Fundamentals of financial derivatives nrp

FUNDAMENTALS OF
FINANCIAL DERIVATIVES

FUNDAMENTALS OF FINANCIAL DERIVATIVES
N.R. Parasuraman
SDM-IMD
SDM Institute for Management Development
Mysore

PREFACE
Few topics in Finance have undergone the type of change that Derivatives have over
the last few years. Dealers and Corporate practitioners have discovered several new
uses to Derivatives, resulting in lowering risk and optimizing return. Many students of
Financial Management and practitioners find the basic tenets of these Derivatives
difficult to understand in the beginning. Standard text books give answers to their
queries but since these are embedded in a cluster of applicable theory, exceptions and
mathematical notations, the beginner is often confounded.
The principal objective behind attempting this work is to help the newcomer to the world
of Derivatives get a grip on various facets in a simple manner. The attempt has been to
dwell on the most important characteristics of these instruments, without going into too
much of mathematical analysis. Let me hasten to add that in the process the work
cannot be a substitute for an advanced text book on the topic. What it seeks to
accomplish is to give the reader a quick and easy approach to understand the basic
complexities in day-to-day situations. Once comfortable with the basics, the reader is
advised to get a deeper understanding of various sub-topics with the help of text books
that cover the full mathematical application.
In completing this work, I am indebted to a number of people who encouraged me and
provided me with support. Dr. Jagadeesha, Professor and Chairman, DOS in
Management, KSOU was instrumental in convincing me that such a work would be useful
and in showing me the type of emphasis I should lay on various topics. Prof.
J.M.Subramanya, Director of SDM-IMD was very helpful at various stages of the work and
in generally reassuring me about the quality. Prof.Vinod Madhavan of SDM-IMD, helped
me by patiently going through the drafts and suggesting a number of changes.
Thanks are also due to my father N.P.Ramaswamy who was a source of inspiration and
encouragement and to my wife Prema, who spent long hours checking the drafts and
helping me in data entry. I also thank my colleagues at SDM-IMD for all their support. I
wish to make particular mention of the support of Mr.M.V.Sunil, Mr.M. Rangaswamy,
Ms.Madhura .S. Narayan and Ms.R. Gayithri in completing the final transcript of the book.
N.R.PARASURAMAN

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BRIEF CONTENTS
Module 1 ............................................................................................ 8
FUTURES AND FORWARDS ......................................................... 8
1 Introduction to Derivatives Markets ......................................... 9
2 Forwards and Futures – a quick look...................................... 17
3 Hedging with Futures .............................................................. 28
4 Pricing of Futures and arbitrage conditions ........................... 41
5 Stock Index Futures ................................................................. 54
Module 2 – ....................................................................................... 69
INTRODUCTION TO OPTIONS ................................................... 69
1 Types of Options ....................................................................... 70
2 Pay off of various Options ........................................................ 79
3 Special applications of Options ................................................ 91
4 Options bounds- Calls............................................................. 104
5 Options bounds -Puts ............................................................. 113
Module 3 ........................................................................................ 121
ADVANCED TOPICS ON OPTIONS .......................................... 121
1 Option combinations............................................................... 122
2 Principles of Option Pricing – Put call parity. ...................... 141
3 The Binomial model for pricing of Options ........................... 153
4 The Black-Scholes model........................................................ 163
5 Volatility and Implied Volatility from the Black-Scholes model
172
Module 4 ........................................................................................ 180
OTHER DERIVATIVES AND RISK MANAGMENT................. 180
1 Introduction to Options Greeks and Basic Delta Hedging... 181
2 Interest Rate Derivatives and Eurodollar Derivatives......... 191
3 Swaps ...................................................................................... 203
4 Credit Derivatives................................................................... 216
5 Risk Management with Derivatives ...................................... 225
References ..................................................................................... 238

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DETAILED CONTENTS
Module 1 ............................................................................................ 8
FUTURES AND FORWARDS ......................................................... 8
1 Introduction to Derivatives Markets ......................................... 9
1.1 Objectives ..............................................................................................9
1.2 Introduction...........................................................................................9
1.3 Derivatives – meaning and definition..................................................9
1.4 Types of Derivatives............................................................................10
1.5 Uses of Derivatives .............................................................................12
1.6 Derivatives in India ............................................................................14
1.7 Summary .............................................................................................15
1.8 Key words ............................................................................................15
1.9 Questions for Self- study.....................................................................15
2 Forwards and Futures – a quick look...................................... 17
2.1 Objectives ............................................................................................17
2.2 Introduction to Forwards and Futures ..............................................17
2.3 Basic hedging practices.......................................................................19
2.4 Cost of carry ........................................................................................23
2.5 Differences between Forwards and Futures......................................25
2.6 Summary .............................................................................................26
2.7 Key words ............................................................................................27
3 Hedging with Futures .............................................................. 28
3.1 Objectives ............................................................................................28
3.2 Introduction.........................................................................................28
3.3 Long hedge and Short hedge ..............................................................29
3.4 Margin requirements for Futures ......................................................32
3.5 Basis risk.............................................................................................34
3.6 Cross hedging......................................................................................37
3.7 Summary .............................................................................................39
3.8 Key words ............................................................................................40
4 Pricing of Futures and arbitrage conditions ........................... 41
4.1 Objectives ............................................................................................41
4.2 Introduction.........................................................................................41
4.3 Basic pricing principles.......................................................................42
4.4 Arbitrage opportunities ......................................................................44
4.5 Empirical evidence on cost of carry....................................................48
4.6 Rolling the hedge forward...................................................................49
4.7 Summary .............................................................................................51
4.8 Key words ............................................................................................52

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5 Stock Index Futures ................................................................. 54
5.1 Objectives ............................................................................................54
5.2 Introduction.........................................................................................54
5.3 Construction of stock indices..............................................................55
5.4 Uses and applications of stock Index Futures ...................................56
5.5 Hedging with stock Futures ...............................................................57
5.6 Beta and the Optimal Hedge Ratio ....................................................61
5.7 Increasing and Decreasing Beta.........................................................63
5.8 Other uses of stock Futures................................................................64
5.9 Illustrations.........................................................................................66
5.10 Summary .............................................................................................67
5.11 Key words ............................................................................................68
Module 2 – ....................................................................................... 69
INTRODUCTION TO OPTIONS ................................................... 69
1 Types of Options ....................................................................... 70
1.1 Objectives ............................................................................................70
1.2 Introduction.........................................................................................70
1.3 Types of Options and option terminology ..........................................71
1.4 The question of exercise......................................................................74
1.5 Options markets..................................................................................75
1.6 Differences between Options and Futures.........................................76
1.7 Summary .............................................................................................77
1.8 Key words ............................................................................................78
2 Pay off of various Options ........................................................ 79
2.1 Objectives ............................................................................................79
2.2 Introduction.........................................................................................79
2.3 Payoff of long and short call ...............................................................80
2.4 Payoff of long and short put................................................................83
2.5 Risk and premium...............................................................................86
2.6 Illustrations.........................................................................................87
2.7 Summary .............................................................................................89
2.8 Key words ............................................................................................90
3 Special applications of Options ................................................ 91
3.1 Objectives ............................................................................................91
3.2 Introduction.........................................................................................91
3.3 Covered Call writing...........................................................................91
3.4 Protective Put strategy .......................................................................97
3.5 Mimicking and synthetic portfolios....................................................99
3.6 Summary ...........................................................................................102
3.7 Key words ..........................................................................................103

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3.8 Questions for Self- study...................................................................103
4 Options bounds- Calls............................................................. 104
4.1 Objectives ..........................................................................................104
4.2 Introduction.......................................................................................104
4.3 Upper bounds of call prices...............................................................105
4.4 Lower bounds of call prices...............................................................106
4.5 Upper bounds of call prices-American Options................................108
4.6 Lower bounds of call prices-American Options................................109
4.7 Summary of principles of American Options pricing ......................110
4.8 Summary ...........................................................................................111
4.9 Key words ..........................................................................................111
5 Options bounds -Puts ............................................................. 113
5.1 Objectives ..........................................................................................113
5.2 Introduction.......................................................................................113
5.3 Upper bounds of put prices...............................................................114
5.4 Lower bounds of put prices...............................................................115
5.5 Upper bounds of put prices-American Options................................116
5.6 Lower bounds of put prices-American Options................................117
5.7 Summary ...........................................................................................119
5.8 Key words ..........................................................................................119
Module 3 ........................................................................................ 121
ADVANCED TOPICS ON OPTIONS .......................................... 121
1 Option combinations............................................................... 122
1.1 Objectives ..........................................................................................122
1.2 Introduction.......................................................................................122
1.3 Straddle .............................................................................................122
1.4 Strangle .............................................................................................125
1.5 Bull spreads.......................................................................................128
1.6 Bear spread .......................................................................................131
1.7 Butterfly spread ................................................................................134
1.8 Box spread .........................................................................................136
1.9 Summary ...........................................................................................139
1.10 Key words ..........................................................................................139
2 Principles of Option Pricing – Put call parity. ...................... 141
2.1 Objectives ..........................................................................................141
2.2 Introduction.......................................................................................141
2.3 Some truisms about Options pricing with small illustrations........142
2.4 Put call parity....................................................................................144
2.5 Exercise of the American Call early.................................................147
2.6 Exercise of the American put early ..................................................150

