with the price of an asset. They use
future or options markets to reduce or eliminate this risk.

Hedgers are the traders who wish to eliminate the risk to which they
are already exposed.

Example: suppose a leading trader buys a large quantity of wheat that

would take two weeks to reach him. Now, he feels that the wheat
prices may fall in the coming two weeks and so wheat may have to sell
at lower prices. The trader can sell futures contracts with a matching
price, to hedge. Thus if the wheat price do fall, the trader would lose
money on the inventory of wheat but will profit from the future
contract which would balance the loss. Similarly, an investor who holds
a large quantity of shares of a company can hedge by selling futures
on them or by buying put option contracts, in case he fears a fall in the
price of that share.

2) Speculators:
If hedgers are the people who wish to avoid the price risk, speculators
are those who are willing to take suck risk.

These are the people who take positions in the market and assume
risks to profit from fluctuations in prices.

Speculators with to bet on future movements in the price of an asset.

Speculators consume information make forecasts about the prices and

put their money in these forecasts.

In this process, they feed information into prices and this contributes
to market efficiency.

By taking positions, they are betting that a price would go up they are
getting that it would go down.

The speculators in the derivatives markets may either by day traders

or position traders.

The day traders speculate on the price movements during one trading
day, open and close positions many times a day and do not carry any
position at the end of the day. Obviously, they monitor the prices
continuously and generally attempt to make profit from just a few ticks
per trade.

The position traders also attempt to gain from price fluctuations but
they keep their positions for longer durations – may be for a few days,
weeks or even months.
 3) Arbitrageurs:
Arbitrageurs are in business to take advantage of a discrepancy
between prices in two different markets.

They see the futures price of an asset getting out of line with the cash
price, they will take off setting positions in the two markets to lock in a
profit.

If a trader believes that the price differential between the future

contracts on the same underlying asset with differing maturities is
more or less than what he/she perceives them to be, then appropriate
positions in them may be taken to make profits.

Derivative Products:

Types of Derivatives:

There are basically four types of derivatives:

1) Forward
2) Future
3) Option
4) Swap

1) Forward Contract:
It is the agreement to purchase or sell any asset at a particular price
on same future maturity date.

A forward contract is a customized contract between two entities,

where settlement takes place on a specific date in the future at today's
pre-agreed

price.
 Essential for forward contracts:

 Two parties with exactly opposite mind set.

 Existence of assets or expectation of existence of assets.
 Both parties should be in a position to except the price.

2) Future Contract:
It is the agreement to buy or sell any asset for agreed price on agreed
future date with existence of third party.

A futures contract is an agreement between two parties to buy or sell

an asset at a certain time in the future at a certain price.

Futures contracts are special types of forward contracts in the sense

that the former are standardized exchange-traded contracts

3) Options:
Options are of two types Calls and Puts:

Calls give the buyer the right but not the obligation to buy a given
quantity of the underlying asset, at a given price on or before a given
future date.

Puts give the buyer the right, but not the obligation to sell a given
quantity of the underlying asset at a given price on or before a given
date.

 Warrants:
 Options generally have lives of upto one year, the majority of
options traded on options exchanges having a maximum
maturity of nine months. Longer-dated options are called
warrants and are generally traded over-the-counter.

 Leaps:
The acronym LEAPS means Long-Term Equity Anticipation
Securities.

These are options having a maturity of upto three years.

 Baskets:
Basket options are options on portfolios of underlying assets.
The underlying asset is usually a moving average of a basket of
assets. Equity index options are a form of basket options.

4) Swaps:
Swaps are private agreements between two parties to exchange cash
flows

in the future according to a prearranged formula. They can be

regarded as

portfolios of forward contracts. The two commonly used swaps are:

Interest rate swaps: These entail swapping only the interest

related cash flows between the parties in the same currency.

Currency swaps: These entail swapping both principal and

interest between the parties, with the cash flows in one direction being
in a different currency than those in the opposite direction.
  Swaptions:
Swaptions are options to buy or sell a swap that will become
operative at the expiry of the options. Thus a swaption is an
option on a forward swap. Rather than have calls and puts, the
swaptions market has receiver swaptions and payer swaptions.

