MAT ABACUS BUSINESS SCHOOL

DERIVATIVES AND RISK HEDGING

Introduction

A firm faces several kinds of risk. Its profitability fluctuates due to

unanticipated changes in demand, selling price, costs, taxes, interest rates,
technology, exchange rate and host of other factors. Managers may not be
able to fully control these risks, but, to some degree, they can decide the risks
that the firm should take. They adopt many strategies to reduce their firms’
risks. They keep several options open; they create operating flexibility that
might bail them out in difficulties.
Managers can reduce their risk exposure by entering into financial contracts.
In this chapter, we shall discuss many more financial contracts that managers
can use to hedge various kinds of risks. We shall also see how firms could use
real options to increase their flexibility and reduce their exposure to risk
Derivative defined
A derivative is a financial instrument whose pay-off and value is derived from
or depend on the behavior of the value of an underlying asset (the
"underlying"). The most common underlying’s assets include shares, bonds,
currencies and interest rates. For example, in case of a stock option, the
underlying asset is share (stock) of a company. The value of the stock option
depends on the value of the share. Derivatives are contracts that give the right
and sometimes the obligation, to buy or sell a quantity of the underlying or
benefit in some other way from a rise or fall in the value of the underlying
Firms do not like risk; they do not consider it desirable. Firms will take risk
only when they are appropriately compensated for it. By reducing risk, they
can avoid cash flow fluctuations and thus, increase value of their assets or
investments. Not surprisingly, firms always look for ways and means of
reducing their risk. As we shall be discussing later in this chapter, derivatives
are tools to reduce a firm’s risk exposure. A firm can do away with
unnecessary parts of risk exposure and even convert exposures into quite
different forms by using derivatives. Hedging is the term used for reducing or
eliminating risk by using derivatives.
Advantages of risk hedging
✓ Increased profitability: Through hedging using derivatives, the firm will
be able to protect itself against potential losses arising from adverse
movement in exchange rates and interest rates thereby enhancing the
firm’s profitability and cash flow position
✓ Increased focus on operations: Financial risk management requires
1 | HEDGING NOTES BY CPA GEOFFREY KARUBANGA
 managers to hedge against possible movements in interest rates and
exchange rates. These factors are not under the control of managers
nor can they predict their behavior.
Hedging against such risks encourages managers to concentrate
their efforts on improving operations rather than worrying about
factors that are not in their control.

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Types of derivatives:
1. Forward contracts.
2. Futures contract.
3. Options.
4. Swaps

Forward Contract

We do frequently enter into arrangements or understanding now for buying

or selling items in the future. These arrangements, when formalized, are
referred to as forward contracts.
A forward contract is an agreement between two parties to buy or sell an asset
at a future date at a predetermined price specified today. Under the forward
contract, both the buyer and the seller are bound by the contract; the buyer
must take the delivery and the seller must make the delivery at the agreed
price on the specified due date
Forward contracts are flexible. They are tailor-made to suit the needs of the
buyers and sellers. You can enter into a forward contract for any goods,
commodities or assets. You can choose your delivery date for any quantity of
goods or commodities. For example, you need to buy 100 barrels of oil after 3
months. You can fix the price of 100 barrels of oil today by an agreement with
an oil supplier forward. In 3-month’s, you will buy 100 barrels of oil by paying
the agreed price irrespective prevailing market price. There should always be
willing buyers and sellers for entering into a forward contract. Though forward
contracts are entered into for different goods, commodities or assets, etc., but
the foreign currencies forwards have the largest trading.
Key features of forward contracts are:
1. Highly customized - Counterparties can determine and define the terms
and features to fit their specific needs, including when delivery will take
place and the exact identity of the underlying asset.
2. All parties are exposed to counterparty default risk - This is the risk
that the other party may not make the required delivery or payment.
3. Transactions take place in large, private and largely unregulated
markets called over-the-counter markets
4. Underlying assets can be commodities such as wheat and orange juice;
energy resources such as oil and gas; and financial assets such as
shares, bonds and foreign currencies.
5. They tend to be held to maturity and have little or no market liquidity.