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2.7 Summary ...........................................................................................151
2.8 Key words ..........................................................................................151
3 The Binomial model for pricing of Options ........................... 153
3.1 Objectives ..........................................................................................153
3.2 Introduction.......................................................................................153
3.3 Binomial one-period model ...............................................................154
3.4 Binomial two-period model...............................................................156
3.5 Extension of the principle to greater number of periods.................160
3.6 Summary ...........................................................................................161
3.7 Key words ..........................................................................................162
4 The Black-Scholes model........................................................ 163
4.1 Objectives ..........................................................................................163
4.2 Introduction.......................................................................................163
4.3 Some preliminary ideas ....................................................................164
4.4 Assumptions under the model..........................................................165
4.5 The formula .......................................................................................166
4.6 Illustration ........................................................................................166
4.7 The model inputs...............................................................................168
4.8 The Black-Scholes calculator............................................................168
4.9 Impact of variables on Options pricing ............................................169
4.10 Summary ...........................................................................................171
4.11 Key words ..........................................................................................172
5 Volatility and Implied Volatility from the Black-Scholes model
172
5.1 Objectives ..........................................................................................172
5.2 Introduction.......................................................................................172
5.3 Importance of Volatility and the concept of Implied volatility .......173
5.4 A discussion.......................................................................................174
5.5 Summary ...........................................................................................178
5.6 Key words ..........................................................................................179
Module 4 ........................................................................................ 180
OTHER DERIVATIVES AND RISK MANAGMENT................. 180
1 Introduction to Options Greeks and Basic Delta Hedging... 181
1.1 Objectives ..........................................................................................181
1.2 Introduction.......................................................................................181
1.3 Delta and uses...................................................................................181
1.4 Delta hedging ....................................................................................183
1.5 Gamma, Theta, Vega and Rho..........................................................187
1.6 Summary ...........................................................................................189

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1.7 Key words ..........................................................................................190
2 Interest Rate Derivatives and Eurodollar Derivatives......... 191
2.1 Objectives ..........................................................................................191
2.2 Introduction.......................................................................................191
2.3 T Bill and T Bond Futures................................................................192
2.4 Hedging with T Bills and T-notes ....................................................193
2.5 Eurodollar Derivatives......................................................................194
2.6 Forward Rate Agreements................................................................195
2.7 Caps ...................................................................................................198
2.8 Floors .................................................................................................199
2.9 Collars................................................................................................200
2.10 Summary ...........................................................................................201
2.11 Key words ..........................................................................................202
3 Swaps ...................................................................................... 203
3.1 Objectives ..........................................................................................203
3.2 Introduction.......................................................................................203
3.3 Plain Vanilla Interest Rate Swaps...................................................204
3.4 Exploiting disequilibrium in interest quotes – the Spread
differential....................................................................................................206
3.5 Currency Swaps ................................................................................209
3.6 Valuing swaps and unwinding .........................................................211
3.7 Collars mimicking swaps..................................................................213
3.8 Summary ...........................................................................................214
3.9 Key words ..........................................................................................215
4 Credit Derivatives................................................................... 216
4.1 Objectives ..........................................................................................216
4.2 Introduction.......................................................................................216
4.3 Common Credit Derivatives .............................................................217
4.4 Credit default swap...........................................................................217
4.5 Total Return Swap............................................................................219
4.6 Collateralized Debt Obligations ( CDOs) .........................................219
4.7 An example of CDO...........................................................................220
4.8 The Indian scenario ..........................................................................221
4.9 Other aspects.....................................................................................223
4.10 Summary ...........................................................................................223
4.11 Key words ..........................................................................................224
5 Risk Management with Derivatives ...................................... 225
5.1 Objectives ..........................................................................................225
5.2 Introduction.......................................................................................225
5.3 Hedging using Greeks.......................................................................226

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5.4 Delta-Gamma hedging......................................................................231
5.5 Discussion on Hedging Policy...........................................................233
5.6 Summary ...........................................................................................236
5.7 Key words ..........................................................................................237
References ..................................................................................... 238

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Module 1
FUTURES AND FORWARDS

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1 Introduction to Derivatives Markets
1.1 Objectives
The objectives of this unit are to
• Introduce Derivative instruments
• Briefly look at the common uses and applications of Derivatives
• Briefly look at the trading of Derivatives in the Indian market
1.2 Introduction
As financial instruments, Derivatives have become very popular over the last
two decades. While the practice of using Derivatives instruments has been
there for even centuries, the formal application of these instruments in
everyday financial management came about only recently. With the
development of appropriate markets for these securities, a lot of academic
research has also been carried out in their various facets. Understanding
their applications, uses and misuses constitutes an important part of the
study of Financial Management
1.3 Derivatives – meaning and definition
Derivatives are instruments in respect of which the trading is carried out as
a right on an underlying asset. In normal trading, an asset is acquired or
sold. When we deal with Derivatives, the asset itself is not traded, but a
right to buy or sell the asset, is traded. Thus a derivative instrument does
not directly result in a trade but gives a right to a person which may
ultimately result in trade. A buyer of a derivative gets a right over the asset
which after or during a particular period of time might result in her buying or
selling the asset.

10
A derivative instrument is based first on an underlying asset. The asset may
be a commodity, a stock or a foreign currency. A right is bought either to buy
or sell the underlying the asset after or during a specified time. The price at
which the transaction is to be carried out is also spelt out in the beginning
itself.
1.4 Types of Derivatives
There are many types of Derivatives in the market and everyday parlance.
Any transaction that results in a right without actually transacting the asset
becomes a derivative instrument. A brief picture of the common Derivatives
is given below:
Futures and Forwards
Contracts under this category relate to transactions entered into on a given
date to become effective after a specified time frame and subject to payment
at rates determined currently but becoming due after that specified time.
Forwards and Futures are entered into by those who wish to be assured of a
price after a specified time in line with the current price. With prices
fluctuating all the time, it is impossible to predict the price levels after a few
months. A Forward or a Futures contract will ensure that the prices are
frozen upon at the time of entering into the contract and the time frame for
the contract is also firmed up. There are several other aspects to Forwards
and Futures which will be discussed in detail in a later section
Options
Option contracts are a step ahead of the Forwards/Futures contract in that
they result in a right being created without a corresponding obligation. The
buyer of an option contract gets the right without the obligation to either buy
or sell the underlying asset. There is a time frame and a price fixed for the
contract. For the privilege of going ahead with the contract as per her desire,
the option buyer has to pay the seller a premium up front. If, ultimately

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prices do not allow the Options to be exercised, then the premium is the only
loss incurred by the buyer of the Option contract. All the detailed aspects of
an Options contract are covered in a later Module.
Swaps
In a swap transaction the two parties thereto exchange their obligations on
predetermined terms. In its simplest version, two companies having different
obligations of interest payments (with one Company obliged to pay a fixed
rate of interest to its bankers and the other Company having to pay a floating
rate of interest), enter into a contract whereby they exchange their
obligations. This exchange of their obligations results in one Company
getting the fixed interest from the other Company to be used for satisfying its
obligation. In exchange this Company passes on a floating rate of interest to
the counterparty Company to satisfy the latter’s floating interest obligation.
The principal amount to be reckoned for the purpose of calculating the two
interests (called the Notional principal), and the benchmark interest rate to
be used for the purpose of determining the floating rate are decided at the
time of entering into the contract. Swaps are dealt with in detail in a later
module
Commodity Derivatives
The most common intuitive use of Derivatives will be in the commodity
segment, where operators fear price rise/fall based on natural weather
conditions. To safeguard their interests these operators can enter into a
buy/sell contract for the required amount of commodity Derivatives.
Typically, like all Derivatives, this does not directly result in the underlying
commodity being traded. Instead, a right or an obligation is established with
respect to the underlying commodity. This type of derivative is also used by
manufacturers and exporters who want to ensure a specified amount of
commodities to meet their business obligations. The principles involved in
these Derivatives are the same as those governing general Options.