A receiver swaption is an option to receive fixed and pay

floating. A payer swaption is an option to pay fixed and receive
floating.

Functions of Derivative Market:

1. Price discovery…Prices in an organized derivatives market reflect the

perception of market participants about the future and lead the prices
of underlying to the perceived future level. The prices of derivatives
converge with the prices of the underlying at the expiration of the
derivative contract. Thus derivatives help in discovery of future as well
as current prices.

2. Derivatives, due to their inherent nature, are linked to the underlying

cash markets with the introduction of derivatives, the underlying
market witness higher trading volumes because of participation by
more players who would not otherwise participate for lack of an
arrangement to transfer risk.

3. Catalyst for new entrepreneurial activity: The derivatives have a

history of attracting many bright, creative, well-educated people with
an entrepreneurial attitude. They often energize others to create new
businesses, new products and new employment opportunities, the
benefit of which are immense.

4. Speculative trades shift to a more controlled environment of

derivatives market. In the absence of an organized derivatives market,
speculators trade in the underlying cash markets. Margining,
monitoring and surveillance of the activities of various participants
become extremely difficult in these kind of mixed markets.
History of Derivative Markets:

Early forward contracts in the US addressed merchants' concerns about

ensuring that there were buyers and sellers for commodities. However 'credit
risk" remained a serious problem. To deal with this problem, a group of
Chicago businessmen formed the Chicago Board of Trade (CBOT) in 1848.

The primary intention of the CBOT was to provide a centralized location

known in advance for buyers and sellers to negotiate forward contracts. In
1865, the CBOT went one step further and listed the first 'exchange traded"
derivatives contract in the US, these contracts were called 'futures
contracts".

In 1919, Chicago Butter and Egg Board, a spin-off of CBOT, was reorganized
to allow futures trading. Its name was changed to Chicago Mercantile
Exchange (CME). The CBOT and the CME remain the two largest organized
futures exchanges, indeed the two largest "financial" exchanges of any kind
in the world today.

The first stock index futures contract was traded at Kansas City Board of
Trade. Currently the most popular stock index futures contract in the world is
based on S&P 500 index, traded on Chicago Mercantile Exchange. During the
mid eighties, financial futures became the most active derivative instruments
generating volumes many times more than the commodity futures. Index
futures, futures on T-bills and Euro-Dollar futures are the three most popular
futures contracts traded today. Other popular international exchanges that
trade derivatives are LIFFE in England, DTB in Germany, SGX in Singapore,
TIFFE in Japan, MATIF in France, Eurex etc.

Exchange traded vs OTC derivatives markets:

Derivatives that trade on an exchange are called exchange traded

derivatives, where as privately negotiated derivative contracts are called OTC
contracts
The OTC derivatives markets have the following features compared to
exchange traded derivatives:

1. The management of counter-party (credit) risk is decentralized and

located within individual institutions,

2. There are no formal centralized limits on individual positions, leverage, or

margining,

3. There are no formal rules for risk and burden-sharing,

4. There are no formal rules or mechanisms for ensuring market stability and
integrity, and for safeguarding the collective interests of market participants

5. The OTC contracts are generally not regulated by a regulatory authority

and the exchange's self-regulatory organization, although they are affected
indirectly by national legal systems, banking supervision and market
surveillance.

The following features of OTC derivatives markets can give rise to instability
in institutions, markets, and the international financial system:

(i) the dynamic nature of gross credit exposures;

(ii) information asymmetries;
(iii) the effects of OTC derivative activities on available aggregate
credit;
(iv) the high concentration of OTC derivative activities in major
institutions;
(v) The central role of OTC derivatives markets in the global
financial system.
Chapter 3 Introduction to Futures and Forwards

Forward contracts

A forward contract is an agreement to buy or sell an asset on a specified date

for a specified price. One of the parties to the contract assumes a long
position and agrees to buy the underlying asset on a certain specified future
date for a certain specified price. The other party assumes a short position
and agrees to sell the asset on the same date for the same price. Other
contract details like delivery date, price and quantity are negotiated
bilaterally by the parties to the contract. The forward contracts are normally
traded outside the exchanges.

The salient features of forward contracts are:

• They are bilateral contracts and hence exposed to counter-party risk.