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Advantages of using forward contract
1. Customizability: Forward contracts are highly customizable, allowing
parties to tailor the terms of the contract to suit their specific needs.
They can specify the underlying asset, quantity, delivery date, and other
contract terms, providing flexibility in meeting their individual
requirements.
2. Price certainty: Forward contracts provide price certainty for both
buyers and sellers. By agreeing on a fixed forward price, the parties can
eliminate the uncertainty associated with future price movements in the
underlying asset. This can be particularly useful for hedging against
price volatility.
3. Tailored risk management: Forward contracts allow businesses to
manage various types of risks, such as commodity price risk, foreign
exchange risk, or interest rate risk. They enable parties to lock in prices
and protect themselves from adverse movements in the market,
providing a risk management tool.
4. No upfront payment: Unlike some other derivative instruments, forward
contracts generally do not require an upfront payment or premium. The
parties involved settle the contract at maturity or as per the agreed
terms, reducing the initial cash outflow.

Disadvantages of forward contract

1. Counterparty risk: Forward contracts expose parties to counterparty
risk, which is the risk that the other party may default on their
obligations. If one party fails to fulfill its side of the contract, the other
party may face losses or difficulties in finding an alternative
arrangement.
2. Lack of liquidity: Forward contracts are often traded over-the-counter
(OTC), meaning they are not traded on centralized exchanges. As a
result, they may lack liquidity, making it challenging to find willing
counterparties to enter into or exit positions. This illiquidity can limit
the flexibility and ease of managing the contract.
3. Limited flexibility: Once a forward contract is entered into, it is a binding
agreement that both parties must honor. There is limited flexibility to
modify or cancel the contract before its expiration. This lack of flexibility
can be a disadvantage if circumstances change or if parties wish to
adjust their positions.
4. Opportunity cost: By entering into a forward contract, parties forego the
opportunity to benefit from favorable price movements in the underlying
asset. If the market price of the asset significantly deviates from the
forward price, one party may miss out on potential gains.

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Futures Contract

A futures contract is a derivative that is an agreement to buy or sell a

commodity or an asset in future at a price agreed today. This "delivery" date
can be several months into the future. The futures contract obligates the
buyer of the future to proceed with the purchase. It also obligates the seller of
the future to proceed with the sale. This derivative is a legally binding
agreement between both parties. Each side agrees to honor the terms, even
though the prices may move in such a way that the terms become very
unfavorable to one party.
Futures Contracts Vs Forward Contracts
✓ Organized futures exchanges: Forward contracts are contracts between
two parties, called the counterparties and they are not traded on any
exchange. Future contracts are traded in the organized future
exchanges. As stated earlier, futures contracts are forward contracts
traded on the futures exchanges.
✓ Standardized contracts: Futures contracts are standardized contracts
in terms of the amount or quantity as well as the quality of the product.
The seller of the futures contract will have to deliver goods or
commodities of a specified quality. Forward contracts are non-
standard. They are custom-made depending on the requirements of the
counterparties
✓ Margin: The buyers and sellers of the futures contracts are required to
deposit some cash or securities as margin. This is done to ensure that
the buyers and sellers honor the deal. In case of forward contracts,
there is no formal requirement of margin
✓ Marked to market: This is a significant difference between forward
contracts and futures contracts. Forward contracts are settled on the
due date. But in the case of futures contracts, there is daily closing and
reopening of the position which is called marked to market. Marked to
market means that profit and loss on a futures contract are calculated
each day, and if there is a profit, you receive profit from the exchange
but if there is a loss, you pay loss to the exchange.
Consider the buyer’s perspective. Suppose the buyer holds a future contract
to buy 1,000 kilograms of aluminum at 3,200 euros. The delivery is for three
months’ time. Over the course of those three months, demand decreases,
lowering the price to 2,500 euros. The buyer would prefer if they did not agree
to the higher price. Nevertheless, the current market price is irrelevant to
deciding if the trade should still be done. Futures are obligations.
Counterparties are not allowed to back out of the future contract.
Likewise, suppose there were a supply disruption instead, hiking the price to
4,000 euros. Now the buyer is happy to pay 3,200, because everyone in the