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Interest rate Derivatives
Here the parties to a transaction fear a rise or fall in interest rates in the
future and enter into a derivative transaction by which one counterparty
compensates the other when interest rises beyond the agreed rate.
Sometimes these transactions are entered into for getting compensation for
interest rate declines. The notional principal, the benchmark interest rate
and the time of reckoning all are decided at the time of entering into the
contract. Interest Rate Derivatives are covered in a later Module.
Credit Derivatives
Bankers and lenders use Credit Derivatives to safeguard themselves against
credit defaults. There are many varieties of these Derivatives involving
sometimes the creation of a third body called a Special Purpose Vehicle.
Credit Derivatives constitute an area of great development in recent years
and many new sophisticated instruments are getting developed by the day.
An introduction to some common Credit Derivatives is given in a later
module.
1.5 Uses of Derivatives
Derivatives are used by companies and individuals wanting to cover their
risks. This is facilitated by a counter party who has the motivation to make
profits out of the premium, or is holding a mirror-image opposite position.
Used this way, Derivatives offer an important tool of risk management,
without which companies and individuals would have been exposed to the
vagaries of price fluctuations. However, the use of Derivatives requires skill
in respect of timing, a strategy regarding the extent of coverage and the need
to be consistent in one’s approach. One of the greatest objections to
Derivatives has been that they encourage speculation. In other words, deals
on Derivative contracts can be entered into even by those who do not have a
risky asset position. It can be entered into by speculators betting on a given
price movement or absence of fluctuations. While this in itself may not seem

13
to be objectionable, if this practice is carried to disproportionate limits, they
are exposed to huge losses and sometimes bankruptcy. Many companies
have been ruined by over-zealous officials recklessly entering into positions
on Derivatives and taking on enormous risk in the hope of gains on favorable
price movements. Since Derivatives instruments are complex and involve
sophistication in pricing and strategy, it is beyond the non-specialist manager
to comprehend the exact risk that the Company is exposed to because of a
series of derivative transactions. In the process the Company concerned is
exposed to great risk.
While recognizing the possible misuse of these instruments, they are
nevertheless proved to be invaluable in safeguarding Company’s income and
profits. Many companies set up their own strategies regarding the extent of
risk they needed to be covered and correspondingly they enter into
appropriate Derivative transactions for the purpose. This process results in
prevention of unnecessary risk and optimization of profits.
To give an example, an exporter to the United States expects to get $50000 in
3 months from now. She will be happy to have this converted at around
Rs.45 per $. However there is great uncertainty in the foreign exchange
market as to the nature of the possible movement after 3 months.
Fortunately for the exporter a bank is willing to enter into a Forward
contract with her for paying Rs.45 per $ after 3 months on her surrendering
$50000 then. If the exporter enters into this Forward contract, she makes
sure that she will be able to get Rs.2,250,000 ($50000 multiplied by Rs.45)
after 3 months. However, if the rates change to her favor say to Rs.48 per $
she will not be able to take advantage of the favorable movement since she is
already committed to the Forward agreement. A detailed discussion on this
aspect and other related topics are covered in a later unit. There are other

14
derivative instruments like Options which enable the trader to have the best
of both worlds, at a price.
1.6 Derivatives in India
In India Derivatives have been actively traded over the last decade. The use
of Derivatives in the commodity segment has been existent over several
years, but these were mostly confined to Futures and Forwards transactions.
Options contracts in the stock markets have become very popular in recent
years and have given a new facet to share portfolio management. In the
foreign exchange market, over-the-counter Forwards have been prevalent for
long, but formalized Futures and Options are yet to take shape. Trading of
Interest Rate Derivatives has been formally introduced in the stock
exchanges but these are yet to capture the imagination of the common
investor. Swap transactions have been reported more on a customized one-to-
one basis rather than being taken as formal standardized instruments.
Credit Derivatives have made an entry but are yet to become very popular.
Stock markets find Derivative instruments very useful and portfolio
managers find a number of uses from these for protecting and enhancing
their stock holdings. The rising volumes of Index-based and individual
securities are an indication of their growing popularity. The fact remains,
however, that most of the deals are speculative in nature and are not
necessarily for risk management. But this by itself need not be taken as an
adverse factor, since in most world markets initial uses of derivative
instruments have been basically speculative. Besides, the existence of a large
number of speculators enables the genuine risk manager to put through his
deals comfortably and volumes will not suffer.
The regulation of the Derivatives segments has been handled by the
Securities & Exchange Board of India and the stock exchanges. Strict

15
margins and deposits are taken from the trading members to avoid defaults
and payment problems.
1.7 Summary
The study of Derivatives involves an approach different from the customary.
In conventional analysis, trading involves buying and selling an asset. In the
Derivatives segment, trading involves not the selling and buying of the asset
itself, but a right on the asset. This right does not carry with it any
obligation and comes at a price called the premium. There are many types of
derivative instruments, the most notable among them being Forwards and
Futures, Options and Swaps. In addition, Interest Rate Derivatives and
Credit Derivatives have become very popular in the US and other countries
in recent years. Derivatives are useful for managing the risk of an
organization. Usually companies develop a strategy for active risk
management using Derivatives. The stock-based Derivatives have become
very popular in India and result in great trading volumes. In India, Forwards
and Futures are in great use in the commodity segment. It is also common to
have Forward contracts in foreign exchange transactions.
1.8 Key words
• Derivatives
• Forwards
• Futures
• Swaps
• Interest Rate Derivatives
• Credit Derivatives
1.9 Questions for Self- study
1) How are Derivative instruments different from regular instruments of
trade?

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2) What are the common type of Derivatives?
3) What categories of investors/traders use Derivatives?
4) Are Derivatives well regulated in India?

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2 Forwards and Futures – a quick look
2.1 Objectives
The objectives of this unit are
• To give a general framework of Forwards and Futures contracts
• To understand the benefits of these contracts
• To understand the principle of cost of carry and its uses in practice
2.2 Introduction to Forwards and Futures
Forwards and Futures constitute the most basic of derivative instruments.
They are widely used and are quite intuitive in nature. The pricing and
payoff follow a pattern that can be easily understood.
A Forward or Futures contract enables one to enter into an agreement to buy
or sell a specified quantity of the underlying asset after a specified time at a
specified price. In other words, a Forward or Futures contract locks up the
rate of the underlying asset and regardless of the actual rate at the time of
expiry, the deal has to be executed at the rate agreed upon. This
arrangement enables the parties to the contract to lock up their receipts or
payments at convenient levels. However, the disadvantage is that if rates
move in the opposite direction to what is feared, it might turn out to be a
mistake to have entered into the contract. For instance a commodity trader
wishes to sell 500 kgs. of a commodity at Rs.50 per kg. He expects the price
to be steady at this level even after 3 months when the crop will be ready, but
fears that some adverse movements in other sectors might result in a fall in
the price. To safeguard himself he enters into a Forward contract for the
quantity at around Rs.50 per kg. The contract in effect means that he is
obliged to surrender 500 kgs of the commodity after 3 months in exchange of
getting Rs.25000 (500 multiplied by Rs.50). Now, if as feared, the prices fall

18
to a level less than Rs.50, the farmer will still get Rs.25000 calculated at
Rs.50 per kg, because that is the agreed rate. However, if the price rises
above Rs.50 to say Rs.60 per kg. the farmer loses on the opportunity profits,
since he is obliged to fulfill his Forward contract at Rs.50 per kg. and will not
be able to participate in the higher profits. Thus, in a Forward/ Futures
contracts, one of the parties to the contract is likely to lose out on the deal in
the final analysis.
To continue an example from the previous unit, an exporter to US expects to
get $.50000 in 3 months time. If the invoicing were made in the Indian
currency, the exporter would not have had any difficulty in estimating her
potential receipt after 3 months. Since the invoicing is in US$, her actual
receipt in terms of Rupees would depend upon the exchange rate at that point
of time. A Forward contract for selling US$ after 3 months at a mutually
acceptable rate would ensure that the exporter gets this rate regardless of the
ultimate actual exchange rate. A Forward contract is thus an agreement to
buy or sell an underlying asset (in this case the US$) for a predestined
quantity, at a predetermined price after a predetermined period. In this case
the buyer of the US$ forward agrees to pay the Indian rupees at the pre-
determined rate. It goes without saying that one of the parties to the
contract will stand to gain more in the final analysis, but what it ensures at
the time of entering into the contract is that the risk element is eliminated.
To take the opposite situation, an importer of goods from the US has to pay
$75000 after 3 months. The invoicing is in $ and so the importer is exposed
to exchange rate risk. The importer is apprehensive that the amount to be
paid may become more in terms of the Indian rupees because of adverse
movements in the foreign exchange market. To ensure that the amount is
frozen, the importer can enter into a Forward agreement to buy $ at a pre-
determined price. At the expiry of the period, the importer pays the agreed

19
amount in Rupees for getting $75000. The amount to be paid in Indian
Rupees does not vary with the then prevailing exchange rate. Even if the
exchange rate movement is adverse, the importer is not affected since the
amount to be paid in exchange has been firmed up in advance. However, like
the contract for selling foreign currency seen earlier, here again one of the
parties would lose opportunity gains in the final analysis depending upon the
exchange rates at the time of expiry, but it ensures that the risk is eliminated
at the time of entering into the contract.
Futures contracts work in exactly the same way as the Forwards, except that
they are better regulated. The quantity of the underlying asset that is to be
contracted is in specified lots and the time of expiry is also pre-fixed. For
instance if the importer wants to sell Rs.50000 worth Forward for a period of
3 months, she has to sell this in an exchange contracts corresponding as
nearly as possible to the amount and the horizon needed. Thus if a standard
Futures contract is for say Rs.10000, 5 such contracts have to be sold and if
the contracts expire in 2 months or 6 months, the former is chosen being the
nearest to the horizon needed. There are other structural differences in
Futures as well like the margin requirement, mark-to-market rules and
settlement. These are dealt with in detail later in the Module.
2.3 Basic hedging practices
The hedging practice can be formalized through a couple of examples:
A commodity farmer expects 10000 kgs of a commodity to be ready after
harvest in 3 months. The price of the commodity as of now is Rs.2.80 per kg.
The farmer would be happy if the price he obtains is around this level.
However, market economics suggest that the price may take a dip and he
may end up getting only say around Rs.2 per kg.