• Each contract is custom designed, and hence is unique in terms of contract

size, expiration date and the asset type and quality.

• The contract price is generally not available in public domain.

• On the expiration date, the contract has to be settled by delivery of the

asset.

• If the party wishes to reverse the contract, it has to compulsorily go to the

same counter-party, which often results in high prices being charged.

LIMITATIONS OF FORWARD MARKETS

Forward markets world-wide are afflicted by several problems:

• Lack of centralization of trading,

• Illiquidity, and

• Counterparty risk

In the first two of these, the basic problem is that of too much flexibility and
generality. The forward market is like a real estate market in that any two
consenting adults can form contracts against each other. This often makes
them design terms of the deal which are very convenient in that specific
situation, but makes the contracts non-tradable. Counterparty risk arises
from the possibility of default by any one party to the transaction. When one
of the two sides to the transaction declares bankruptcy, the other suffers.
Even when forward markets trade standardized contracts, and hence avoid
the problem of illiquidity, still the counterparty risk remains a very serious
issue.

INTRODUCTION TO FUTURES

A futures contract is an agreement between two parties to buy or sell an

asset at a certain time in the future at a certain price. But unlike forward
contracts, the futures contracts are standardized and exchange traded. To
facilitate liquidity in the futures contracts, the exchange specifies certain
standard features of the contract. It is a standardized contract with standard
underlying instrument, a standard quantity and quality of the underlying
instrument that can be delivered, (or which can be used for reference
purposes in settlement) and a standard timing of such settlement. A futures
contract may be offset prior to maturity by entering into an equal and
opposite transaction. More than 99% of futures transactions are offset this
way.

The standardized items in a futures contract are:

Quantity of the underlying

Quality of the underlying

The date and the month of delivery

The units of price quotation and minimum price change

Location of settlement

Futures terminology

Spot price: The price at which an asset trades in the spot market.

Futures price: The price at which the futures contract trades in the
futures market.

Contract cycle: The period over which a contract trades. The index
futures contracts on the NSE have one- month, two-months and three
months expiry cycles which expire on the last Thursday of the month.

Thus a January expiration contract expires on the last Thursday of January

and a February expiration contract ceases trading on the last Thursday of
February. On the Friday following the last Thursday, a new contract having a
three- month expiry is introduced for trading.

Expiry date: It is the date specified in the futures contract. This is the
last day on which the contract will be traded, at the end of which it will cease
to exist.

Contract size: The amount of asset that has to be delivered under one
contract. Also called as lot size.29

Basis: In the context of financial futures, basis can be defined as the

futures price minus the spot price. There will be a different basis for each
delivery month for each contract. In a normal market, basis will be positive.
This reflects that futures prices normally exceed spot prices.
Cost of carry: The relationship between futures prices and spot prices
can be summarized in terms of what is known as the cost of carry. This
measures the storage cost plus the interest that is paid to finance the asset
less the income earned on the asset.

Initial margin: The amount that must be deposited in the margin

account at the time a futures contract is first entered into is known as initial
margin.

Marking-to-market: In the futures market, at the end of each trading

day, the margin account is adjusted to reflect the investor's gain or loss
depending upon the futures closing price. This is called marking-to-market.

Maintenance margin: This is somewhat lower than the initial margin.

This is set to ensure that the balance in the margin account never becomes
negative. If the balance in the margin account falls below the maintenance
margin, the investor receives a margin call and is expected to top up the
margin account to the initial margin level before trading commences on the
next day.

Difference between forward and future

Specification Forward Future

Standardisation No standardisation in- Future contract are
terms of quantity, standardise for quality,
maturity date, quality quantity, price, maturity
place of delivery, etc date and place of delivery
Liquidity There is no liquidity or High liquidity with
secondary market for this secondary market in this
Conclusion of Contract Generally forward Future contracts can be
contracts are concluded concluded with either
with delivery of the asset delivery or cash
Margins Forward contracts does Future contracts requires
not require margins margins
Third party No existence Existence

Example:

Types of Future contracts:

 1) Commodity Futures:
The commodity futures contracts involve a wide range of
agricultural and other commodities, including precious metals

2) Financial Futures:
Financial futures involve financial assets / tools as against
commodities. Contracts are written over a wide range of the
underlying assets.