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current market is paying 25% higher. The seller, on the other hand, would
prefer to sell at the higher price. Again, each side must honor the obligation.
Indeed, only a small amount of money changes hands until the future is
actually traded. These upfront costs are related to initial margins and
exchange fees. However, if the current price moves away from the delivery
price, then one side may require significant amounts of cash to meet margin
calls. This process of requiring margin is part of the responsibility of the
futures exchange. In our first example, the buyer is going to have to take a
loss and will have to post margin to cover that loss (2,500 - 3,200 = 700 euros).
This extra cash ensures the buyer can proceed with the transaction on the
delivery date. Why does the exchange require this margin? The reason is the
credit risk. Without the margin, a buyer could theoretically show up on the
delivery date with insufficient funds to proceed with the trade. Consistent
margin calls by the exchange ensure this doesn’t happen. In doing so,
exchanges help to mitigate credit risk.
Advantages of Futures Contracts:
1. Standardization and Liquidity: Futures contracts are typically traded
on regulated exchanges, which ensures standardization in terms of
contract size, expiration dates, and settlement procedures. This
standardization leads to high liquidity, making it easier to enter and
exit positions. Traders can readily find counterparties in the market,
enhancing market efficiency.
2. Price Discovery: Futures contracts facilitate price discovery by providing
a transparent platform where buyers and sellers can openly trade. The
continuous buying and selling activity in the futures market helps
establish fair and efficient prices that reflect market sentiment and
supply-demand dynamics. This price discovery mechanism benefits
market participants by providing reliable price references.
3. Risk Management: Futures contracts serve as effective risk
management tools, allowing market participants to hedge against price
fluctuations. By taking opposite positions in the futures market to their
underlying assets, businesses can protect themselves from adverse
price movements. Hedging with futures can help mitigate risks
associated with commodities, currencies, interest rates, and stock
market indices.
4. Leverage: Futures contracts allow traders to gain exposure to
underlying assets with a relatively small initial investment. This
leverage amplifies potential returns, enabling market participants to
control larger positions with a smaller capital outlay. However, it's
important to note that leverage also increases the potential for losses.
5. Clearinghouse and Counterparty Risk Mitigation: Futures contracts
utilize clearinghouses, which act as intermediaries between buyers and
sellers. The clearinghouse becomes the counterparty to every trade,

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 reducing counterparty risk. The clearinghouse also manages margin
requirements and guarantees the performance of contracts, ensuring
the fulfillment of obligations.

Disadvantages of Futures Contracts:

1. Margin Requirements: Futures contracts typically require traders to
maintain margin accounts, which necessitate posting initial margin and
potentially additional variation margin. Margin requirements tie up
capital and can result in significant margin calls if positions move
against traders, requiring additional funds to meet margin obligations.
2. Contract Size: The standardized contract sizes of futures contracts may
not suit the needs of all market participants. For some assets, the
contract size might be too large or not align with specific requirements,
leading to difficulties in effectively managing positions or hedging
exposures.
3. Limited Flexibility: Futures contracts have predefined expiration dates
and settlement procedures, limiting flexibility compared to over-the-
counter (OTC) derivatives. It may not be possible to precisely align the
futures contract terms with specific hedging needs or desired trading
strategies.

Options

An option is a derivative that provides the holder with optionality, or choice,

in whether to trade a security in the future at the exercise price. The two basic
types of options are calls and puts. A call option grants the holder the right,
but not the responsibility, to purchase the underlying asset at a fixed price.
That is, the holder calls in the underlying at the price that was fixed. A put
option grants the holder a limited-time offer: the right, but not the
responsibility, to sell the underlying asset at a fixed price. That is, the holder
puts the security on sale at a fixed price. This fixed price is known as the
strike price or exercise price. Where does the option buyer purchase the
option? This would most likely be on an options exchange
An option costs money to purchase. The price of any option is known as the
premium. The premium is the price the buyer pays to enter a call. Likewise,
the premium is the term used for the price the buyer pays to enter a put. Note
these are not the same value, but the term applies to both calls and puts.
Once an option buyer pays the premium, they own the option. They will also
never lose more money than the premium