20
A trader in the same area is prepared to get into an agreement with the
farmer to buy the harvest from him at a rate of Rs.2.90 per kg., provided the
farmer commits to the quantity and price today. In other words, the farmer
would be obliged to sell 10000 kgs of the commodity after 3 months at a price
of Rs.2.90 per kg. and the trader would be obliged to buy the quantity at the
price. This will be regardless of what the final price of the commodity is to be
at the end of 3 months.
If he accepts this offer today, the farmer is able to make sure that he gets
Rs.2.90 per kg and he can stop worrying about any possible fall in prices in
the interim. However, he has to continue to worry about obtaining the
harvest of 10000 kgs In case the harvest is not as successful as anticipated
and he ends up having only less than 10000 kgs. ready, he will be forced to
buy from the market the difference in quantity and meet his obligation to the
trader.
As far as the trader is concerned he has ensured that he will get a supply of
10000 kgs. of the commodity at a pre-determined price, and he does not now
have to worry either about changes in prices in the interim, nor about the
availability of the quantity.
This is an example of a short hedge as far as the farmer is concerned and a
long hedge as far as the trader is concerned. In a short hedge the individual
is concerned about fall in prices and sells the commodity in advance at a pre-
determined price. In a long hedge the individual is concerned about the rise
in prices and ensures the price by buying the commodity at the pre-
determined price. In either case the quantity is frozen.
The short hedge has enabled the farmer to reduce his anxiety about the
prices. Now regardless of the actual movement of prices in the market the

21
farmer will get Rs.2.90 per kg. If the price at the end turns out to be Rs.2.40
say, he can congratulate himself for having entered into the Forward
agreement, enabling him to force the counterparty to buy from him at 2.90
per kg. On the other hand, if the price rises beyond 2.90 to say Rs.3.20 per
kg. the farmer might feel let down in that he would have been better off
without the Forward contract and would then have been able to sell at
Rs.3.20 per kg. The Forward would force him to sell at Rs.2.90, even though
the actual rate at that time is Rs.3.20. This is the price he pays for ensuring
a minimum amount. He will not be able to participate in upward movement
of prices
The long hedge has enabled the trader to reduce his anxiety about prices.
Now regardless of the actual movement of prices in the market the trader
will have to pay only Rs.2.90 per kg. If the price at the end turns out to be
Rs.3.20 say, he can congratulate himself for having entered into the Forward
agreement, enabling him to force the counterparty to sell to him at 2.90 per
kg. On the other hand, if the price falls to say Rs.2.40 per kg. the trader
might feel that he would have been better off without the Forward contract
and would then have been able to buy from the market at Rs.2.40 per kg. The
Forward would force him to buy at 2.90, even though the actual price at that
time is Rs.2.40. This is the price he pays for ensuring a Forward amount. He
will not be able to take the benefit from downward movement of prices.
In the following example the possible payoff from a short hedge can be seen.
The situation involves selling Forward at Rs.101.51 for 3-month duration. If
the price ends up at Rs.95, he will gain Rs.6.51 on the Futures, but will be
able to sell in the market at Rs.95, making totally Rs.101.51.

22
If the price is Rs.103, he will lose Rs.1.49 on the Futures, but can sell at
Rs.103 in the market, thereby making a total of Rs.101.51.
Table I.2.1 Short-hedge payoff (Amount in Rs.)
Price after
3 months
Gain in
Forwards
Total
proceeds
95 6.51 101.51
96 5.51 101.51
97 4.51 101.51
98 3.51 101.51
99 2.51 101.51
100 1.51 101.51
101 0.51 101.51
102 -0.49 101.51
103 -1.49 101.51
104 -2.49 101.51
105 -3.49 101.51
106 -4.49 101.51
Taking the same situation the position in respect of a long hedge is shown
below. Here again the long hedge has been made at Rs.101.51 for 3-month
duration:
If the price ends up at Rs.97, the long hedge would have to suffer a loss of
Rs.4.51 on the Futures contract, but can get the product at Rs.97 from the
market; thereby the total cost will be Rs.101.51. If the price ends up at
Rs.103, he gains Rs.1.49 on the Futures contract, but has to buy from the
market at Rs.103, resulting in a net position of Rs.101.51.

23
Table I.2.2 Long hedge pay off (Amount in Rs.)
Price after
3 months
Gain in
Forwards
Total
proceeds
97 -4.51 101.51
98 -3.51 101.51
99 -2.51 101.51
100 -1.51 101.51
101 -0.51 101.51
102 0.49 101.51
103 1.49 101.51
104 2.49 101.51
105 3.49 101.51
106 4.49 101.51
2.4 Cost of carry
The principles governing fixation of Forward prices are based on interest
rates computation. In Derivatives pricing the universal method is to use
continuous compounding. In other words, the final value based on interest
for a period of 6 months on an investment of Rs.100 at 10% interest is
calculated as 100 multiplied by ℮ raised to the power of the interest rate
multiplied by the time. The ℮ here is the natural logarithm which has a
value of 2.71828. In the above example, the value will be 100 multiplied by ℮
raised to (.10 multiplied by .5). We get 105.13. It may be noted that the
calculations using the ℮ operator can be easily accomplished by a scientific
calculator, or by using the excel function “EXP”. The Excel formula in the
above case will be (=100* EXP (.10*0.5)).
Forward pricing is based on interest rate computation and the principles of
arbitrage. Financial theory has the support of the principle of arbitrage for
various postulates. If the prices as per the postulate do not hold good, it
would be possible for alert operators to buy one type of instrument and sell

24
another type of instrument at a risk-free profit. Arbitrage ensures that prices
reach their equilibrium levels in an ideal market.
The theoretical correct price for a Forward contract which has 3 months to go
on a spot price of Rs.40 and an interest rate of 5% can be calculated using the
formula given below:
Forward Price = Spot Price * ℮rt, where
* signifies the multiplication symbol
℮ is the natural logarithm
r is the risk free rate of interest
t is the period of the contract reckoned as a fraction of 1 year
So in the example the Forward price will be:
40*℮. 05*.25 = 40.50
The ideal price for the Forward has to be Rs.40.50.
If the price is greater than 40.50 say Rs.40.75, then an alert operator will sell
the Forward at 40.75 and buy the spot asset at 40. The spot asset will be
bought using borrowed funds which will necessitate an interest payment of
Rs.0.50 for the 3 month period. (Interest is calculated on a principal of Rs.40
for a period of 3 months at an interest rate of 5%, using the continuous
compounding method). On the expiry of the period, the operator will sell the
asset (which he had bought originally in the spot market using borrowed
funds) at Rs.40.75 based on the Forward contract. He will repay Rs.40.50 for
the loan (Rs.40 principal and Rs.0.50 interest) and pocket the difference of
Rs.0.25 as risk-free profit. As more and more operators do this the price will
come back to its equilibrium level of Rs.40.50 for the Forward contract.
If the price is less than Rs.40.50 say Rs.40.25, then an alert operator will sell
the asset in the spot market at Rs.40 and buy the Forward at Rs.40. 25. The
amount got out of selling the spot asset (Rs.40) will be invested in risk-free

25
securities earning an interest of 5% for 3 months. i.e. Rs.0.50 (The interest is
calculated on a principal of Rs.40 for a period of 3 months at an interest rate
of 5%, using the continuous compounding method). On the expiry of the
period, the operator will buy the asset based on his Forward contract by
paying Rs.40.25. He will receive Rs.40.50 from his investment (Rs.40 he got
out of sale of the spot asset plus the interest of Rs.0.50 for the 3-month
period), thereby resulting in a net gain of Rs.0.25. As more and more
operators seize on this risk-free opportunity, the prices will reach back the
equilibrium level of Rs.40.50 for the Forward contract.
More aspects of the cost of carry principle and the risk-free arbitrage are
covered in a later unit.
2.5 Differences between Forwards and Futures
The following are the broad differences between a Forward and a Futures
contract:
1. A Futures contract is standardized in terms of the quantity per
contract and the time of expiry. A Forward contract, on the other
hand, is customized based on the needs of the two parties to the
contract.
2. There will be no default risk in a Futures contract since it is exchange-
oriented, whereas the possibility of default exists in Forward contract.
In a Futures contract the buyer and the seller do not directly interact
and the exchange is the effective counterparty for each of the dealers
3. A Futures contract will entail a margin for avoidance of default and
this amount has to be remitted from time to time to the exchange
based on extant regulations. In a Forward contract, there is no
standardized margin but this can also be incorporated as a condition to
the contract by the parties concerned.