Difference between Commodity future and Financial Futures

Specification Commodity Future Financial Future

Quality It is an important aspect It is not at all consider
and hence to be followed
strictly
Cash Settlement No cash settlement and Cash settlement
delivery of commodity is
essential
Contract life Small contract life Long contract life
Maturity date Generally maturity date Maturity date is fix and
depends upon nature of does not depend on
commodity nature of underlying
Place of delivery Depends on nearness of Place of delivery does not
the market depend on market.

Clearing house and its role

A clearing house plays a pivotal role in the trading of futures contracts. It

acts as an intermediary for each contract. It is natural to ask “what happens
if someone decided not to pay for the commodity as promised in the contract
or if someone is unable to deliver the commodity?”

Obviously, if it were possible to back out to the contract without fulfilling

one’s part, the futures exchanges would die very soon. People would lose
confidence in the system and it would provide no attraction to hedgers or
speculators. Elimination of this uncertainty is essentially the job of a clearing
house.
Once a futures price a agreed upon between the buyer and the seller and
trade is completed, the clearing house of the exchange becomes the opposite
party to each one of the parties. Thus, when an investor goes long a futures
contract, he/she effectively buys it from the cleaning house and similarly,
when one goes short a futures contract, one is in fact liable to the clearing
house only. In effect then, the clearing house ensures the integrity of each
futures contract by interposing itself between each buyer and seller.

Thus, whereas in forward contracts, each party faces the risk that the
opposite party may default, the parties in a contract have no worry because
of the guarantee of the clearing house.

It may be noted that while the clearing house takes the opposite position to
each side of the trade, it does not by itself buy or sell contracts. Its role
begins only when two parties to a contract agree to enter into a contract. The
clearing house is always neutral; maintaining both long and short, an
identical number of contracts.

A clearing house is associated with a futures exchange and is concerned with

matching, processing, registering, confirming, settling and reconciling
besides quarantining the trades on the futures exchanges. An adjunct of the
exchange and an intermediary in all the futures transactions, a clearing
house has a number of members.

The brokers, who are not members, have to channel their business through a
member. Since the main function of a clearing house is to eliminate, as far as
possible, the risk that someone at some stage would fail to honour their part
of the commitment, with the possibility that such a happening could wipe out
many of the participants from the exchange, it is no surprise that the
members of a clearing house are usually the most financially secure firms of
an exchange. In some cases, the clearing house might have the backing of
the government as well.

Margins
Page number 33 to 37

Limits

Page number 37 & 38

Divergence of future and spot prices: The Basis

The difference between the future prices and the current price is known as
the basis

Basis = Pt - Po

Pt = Price of the commodity at same future point in time

Po = spot market price

In a “Normal Market” the future price would be greater the spot price so
basis will be positive.

In “Inverted Market” basis is negative since the spot price exceeds the future
price.

F > So The future price is expected to be higher than spot price because of
carrying costs associated with a particular commodity - storage costs,
insurance, cost of funds invested in them and other costs incurred to keep
the commodity in inventory until its delivery date.

The longer the time to maturity the greater the carrying costs. As the
delivery month approaches, the basis declines until the spot and futures
prices are approximately the same. This phenomenon is known as
convergence.

So > F Basis is negative, when future prices are lower than spot prices.
Example when commodity may be in short supply now then that case spot
price may be higher.

If the supply is likely to improve in the course of time and to good monsoons
then the future prices may indeed be lower.

Convergence: The future prices equals or is very close to the spot price,
when the delivery period is reached.

Future price is higher than spot price; an investor would do well to

a) Short sell a future contract

b) Buy the asset
c) Make the delivery to reap a profit equal to the excess of the
futures price over the spot price
As traders exploit this opportunity, the price of future would drop.

Similarly, the future price is lower than the spot price. An investor who is
interested in buying an asset would be inclined to buy a future contract and
take the delivery.

Expected Basis:

The basis is primarily attributable to the carrying costs of a commodity. If

there is no carrying costs involved and if there were no uncertainty, then the
futures today would command a price equal to the expected spot price in the
future.

In case of certainty, then the expected basis would be equal to zero. In case
of uncertainty there are three hypotheses to explain the expected basis

1) Normal backwardation F<so

I. The theory is dependent on the assumption that

speculators usually buy i.e. net long, the contracts while
the hedgers usually write i.e. net short, the contracts.