26
4. A Futures contract is monitored on a regular basis by the regulating
authority and hence entails a mark-to-market margin. Thus, if a
trader has bought a Forward contract of 3 months for a commodity at
Rs.50 and if the price is Rs.40 after a week of entering into the
contract, the exchange may require him to pay up the difference of
Rs.10 on each contract. This is because the adverse price movement
might result in ultimate default and the mark-to-market enables the
contract to be scaled up or down to the current market levels. Mark-to-
market margins are generally not insisted upon in a Forward contract.
5. A Futures contract is cash settled. This means that that the final price
of the underlying is compared to the rate agreed upon and the
difference either paid or received from the parties concerned. Actual
delivery of the underlying is not done. A Forward contract, on the
other hand, can be of cash settled or based on physical delivery.
2.6 Summary
Forwards and Futures constitute the simplest of derivative instruments.
They are in wide use for risk management. A Forward contract facilitates
the buying or selling of an underlying asset at a pre-determined price after a
specified period. A Futures is similar in operation to Forwards except for
structural variations on account of contractual specifications, margins and
mark-to market. A long Forward contract obliges the buying of the
underlying asset and a short Forward contract obliges the selling of the
underlying asset.
Forwards and Futures are used widely in the hedging of price risks. While
the practice of hedging enables the avoidance of price risk, traders find that
occasionally they lose out on opportunity gains because of price movements
which turn out to be more favorable than expected.

27
The pricing of Forwards and Futures follow the cost of carry principle.
According to this, the price at the time of its inception has a definite
relationship with the spot price and is generally represented by the interest
for the period involved. If the price does not conform to this pattern it is
possible to enter to arbitrage and make risk-free profits. That fact that a
number of operators will embark upon this arbitrage will result in the prices
once again coming to the equilibrium levels.
2.7 Key words
• Forwards
• Futures
• Cost of carry
• Hedge
• Arbitrage
1) A rice farmer is happy to note that the price per kg. for the type of rice
that his farm produces is around Rs.15 now. However, he will get his
crop only after 2 months. He fears that the prices might fall in the
meantime. How can the farmer use Forwards to reduce his risk?
2) Will Forwards always result in profits? Under what circumstances
will a trader feel that he would have been better off without the
Forward?
3) What is margin? Is this applicable only to Futures contracts?
4) If the Forward rate is lower than the rate dictated by the cost of carry
principle, how would an arbitrage be possible?

28
3 Hedging with Futures
3.1 Objectives
The objectives of this unit are:
• To introduce a framework for hedging
• To look at the risks associated with hedging
• To look at the overall merits and demerits of dealing with Futures
3.2 Introduction
The basic hedging model was introduced in the previous unit. Hedging is an
important requirement for all mangers. Based on price fluctuations and
market behavior, managers will tend to hedge their exposures short or long.
We recapitulate the principal factors in respect of hedges below:
1. Hedges using Forwards and Futures can be long or short. A long
hedge is one that goes long the Forwards or Futures (long means buy).
This is entered into by those fearing price rises. By buying the
Forward, they seek to freeze the price’s upper limit. A short hedge is
used by those fearing price falls. A short hedge signifies selling of the
Forward or Futures. By selling the Forward or Futures they seek to
freeze the prices at a level.
2. Hedging is a double-edged sword. While the hedge does offer
protection, it would also mean that if the prices do not move in the
direction feared, one might lose a chance for bonanza profits. Thus in
a long hedge if the price ends up well below the level expected, the
hedge would force the operator to the Forward/Futures price, resulting
in an opportunity loss. In the same way, if the price is greater than
anticipated then the short hedger suffers an opportunity loss.
However, in both the cases, the operators would have frozen upon a

29
level of prices which is acceptable to them. It is only the opportunity of
higher gains that they lose in the process.
3. Hedging is a part of the strategic process of companies. They generally
have a policy as to how much of their exposure to hedge and the price
bands at which these should be carried out. Companies tend to leave a
portion of their exposure open. The farmer expects to produce 10000
kgs in 3 months might decide to hedge only 6000 kgs and leave the
remaining 4000 kgs unprotected.
4. Hedging with Forwards and Futures result in almost identical
coverage. However, a Futures contract might entail the payment of
periodic margins and mark-to-market margins, resulting in some
differences in cash flow analysis.
5. The cost of carry principle introduced in the earlier unit needs to be
modified in respect of Futures contract factor because of margins and
deposit money.
6. There will be little default in a Futures contract whereas a Forward,
being a one to one contract might result in defaults.
3.3 Long hedge and Short hedge
Let us take the example of an exporter who has sold a commodity to a foreign
country to be delivered three months hence. In order to make reasonable
profits from the deal the exporter has to acquire this commodity from the
domestic market at Rs.245 per unit. The current price is Rs 245 per unit and
the Futures price currently going in the market is Rs.256 per unit. Readers
would have noticed that the Futures price does not conform to the cost of
carry principle introduced in the previous unit. The possible reasons for this
are discussed in the subsequent Chapters. The exporter will buy Futures at
Rs.256 per unit to the extent required.

30
Let us assume that the actual price of the commodity has gone up to Rs.290
per unit after three months. This was what the exporter had feared.
However, since he had the foresight to go in for a Futures contract his
interests are protected. Now he gets delivery of the commodity at Rs.256 per
unit which is the agreed price under the Futures contract. The actual price of
Rs.290 does not affect him.
One important difference between Forwards and Futures in respect of final
settlement may be noticed in this context. If the exporter had entered into a
Forward contract he would have got actual delivery by paying Rs.256 per unit
to the counter party. But if he had entered into a Futures contract for his
hedge he will instead be paid the difference between the prevailing
commodity price of Rs.290 per unit and his Futures contract price of Rs.256
per unit. This means that he collects the difference of Rs.24 from the counter
party and buys the commodity in the spot market at Rs.290 per . He was in
any case prepared to pay Rs.256 per unit which is the difference between the
two.
If, however, the price in the spot market ends up at Rs.240 per unit, the
exporter loses Rs.16 on the Futures contract (Rs.256 per unit which is his
contracted price minus the actual spot price of Rs.240). In such a case the
exporter is not able to take advantage of the fall in the prices and still ends
up paying Rs.256 per unit. As we have seen this is the sacrifice he makes for
seeking a hedge using Forwards or Futures.
To take an example a Company fears that the price of its output will come
down. It expects 5000 units of output by the end of next 3 months. The
current output price is Rs.50 per unit. The Futures for the output asset are
currently going at Rs.54 per unit for a 3 month contract. Again the reader

31
would have noticed that the Futures price does not strictly conform to the
cost of carry principle. This will be looked at in detail in the next unit.
Similarly, in a short hedge the operator can use Futures by selling the
Futures today. As we have seen the Futures will conform more or less to the
cost of carry principle. After the specified period of contract he will be able to
get the difference between the price at which he sold the Futures, from the
commodity exchange and the actual price in the spot market. If, however, the
spot price in the end happens to be higher than the price at which he sold the
Futures, he will have to pay the difference to the exchange.
Fearing a decline in prices our Company goes for a short hedge by selling the
3 months Futures at Rs.54 per unit for the required quantity. After 3 months
if the final price is say Rs.40 the Company is protected by the Futures
contract. The cash settlement from the Futures contract will give the
Company Rs.14 per unit (Rs.54 contracted in the Futures minus the end spot
price of Rs.40) .The Company will sell the commodity in the market atRs.40
and along with the Rs.14 got from the Futures exchange will be able to pocket
Rs.54 per unit. Here again the procedural difference between a Forward
contract and a Futures contract may be noticed. In the Forward contract the
Company would have been able to sell to the counter party the quantity
contracted at Rs.54 directly. In the Futures contract because it is cash settled
only the difference between the Futures contracted price and the actual price
in the end is handed over to the Company. The actual buying of the
commodity has to be done by the Company through the regular market.
Continuing the example if it so happens that the final price were Rs.70 per
unit the Company will not be able to take advantage of the increase in prices.
Although it will still be able to sell the output at Rs.70 per unit to the
market, it will suffer a loss of Rs.16 (final price of Rs.70 minus contracted

32
price of Rs.54) in the Futures market resulting in a net inflow of only Rs.54
per unit. This is the sacrifice for freezing a price using Forwards or Futures.
3.4 Margin requirements for Futures
We have seen that Futures are different from Forwards in many aspects.
One such aspect is that Futures are well regulated by stock exchanges.
Every person entering into a Futures contract is actually doing so with the
stock exchange as the counter party
In order to ensure that the parties entering into a Futures contract meet the
stringent requirements of payment, stock exchanges insist on margins.
Margins are amounts required to be paid by dealers in respect of their
Futures positions. There will be initial margins, some special margins
imposed from time to time and mark to market margins.
Mark to market margins result in the dealer having to pay margins specially
for meeting the adverse movement in underlying positions as a result of
changes in spot prices. Some exchanges insist on maintenance margins
which mean that the trader is required to pay additional margin when the
mark to market position falls below a trigger point.
Margin requirements vary from exchange to exchange and sometimes from
time to time. The following illustration shows a typical position and its
impact on the trader based on certain assumptions regarding maintenance
margin and mark to market margin. In the example the initial margin is
Rs.1000 and the maintenance is Rs.750. If the initial margin as adjusted by
adverse mark to market, falls below the Rs.750 level additional margin has to
be remitted by the trader. The table below shows the position:1
1
Example adapted from Derivatives – Valuation and Risk Management, by Dubofski and Miller, Oxford
Press

33
Table I.3.1 Example of margin liability –Amount in Rs.
Date Sett.Price
Initial
cash
Mark to
Market
Equity
Maintenance
margin
Final
cash
Final
equity
6-Nov 286.4 1000 -140 860 0 1000 860
7-Nov 288.8 1000 -240 620 380 1380 1000
10-Nov 289 1380 -20 980 1380 980
11-Nov 288.6 1380 40 1020 1380 1020
12-Nov 290.7 1380 -210 810 1380 810
13-Nov 292.8 1380 -210 600 400 1780 1000
14-Nov 292.8 1780 0 1000 1780 1000
17-Nov 292.7 1780 10 1010 1780 1010
18-Nov 295.8 1780 -310 700 300 2080 1000
19-Nov 296.1 2080 -30 970 2080 970
20-Nov 297.1 2080 -100 870 2080 870
21-Nov 296.4 2080 70 940
On 5th Nov when the contract was entered into the price was Rs.285. The
trader had taken a short position in Futures and has paid an initial margin of
Rs.1000. On 6th Nov the price fell to Rs.286.40 resulting in his effective
margin amount falling to Rs.860, as shown under column “Equity”. The next
day the spot price was Rs.288.80 resulting in a further depletion of Rs.240.
As shown under the Equity column the margin now is only Rs.620. Since the
maintenance margin is Rs.750 and the Equity has fallen below that level the
trader is required to replenish the margin to the original level of Rs.1000.
This entails a payment of Rs.380 from his side as shown under the column
“Maintenance margin”. This procedure goes on till the end of the contract.
The last two columns show the effective Equity position and the margin
position from time to time.

34
3.5 Basis risk
Basis risk refers to the changing difference between the spot asset prices and
the Futures prices. Basis risk has greater application in Futures contracts
because here the contracts do not expire at the time when the trader
requires. In a Forward contract it is possible to customize the contract to the
appropriate time frame.
Basis refers to the difference between the Spot price of the asset and the
Futures price of the asset. Let us work with the following symbols
S1 = Spot at time t1
S2 = Spot at time t2
F1 = Futures at time t1
F2 = Futures at time t2
B1 = Basis at time t1
B2 = Basis at time t2
Suppose the spot price of an asset at inception was Rs.2.50 and the Futures
price at that time was Rs.2.20. After 3 months, the spot price becomes 2.00
and the Futures price at that time is Rs.1.90. Here the basis at the beginning
(B1) is - 0.30 and the basis at the end (B2) is - 0.10
When the spot increases by more than the Futures, Basis strengthens
When the Futures increases by more than the Spot the Basis weakens
In a hedge the final price =
S2 + F1 – F2
This is equal to F1 + B2
F1 is known at inception
If we can correctly estimate B2, the hedge can be perfect

35
The above situation can be elaborated a little more. At the time when the
contract is entered into there is a spot price for the asset and a corresponding
Futures price. Normally, the Futures will conform to the cost of carry
principle and can be determined fairly accurately. But the actual Futures
price may not sometimes conform to the cost of carry rule for various reasons
discussed in the next unit. The difference between the spot and the Futures
price at the start is the basis at the beginning. As time goes on, the spot
price changes and the Futures price also changes. At the expiry time, which
is the next crucial juncture, the Spot and the Futures are at another level of
difference between each other. Occasionally, the difference levels will remain
identical. Sometimes the differences go up or come down. When the
difference goes up (Spot – Futures goes up), the basis is said to have
strengthened. When the difference comes down (Spot-Futures comes down)
the basis is said to have weakened.
A short hedger expects prices to come down and hence has sold the Futures.
At expiry, he settles his position with the Futures price at the end. If the
Futures price at the end is in line with the Spot at the end in exactly the
same level as the difference at the beginning, the basis is the same. In such a
case, the hedge will be perfect. He will realize exactly what he sought out to
do. For instance, the spot at the beginning was Rs.100, and the Futures were
102. He had sold Futures at Rs.102. At the end the spot is Rs.85 and the
Futures is Rs.87. Now he realizes Rs.15 (the difference between 102 and 87).
Suppose the basis had strengthened, the Futures price would have been less
than Rs.87, say Rs.86. the basis at the beginning was (-2) and it has now
become (-1). The (-2) difference has become (-1). The hedger would make
Rs.16 (difference between 102 and 86). Here the short hedger has benefited
from the basis strengthening.

36
On the other hand, if the basis had weakened and if the final Futures price is
88, the hedger would have made only Rs.14 (102-88). He would have lost out
of the basis weakening. The basis originally was (-2) and it has become (-3).
The converse is true of a long hedge. Here the hedger had bought the
Futures. If the spot initially was Rs.200 and the corresponding Futures was
Rs.203 at inception. At expiry the spot is Rs.240 and the Futures Rs.243.
Here the basis remains the same at the start and at the end. So there is no
gain or loss from basis risk. The hedger would get Rs.40 (243-203) from the
long hedge. If the basis had strengthened the Futures would be Rs.242 for
instance. The basis originally was -3 and it has become -2. Here the hedger
would get only Rs.39 (242-203). He has lost on the basis strengthening. If,
the basis had weakened and the Futures price at the end was say 244, he
would have gained Rs.41 (244-203). Here the original basis was (-3) and the
final basis was (-4), so the basis had weakened.
Basis risk arises because the hedge position cannot possibly go on up to the
last date desired. If the horizon required by the hedger exactly conforms to
the horizon of the Futures contract then there will be only negligible basis
risks. The principle of convergence which avoids basis risks is discussed in
detail in the next unit.
A change in basis can result from a change in the risk free rate of interest,
change in floating funds, change in the availability position of the assets and
a phenomenon called convenience yield.
Cross hedges have greater basis risks because the underlying assets are
different.

37
Even though basis risk is a very real factor, the fact that un-hedged positions
carry greater risks means that former can be ignored as a factor in hedging.
3.6 Cross hedging
In all the examples that we have discussed so far there were ready-made
Forward/Futures contracts available for the asset concerned. The Company,
trader or operator was able to directly use the Forwards/Futures contract for
his hedging purposes. In actual practice we may not have ready
Forward/Futures contract available for an asset of our interest.
In such cases the Company has to identify another asset which is covered by
going Forward/Futures contracts. Having identified the asset which is similar
in nature to the asset on which the Company wants a hedge, the extent of the
relationship should be estimated. There is no hard and fast rule for this
estimation.
To take an example a farmer expects 500 kgs of onion in the next three
months. He would be happy to be able to sell the output at around the price of
Rs.15 prevalent today. Unfortunately for him there is no Forward/Futures
contract which would have enabled him to go in for a short hedge. He
however, notices that there are running contracts of Forwards/Futures in
respect of potatoes. Based upon his past experience he feels that the price
movements of potatoes and onions are perfectly co related. In other words a
Re 1 increase in the price of onion is generally concurrent with a Re.1
increase in the price of potato. So he performs his short hedge using his
Futures potato contracts. He will short hedge by selling 500 kgs of potatoes in
the Futures market at Rs.15 per kg. If after 3 months the price of potatoes
falls to say Rs.12 he will be able to enforce his Forward/Futures contract by
being able to get a selling price of Rs.15 per kg. What this effectively means is
he will be able to buy from the spot market at Rs.12 per kg then and sell it at

38
Rs.15 to Forward/Futures market at Rs.15 per kg thereby making a gain of
Rs.3 per kg. Corresponding to the decline in potato prices the onion price
would have also be fallen to around Rs.12 per kg. But the farmer is protected
in the sense that he will be able to sell the onions at Rs 12 per kg in the
market and get Rs.3 per kg as gains from the potato future contract.
Effectively he is able to get Rs.15 per kg which was what he wanted in the
first place.
The practice of hedging with a different asset to the asset of interest to the
hedger is called cross hedging. The following aspects may be noted in respect
of cross hedges.
1. The first step in a cross hedge is to identify another asset similar in
nature to the asset of interest.
2. The exact extent of relationship should be estimated based on past
experience or historical records. In the above example onions and
potatoes were estimated to be perfectly correlated.
3. If there exists a relationship but the extent of correlation estimated is
less than 1, the Company must make adjustments to the quantity to be
hedged so that the hedge becomes near perfect.
4. If in the above example a Re1 change in onion prices is generally
estimated to accompany with a Re.1.50 change in potato prices, the
extent of contracts required in the Futures market is calculated by
Beta approach. Here the Beta of onions to potatoes is 0.67(1 divided by
1.50). Therefore if 500kgs of onions are to be covered, approximately
333 contracts of potatoes will be needed.(0.67 multiplied by 500)
5. The principle here is that if onion prices fall by Rs.2 potatoes prices
would have to fall by Rs.3. The 333 kgs of potato Futures contract will
effectively safeguard the fall for 500kgs of onions (333kgs of potatoes