II. According to this hypothesis, the expected basis is

negative as the future price tends to be a downward
estimate of spot price in the cash market at the contract’s
maturity date.

III. This hypothesis was propounded by J. M Keynes who

argued that the future market is dominated by short
hedgers who attempt to avoid the risk associated with a
decrease in the price of commodity.

2) Contango: F>So
 I. This hypothesis assumes the other possibility - the
hedgers generally buy the contracts, which the speculators
generally sell the contracts.

II. The speculators because of their knowledge and expertise

about the future market, and the in efficiencies of the
market, are largely willing to assume the price risk in
anticipation of earning profit.

III. They bid up prices of the commodity contracts, which

results in a positive basis. They can make profit only if
they net short and future price declines.

IV. With the price of futures contracts above the spot price
initially and then declining over time, this phenomenon is
termed as contango.

3) Expectation principle:

I. This theory assumes that futures prices are an unbiased

estimate of expected future spot prices, as would be
expected in an efficient market.

II. Theory postulates that the expected basis would be equal

or zero.

III. There is no room for any excess returns or the speculators.

Hedging using futures contracts:

Hedging is the process of reducing exposure to risk. A hedge is any act that
reduces the price risk of a certain position in the cash market.

Future contracts are important means of hedging as they enable the market
participants to other the risks from unexpected adverse price changes.

Page number 43 & 44.

Chapter 5 Options

INTRODUCTION TO OPTIONS

In this section, we look at the next derivative product to be traded on

the NSE, namely options.

Options are fundamentally different from forward and futures

contracts. An option gives the holder of the option the right to do
something. The holder does not have to exercise this right. In
contrast, in a forward or futures contract, the two parties have
committed themselves to doing something.

Whereas it costs nothing (except margin requirements) to enter into a

futures contract, the purchase of an option requires an up-front
payment.

OPTION TERMINOLOGY

· Index options: These options have the index as the underlying.

Some options are European while others are American. Like index
futures contracts, index options contracts are also cash settled.

· Stock options: Stock options are options on individual stocks.

Options currently trade on over 500 stocks in the United States. A
contract gives the holder the right to buy or sell shares at the specified
price.

· Buyer of an option: The buyer of an option is the one who by

paying the option premium buys the right but not the obligation to
exercise his option on the seller/writer.

· Writer of an option: The writer of a call/put option is the one who

receives the option premium and is thereby obliged to sell/buy the
asset if the buyer exercises on him.

There are two basic types of options, call options and put options.

· Call option: A call option gives the holder the right but not the
obligation to buy an asset by a certain date for a certain price.

· Put option: A put option gives the holder the right but not the
obligation to sell an asset by a certain date for a certain price.
· Option price/premium: Option price is the price which the option
buyer pays to the option seller. It is also referred to as the option
premium.

· Expiration date: The date specified in the options contract is known

as the expiration date, the exercise date, the strike date or the
maturity.

· Strike price: The price specified in the options contract is known as

the strike price or the exercise price.

· American options: American options are options that can be

exercised at any time upto the expiration date. Most exchange-traded
options are American.

· European options: European options are options that can be

exercised only on the expiration date itself. European options are
easier to analyze than American options, and properties of an
American option are frequently deduced from those of its European
counterpart.

· In-the-money option: An in-the-money (ITM) option is an option

that would lead to a positive cash flow to the holder if it were
exercised immediately. A call option on the index is said to be in-the-
money when the current index stands at a level higher than the strike
price (i.e. spot price > strike price). If the index is much higher than
the strike price, the call is said to be deep ITM. In the case of a put,
the put is ITM if the index is below the strike price.

· At-the-money option: An at-the-money (ATM) option is an option

that would lead to zero cashflow if it were exercised immediately. An
option on the index is at-the-money when the current index equals the
strike price (i.e. spot price = strike price).

· Out-of-the-money option: An out-of-the-money (OTM) option is an

option that would lead to a negative cashflow if it were exercised
immediately. A call option on the index is out-of-the-money when the
current index stands at a level which is less than the strike price (i.e.
spot price < strike price). If the index is much lower than the strike
price, the call is said to be deep OTM. In the case of a put, the put is
OTM if the index is above the strike price.