39
multiplied by Rs.3 is approximately equal to 500 kgs of onions
multiplied by 2).
6. Basis risk constitutes an important disadvantage of cross hedges
7. Besides the time horizon of the Futures contract may not conform to
the required time horizon of the Company. For instance in the above
example potato Futures contract may not be available for 3 months but
may be available for only 6 months. Since the farmer requires coverage
only for 3 months in respect of his onions output a suitable Futures
contract will be difficult to find.
3.7 Summary
Hedging constitutes the most important use of Futures. Hedges are either
long or short. In a long hedge a Futures contract is bought and in a short
hedge a Futures contract is sold. Hedging is a double-edged sword. It offers
protection for adverse movement of prices but prevents the hedger from
participating in extreme favorable movements.
When a Futures contract is not available for a specific underlying asset it
may become necessary to hedge with another asset similar in nature to the
desired underlying. This is called cross hedge. Here the hedger will have to
estimate the extent of change likely to occur in the second asset for a change
in the underlying asset. Accordingly the desired number of contracts is
entered into on the second asset - either as a short hedge or a long edge. If
the estimated movements of the two assets are in conformity with
expectations, a cross hedge will perform as efficiently as a regular hedge.
In entering into transactions with Futures a trader is exposed to basis risk.
A basis risk refers to the difference between the spot and the Futures prices
at the beginning and the changes thereto at the end. A short hedger benefits

40
from strengthening of the basis and a long hedger benefits from the
weakening of the basis.
Futures contracts entail margins as imposed by stock exchanges. The
practices of levying margins are different in various exchanges and usually
have components like an initial margin, some maintenance margin and
mark-to-market margins. Traders are required to keep paying the necessary
amounts to the exchange. This procedure enables the exchange to manage
the risk and prevent default. It must be remembered that in Futures
contracts, the counterparty is always the Exchange.
3.8 Key words
• Hedging
• Basis
• Cross-hedge
• Strengthening of basis
• Weakening of basis
• Margin
• Mark-to-market
1) What is basis risk? Is it important in hedging?
2) In a long hedge, if the price of the underlying falls, what will be the
nature of the final payoff for the hedger?
3) How is the appropriate asset chosen in a cross-hedge, where the
desired underlying is not traded in the market?
4) How are margins levied by stock exchanges? What is the role of mark-
to-market in this?

41
4 Pricing of Futures and arbitrage conditions
4.1 Objectives
• To understand the principle of arbitrage and how it ensures a fair
Futures price
• To look at the general rules and exceptions to these rules in respect of
Futures pricing
• To understand the evidence from the market on the principle of cost of
carry
4.2 Introduction
The basic principle of cost of carry and its application in Forward/Futures
pricing was introduced in an earlier unit. Operators in a stock exchange
should be well-versed with principles governing the pricing of Derivatives, to
enable them to use these in the right manner. These postulates in pricing
are intricately connected to basis risk and other economic imbalances that
might be existent.
Many economic conditions exist which prevent absolute perfection of
markets. These imperfections result in the absence of perfect arbitrage and
consequent in equilibrium in pricing
Lastly, an efficient market will necessarily want the Futures to be priced as
close to the cost of carry principle as possible, so that other strategies like the
calendar spread and rolling the hedge Forward can be practiced.

42
4.3 Basic pricing principles
We have seen the following equation for Forwards pricing. The same applies
to Futures pricing as well, with modifications to provide for margin
remittance.
Futures Price = Spot Price * ℮rt, where
*signifies the multiplication symbol
r is the risk free rate of interest : t is time to maturity
The basic model applies where there is no expected income from the
underlying during the tenure of the contract.
In case a specific dividend is expected from the asset during the period of the
contract, the formula has to be modified as follows:
Futures Price = (Spot Price- Expected Dividend) * ℮rt, where
r is the risk free rate of interest : t is time to maturity
The principle behind the modification is that if there is to be a dividend, then
the cost of carry will be lesser by that amount. The amount required for
investment will be lower by the extent of dividend.
Sometimes, instead of a specific dividend the underlying has an expected
dividend yield. This is particularly applicable for Index based Futures, which
is the subject matter of the next unit. The formula for the Futures price will
then be:
Futures Price = Spot Price * ℮(r-y) t, where
r is the risk free rate of interest: t is time to maturity
y is the expected dividend yield.

43
Here the dividend is not a specified amount, but a specified yield, and so the
adjustment is made to the rate of interest. The cost of carry principle works
on the principle of interest carry and the interest levy will be lower when
there is a specific income yield
The impact of the time and rate of interest on the cost of carry can be seen
with an example.
Assuming the asset price today is Rs.200, the time to expiry of the Futures
contract is 3 months and the interest rate to be 6% p.a, the Futures price
should be Rs.203.02, based on the cost of carry principle. This is calculated
as
200 * ℮ (0.06 * 0.25)
The time has been taken as 0.25 (3 months), and the rate of interest in
decimals is 0.06.
The impact of changes in interest rates and time to expiry on the Futures
contract is shown in the following table:
The table shows the relative impact of interest rates and the time to expiry
(both expressed in decimals). As the interest rates go up, keeping the time
constant, the Futures price goes up. This is intuitive in that a greater rate of
interest results in greater cost of carry. Similarly, as time goes up, keeping
the interest rate constant, the cost of carry goes up. This again is intuitive in
that the greater the time period, the greater the carry element.

44
Table I.4.1 Cost of carry and formula-based prices
Interest
Time 0.25 0.35 0.45
6.00% 203.0226 204.2444 205.4736
6.25% 203.1495 204.4232 205.7048
6.50% 203.2765 204.6022 205.9364
6.75% 203.4036 204.7813 206.1682
7.00% 203.5308 204.9605 206.4003
7.25% 203.6581 205.1399 206.6326
7.50% 203.7854 205.3195 206.8652
7.75% 203.9128 205.4992 207.0981
8.00% 204.0403 205.6791 207.3312
8.25% 204.1678 205.8592 207.5645
8.50% 204.2955 206.0394 207.7982
8.75% 204.4232 206.2198 208.0321
9.00% 204.551 206.4003 208.2663
In Futures contracts, the cost of carry has to factor in the possible margins
and the interest thereon into the calculations.
Another factor that has occasionally resulted in differences is the rate of
interest to be reckoned for calculation. Academics are divided as to the most
relevant rate of interest for this purpose. The broad consensus is that the
Treasury Bill rate, which is risk-free, should be reckoned for calculation. The
logic behind this is that the theoretical prices are held in a place by arbitrage
and unless the arbitrage is risk-free, it cannot be universal. A risk-free
arbitrage can have only a cost of carry of the risk-free rate.
4.4 Arbitrage opportunities
The possibility of arbitrage makes the theoretical price rigid.

45
Let us take the case of an asset having a price of Rs.100. The Futures to
expire 3 months from date will have a price of Rs.101.51, if the interest rate
is 6%. (100 * ℮ (0.06 * 0.25)).
Suppose the price is only Rs.101, operators will seize the opportunity to buy
Futures at 101 and sell the spot asset at Rs.100. In 3 months they will earn
an interest of Rs.0.51 on the sale proceeds of the spot asset, and use Rs.101 of
this to buy back the asset. The difference of Rs.0.51 is their risk-free profit.
There will be a gain regardless of the final price of the asset. This is shown
in the following table:
Table I.4.2 Arbitrage when actual Futures is less than theoretical price
If final
price is
gain on
spot
gain on
Futures
total
gain
interest
net
gain
-2 1 -1 1.51 0.51
99 1 -2 -1 1.51 0.51
100 0 -1 -1 1.51 0.51
101 -1 0 -1 1.51 0.51
102 -2 1 -1 1.51 0.51
103 -3 2 -1 1.51 0.51
104 -4 3 -1 1.51 0.51
105 -5 4 -1 1.51 0.51
106 -6 5 -1 1.51 0.51
107 -7 6 -1 1.51 0.51
108 -8 7 -1 1.51 0.51
109 -9 8 -1 1.51 0.51
110 -10 9 -1 1.51 0.51
As more and more operators take this opportunity, the price of the Futures
will get automatically adjusted to the theoretical level of Rs.101.51.

46
In the same way, if the final price is Rs.102 (greater than the theoretical
level), alert operators will sell Futures and buy the asset in the spot market
at Rs.100. They will borrow Rs.100 for buying the asset spot. They will incur
an interest of Rs.1.51 over the 3-month period for the borrowed money. They
will, however, realize Rs.102 from the sale of The Futures. This will be
available to them after 3 months, and they will use Rs.101.51 of that for
repaying the interest and principal of the loan, thereby pocketing the
difference as risk-free profits. The risk-free profits will be available to them
regardless of the final spot price, as shown in the following table:
Table I.4.3 Arbitrage when actual Futures is greater than theoretical price
End
spot
gain
spot
gain F tot gain int. loss
net
gain
99 -1 3 2 1.511306 0.49
100 0 2 2 1.511306 0.49
101 1 1 2 1.511306 0.49
102 2 0 2 1.511306 0.49
103 3 -1 2 1.511306 0.49
104 4 -2 2 1.511306 0.49
105 5 -3 2 1.511306 0.49
106 6 -4 2 1.511306 0.49
107 7 -5 2 1.511306 0.49
108 8 -6 2 1.511306 0.49
109 9 -7 2 1.511306 0.49
110 10 -8 2 1.511306 0.49
As more and more operators take this opportunity, the price of the Futures
will get automatically adjusted to the theoretical level of Rs.101.51.
While the possibility of arbitrage ought to restore the Futures prices to their
correct theoretical position, the following factors need to be considered:

47
1. For arbitrage to be possible, the information regarding Futures prices
should be constantly available to all possible dealers. In India, stock
markets work on an online real-time basis andhence the information
will be readily available. In the commodity segment and the foreign
exchange segment, information may not be that easily available and
this factor will create the difficulty for arbitrage.
2. When the actual Futures prices are greater than the theoretical level,
the dealer will seek to sell Futures and buy the Spot. For buying the
Spot, he may need borrowed money. Depending upon the economic
situation in the country, money supply may or may not be freely
available. If it is available only at high rates of interest, the arbitrage
may not work out to any benefit.
3. The basic assumption under the principle of cost of carry is that both
borrowing and lending can be done at the risk-free rate of interest.
This assumption itself is not far-fetched since any model requires a
stable assumption to proceed. The Capital Assets Pricing Model and
the Modigliani-Miller propositions all assume this. However, the
availability of ready funds at this rate is crucial.
4. When the actual Futures price is less than theoretical level, the dealer
will seek to buy Futures and sell the Spot. Many times this may not be
possible because of the absence of ready stock of the Spot to sell.
Shortage of delivery position in the market might result in this
difference not being exploited.
5. Sometimes, the phenomenon of convenience yield prevents selling the
Spot even though an arbitrage opportunity exists. The holder of the
Spot might feel inclined to hold on to the stock for the convenience of
having the stock ready, rather than selling and getting back the stock
after some time. This is particularly true of export traders who like to
keep their stock ready for possible exports in the future.

48
6. Sometimes regulatory provisions may create obstacles to smooth
arbitrage. Prohibition of short selling and disclosure of open positions
might result in arbitrage opportunities not being exploited. This could
also be because of the need to have a minimum quantity of buy or sell
in the Futures segment. The amount required as margins may not be
easily forthcoming and this again could prevent arbitrage.
4.5 Empirical evidence on cost of carry
Empirical evidence on the cost of carry principle has not been consistent.
Most studies have shown that cost of carry does not perfectly hold in markets
over a length of time. Many economic reasons including those listed above
are given for the phenomenon.
A typical study on the behavior of Futures prices to changing spot prices
would look at spot rates at various dates and compare these with the
corresponding Futures rates. Over a period of time, the interest rate taken
by the market can be intuitively understood. When this rate is steady over a
period of time, we can gather that the cost of carry principle holds and that
the interest rate attributed is rigid. However, studies of this nature have
found the cost of carry varying from week to week and sometimes even day to
day.
Studies have also found the existence of backwardation in Futures prices.
Backwardation refers to the Futures price being lower than the Spot price.
This is counter-intuitive to the cost of carry principle. But the predominant
weight of the other economic factors and market imperfections do lead to this
from time to time.

49
4.6 Rolling the hedge forward
In all the examples discussed so far the hedging was successful because the
time horizon of the hedger matched substantially with the time horizon of the
Futures contract. There will be a great deal of difficulty if this matching is
not perfect.
Let us take the case of a trader wanting a long hedge for a period of 6
months. He fears that the prices will go up for the commodity in 6 months
and wants to lock in the price at a level. He understands that a Futures
contract on the commodity will be the ideal answer to his problem. However,
he finds that Futures contracts exist only for a 3-month horizon, while his
requirement is for 6 months.
In such a case, he can go in for rolling the hedge forward. We can look at this
phenomenon through an example (all figures in Rs.):
The rolling of the hedge forward involves first taking Futures of a suitable
duration and just before its expiry, squaring it off and going in for a new
Futures contract. If the horizon is not met even after the second set of
Futures, the process is repeated as many number of times as makes the
horizon match.
In the above example it would not have mattered if the Spot price at the end
of 3 months had actually been less than the original. The square off will be
done at the spot price at end of 3 months ( the Futures will be very near this
spot price near expiry because of the principle of convergence and low cost of
carry) and the new Futures will be taken up at around the same price.

50
Table I.4.4 Rolling the hedge forward (Amount in Rs.)
Spot price 100
Risk-free rate of
interest 6%
Time 3 months 0.25
Futures price 101.51
The trader goes in for a Long hedge buying Futures at 101.51,
to safeguard against rise in prices
The requirement being for 6 months the hedge is not
complete even after 3 months.
After 3 months
Let us say the
Spot is 110
At this time, the Futures for other 3-months will be available
at Rs.111.66. The trader will square off his original position
at the converged spot price.
Long new Futures 111.66
The trader’s gain (110-
101.51) on the old Futures 8.49
After 6 months
Spot is say 120
Gain (120-111.66) on new
Futures 8.34
Total gain 16.83
Spot trader was ready to pay 100
Interest saved 3.05
Total he can now pay 119.87
actual spot 120
Sometimes tracking errors can occur and there could be resultant differences
in the prices of square off and new Futures. So the gains listed above may not
always occur accurately. But hedging is all about approximation and the
trader will continue to be covered by and large.

51
One great disadvantage of rolling the hedge forward is the high amount of
transaction costs that are likely to be incurred in the process. Ultimately, like
a single-period hedge, a rolling of hedge is also a double-edged sword. If the
price movements are not as anticipated, the potential for participation in
these will be foregone. However, the rolling process can be reviewed at the
end of each interval.
4.7 Summary
The principles of hedging for Forwards and Futures are substantially the
same. In Futures, we also have to reckon the question of margins. A long
hedge involves going in for buying the Futures contract fearing a price rise.
A short hedge involves going in for selling the Futures contract fearing a
price fall. In either case, the hedge is based on the going Futures prices,
which, in turn is expected to conform to the cost of carry principle.
The pricing of Futures is basically by the Cost of carry principle. Continuous
compounding is used in calculating interest rates and for this the natural
logarithm is taken as the basis. In case the asset has a known return during
the period of the contract, the amount of such known income is deducted from
the Spot price today to calculate the cost of carry. The principle behind this
is that the known income reduces the amount involved in the spot
investment. In the same way, sometimes, it is a known dividend yield that
comes about and not a fixed known income. Here, the dividend yield is
deducted from the risk free rate in calculating the cost of carry.
The principle of Futures pricing is based on arbitrage. If equilibrium does
not hold then alert investors will invoke arbitrage and reap risk-free profits.
As the number of such operators goes up, the prices come back to the
equilibrium level. However, there are certain economic factors which may

52
prevent perfect arbitrage. These could also be sometimes regulatory in
nature. In such cases the Futures prices will not conform to the cost of carry
principle and what is more, may sometimes go into backwardation.
Empirical evidence on these prices has not established that cost of carry as a
principle will always hold good. In fact, the evidence has been weighed in
favor of its not holding most of the time.
When the horizon of the hedger does not match the horizon of a Futures
contract, the hedge can be rolled forward. This involves taking a Futures
shorter than the required horizon at first and then shifting this to other
Futures on the expiry of the first. By and large this will result in a perfect
hedge. However, high transaction costs and occasional tracking errors might
result in some losses. Like a regular hedge the rolling hedge will also be able
to ensure around the agreed price regardless of the movement of spot prices.
By the same token, any unforeseen profits cannot be participated in because
a Futures contract is a commitment at a particular level.
4.8 Key words
• Long and Short hedges
• Cost of carry
• Continuous compounding
• Arbitrage
• Rolling the hedge Forward
• Convenience Yield
• Backwardation
1) How is the theoretical Futures price computed when it is known that
the asset is expected to yield a dividend of 5% during the period of the
Futures contract?

53
2) Does the principle of rolling the hedge Forward always result in a
perfect hedge?
3) What circumstances prevent the Futures prices from following the cost
of carry principle?
4) Explain the intuition behind using the risk-free rate for calculating the
Futures prices based on cost of carry.

54
5 Stock Index Futures
5.1 Objectives
• To introduce the concept of Index Futures
• To see the applicability of pricing theories to Index Futures
• To see some of the higher uses of Index Futures
• To understand the principle of cross hedges as applicable to Index
Futures.
5.2 Introduction
Several years after Futures trading got into full swing in both the NSE and
the BSE, the investing community in India does not appear to be fully aware
of all the possibilities that these offer. This unit attempts to draw attention
to common uses and certain possibilities of sophisticated uses of Index
Futures. At a given point of time, there are three Futures being traded in
each exchange – one expiring on the last Thursday of the third month from
the date of the trade, another expiring on the last Thursday of the second
month from the date of the trade and yet another one expiring on the last
Thursday of the month succeeding the date of the trade. The 3-month
Futures of today will thus become the 2-month Futures after one-month and
a 1-month Futures after 2 months. The permitted lot size of S&P CNX Nifty
Futures contracts is 200 and multiples thereof. The spot price at the expiry
of the last trading day will be reckoned as the converged Futures price for
settlement purposes. In other words, this means that as on the last date of
trading of the Index Futures, the Futures price will equal the spot price. For
instance, if Futures are expiring on 27th July 2000, the final rate for this will
be exactly equal to the spot Index rate on that day. As a corollary, it means

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