· Intrinsic value of an option: The option premium can be broken

down into two components - intrinsic value and time value. The
intrinsic value of a call is the amount the option is ITM, if it is ITM. If
the call is OTM, its intrinsic value is zero. Putting it another way, the
intrinsic value of a call is Max[0, (St — K)] which means the intrinsic
value of a call is the greater of 0 or (St — K). Similarly, the intrinsic
value of a put is Max [0, K — St], i.e. the greater of 0 or (K — St). K is
the strike price and St is the spot price.

· Time value of an option: The time value of an option is the

difference between its premium and its intrinsic value. Both calls and
puts have time value. An option that is OTM or ATM has only time
value. Usually, the maximum time value exists when the option is
ATM. The longer the time to expiration, the greater is an option's time
value, all else equal. At expiration, an option should have no time
value.

Call Put

Buyer Right to buy Right to sell

security security

seller Obligation to sell Obligation to buy

security security
PROPERTIES OF STOCK OPTIONS

In this chapter we look at the factors affecting stock option prices. We

use a number of different arbitrage arguments to explore the
relationships between European option prices, American option prices,
and the underlying stock price. The most important of these
relationships is put-call parity, which is a relationship between
European call option prices and European put option prices.

The chapter examines whether American options should be exercised

early. It shows that it is never optimal to exercise an American call
option on a non-dividend-paying stock prior to the option's expiration,
but the early exercise of an American put option on such a stock can
be optimal.

8.1 FACTORS AFFECTING OPTION PRICES

There are six factors affecting the price of a stock option:

1. The current stock price, So
2. The strike price, K
3. The time to expiration, T
4. The volatility of the stock price, a
5. The risk-free interest rate, r
6. The dividends expected during the life of the option

In this section we consider what happens to option prices when one of

these factors changes with
jill the others remaining fixed. The results are summarized in Table
8.1.' Figures 8.1 and 8.2 show how the price of a European call and
put depends on the first five |ktors in the situation where So = 50, K
— 50, r = 5% per annum, a = 30% per annum,1"= 1 year, and there
are no dividends. In this case the call price is 7.116 and the put price
is 4.677.

Stock Price and Strike Price

If a call option is exercised at some future time, the payoff will be the
amount by which the stock price exceeds the strike price. Call options
therefore become more valuable as the stock price •tocreases and less
valuable as the strike price increases. For a put option, the payoff on
exercise is the amount by which the strike price exceeds the stock
price. Put options therefore behave in the ^Opposite way from call
options. They become less valuable as the stock price increases and
more valuable as the strike price increases. Figures 8.1a, b, c, d
illustrate the way in which put and call prices depend on the stock
price and strike price.
Table 8.1 Summary of the effect on the price of a stock option of
increasing one variable while keeping all others fixed.*

Variable European call European put American call American put

Current stock price + — + —
Strike price — + — +
Time to expiration ? ? + +
Volatility + + + +
Risk-free rate + — + —
Dividends — + — +
* + indicates that an increase in the variable causes the option price to
increase; — indicates that an increase in the variable causes the
option price to decrease? Indicates that the relationship is uncertain.

Time to Expiration

Consider next the effect of the expiration date. Both put and call
American options become more valuable as the time to expiration
increases. Consider two options that differ only as far as the expiration
date is concerned.

The owner of the long-life option has all the exercise opportunities
open to the owner of the short-life option—and more. The long-life
option must therefore always be worth at least as much as the short-
life option. Figures 8.1e, f illustrate the way in which calls and puts
depend on the time to expiration.

Although European put and call options usually become more valuable
as the time to expiration increases, this is not always the case.
Consider two European call options on a stock: one with an expiration
date in one month, and the other with an expiration date in two
months. Suppose that a very large dividend is expected in six weeks.
The dividend will cause the stock price to decline, so that the short-life
option could be worth more than the long-life option.

Volatility

The precise way in which volatility is defined is explained in Chapters

11 and 12. Roughly speaking, the volatility of a stock price is a
measure of how uncertain we are about future stock price movements.
As volatility increases, the chance that the stock will do very well or
very poorly increases. For the owner of a stock, these two outcomes
tend to offset each other. However, this is not so for the owner of a
call or put.
The owner of a call benefits from price increases but has limited
downside risk in the event of price decreases because the most the
owner can lose is the price of the option. Similarly, the owner of a put
benefits from price decreases, but has limited downside risk in the
event of price increases. The values of both calls and puts therefore
increase as volatility increases.

Risk-Free Interest Rate

The risk-free interest rate affects the price of an option in a less clear-
cut way. As interest rates in the economy increase, the expected
return required by investors from the stock tends to increase.

Also, the present value of any future cash flow received by the holder
of the option decreases. The combined impact of these two effects is
to decrease the value of put options and increase the value of call
options

Properties of Stock Options

It is important to emphasize that we are assuming that interest rates

change while all other variables stay the same. In particular, we are
assuming that interest rates change while the stock price remains the
same. In practice, when interest rates rise (fall), stock prices tend to
fall (rise).

The net effect of an interest rate increase and the accompanying stock
price decrease can be to decrease the value of a call option and
increase the value of a put option.

Dividends

Dividends have the effect of reducing the stock price on the ex-
dividend date. This is bad news for the value of call options and good
news for the value of put options.

The value of a call option is therefore negatively related to the size of

any anticipated dividends, and the value of a put option is positively
related to the size of any anticipated dividends.

8.2 ASSUMPTIONS AND NOTATION

In this chapter we will make assumptions similar to those made for
deriving forward and futures prices in Chapter 3. We assume that
there are some market participants, such as large investment banks,
for which

1. There are no transactions costs.

2. All trading profits (net of trading losses) are subject to the same tax
rate.
3. Borrowing and lending are possible at the risk-free interest rate.

We assume that these market participants are prepared to take

advantage of arbitrage opportunities as they arise. As discussed in
Chapters 1 and 3, this means that any available arbitrage
opportunities disappear very quickly.

For the purposes of our analyses, it is therefore reasonable to assume

that there are no arbitrage opportunities.

We will use the following notation:

So: Current stock price
K: Strike price of option
T: Time to expiration of option
ST: Stock price at maturity
r: Continuously compounded risk-free rate of interest for an
investment maturing in time T
C: Value of American call option to buy one share
P: Value of American put option to sell one share
c: Value of European call option to buy one share
p: Value of European put option to sell one share
It should be noted that r is the nominal rate of interest, not the real
rate of interest. We can assume that r > 0; otherwise, a risk-free
investment would provide no advantages over cash. (Indeed, if r < 0,
cash would be preferable to a risk-free investment.)
Put call Parity

Put call parity describes how the values of call and put are related
and can be stated in terms of each other.

A call and the underlying equity can be combined so that they have
the same pay- off as a put.

Similarly a put and a underlying stock can be combined to yield the

same pay-off as a call.

This permits the put and call to be priced in terms of the other
security.

Portfolio P1

1 Buy one European call option.

2 Cash for an amount for Ee -rt , buy bonds maturing at the

expiration date of the call and which at the date will have equal to
the exercise price.

C = current price of call E = exercise price

r = rate of Interest (per year) t = time period
Ee-rt = Amount needed to purchase bonds
Portfolio P2

1 Buy one European put option.

2 Buy share of stock worth So

where,
So = current price of share purchase
S 1 = stock price at expiration date
P = current price of put

Determination of Terminal values of portfolios

Thus, the principle of put call parity states that the prices of call
and put options on an asset are related and given the value of one,
the value of the other can be obtained.

The relationship can be expressed as,

 C + Ee-rt = P + So

Port Cash flow at t S1 > E S1 < = E

folio =0

P1 C S1 - E 0

Ee-rt E E

Total S1 E

P2 P 0 E- S1

So S1 S1

Total S1 E

The Black and Scholes Model

Assumptions,

There are no transaction costs and there are no taxes.

The risk free interest rate is known and constant over the life of the
option.

The market is an efficient one.

The underlying security pays no dividends during the life of the option.

The volatility of the underlying instrument is known and is constant

over the life of the option.

The distribution of the possible share prices at the end of a period of

time is long normal or, in other words, a share’s continuously
compounded rate of return follows a normal distribution.