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Validity of using option contract
Advantages of Option Contracts:
1. Flexibility: Options offer a high degree of flexibility for market
participants. As the holder of an option, you have the choice to exercise
or not exercise the contract depending on market conditions and your
desired outcome. This flexibility allows for various trading and hedging
strategies to be implemented.
2. Limited Risk: The risk for an option buyer is limited to the premium
paid for the option contract. This means that if the market moves
unfavorably, the maximum loss is known upfront. This limited risk can
be advantageous for risk-averse individuals or when seeking downside
protection in a portfolio.
3. Leverage: Options provide the potential for amplified returns through
leverage. With a smaller investment (premium) compared to purchasing
the underlying asset, option buyers can control a larger position. This
leverage allows for potentially significant gains if the market moves
favorably.
4. Hedging and Risk Management: Options serve as valuable tools for
hedging and risk management. They can be used to protect against
adverse price movements in the underlying asset, providing insurance
against market volatility. Option contracts enable market participants
to mitigate risks associated with commodities, currencies, interest
rates, and stock market indices.
5. Income Generation: Option sellers, also known as option writers, can
generate income through the collection of option premiums. By
assuming the obligation to buy or sell the underlying asset, sellers earn
premiums from buyers. This income generation strategy can be
employed in range-bound or declining markets.

Disadvantages of Option Contracts:

1. Limited Time: Options have expiration dates, after which they become
worthless. This limited time frame puts pressure on option holders to
be accurate in their market timing. If the underlying asset doesn't move
in the anticipated direction within the specified period, the option may
expire worthless, resulting in a loss of the premium paid.
2. Price Decay: Options experience time decay, meaning the value of the
option decreases as time passes. This decay accelerates as the
expiration date approaches. This decay erodes the value of the option,
particularly if the underlying asset price remains stagnant or moves
against the anticipated direction.
3. Complexity: Option contracts can be complex financial instruments,
involving different strike prices, expiration dates, and option strategies.
Understanding and effectively utilizing options require a good grasp of

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 their characteristics and potential risks. Novice investors may find
options challenging to comprehend and navigate.
4. Liquidity and Bid-Ask Spread: Some options may have lower liquidity
compared to more actively traded securities. This can result in wider
bid-ask spreads, making it more difficult to enter and exit positions at
desired prices. Illiquid options can also lead to challenges in executing
complex strategies or adjusting positions.
5. Risk of Total Loss: Option buyers risk losing the entire premium paid if
the option expires worthless or if the market doesn't move in the
anticipated direction. Unlike option sellers who receive the premium
upfront, option buyers face the potential for a complete loss of their
investment.

Swaps

Swaps are similar to futures and forwards contracts in providing hedge

against financial risk. A swap is an agreement between two parties, called
counterparties, to trade cash flows over a period of time. Swaps arrangements
are quite flexible and are useful in many financial situations. The two most
popular swaps are currency swaps and interest-rate swaps. These two swaps
can be combined when interest on loans in two currencies are swapped. The
interest rate and currency swap markets enable firms to arbitrage the
differences between capital markets. They make use of their comparative
advantage of borrowing in their domestic markets and arranging swaps for
interest rates or currencies that they cannot easily access
Unlike futures and options, swaps are not traded on exchanges but over-the-
counter. In addition, counterparties in swaps are usually companies and
financial organizations and not individuals, because there is always a high
risk of counterparty default in swap contracts.
Currency swap involves an exchange of cash payments in one currency for
cash payments in another currency. Most international companies require
foreign currency for making investments abroad. These firms find difficulties
in entering new markets and raising capital at convenient terms. Currency
swap is an easy alternative for these companies to overcome this problem.
Interest rate swap allows a company to borrow capital at fixed (or floating)
rate and exchange its interest payments, with interest payments at floating
(or fixed) rate. That is, Counterparties agree to exchange one stream of future
interest payments for another, based on a predetermined notional principal
amount. Generally, interest rate swaps involve the exchange of a fixed interest
rate for a floating interest rate.

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Advantages of using currency swap
✓ Reduced cost of borrowing: Currency swaps can enable company
borrow cheaply in a foreign currency than it would i.e. currency swaps
allow companies to take advantage of the global markets more
efficiently by bring two parties that have an advantage in different
market
✓ Flexibility: Swaps offer a high level of flexibility in terms of size and
reversibility. They can be tailored to meet the specific needs and
objectives of the transacting parties. Swaps can be arranged in any
desired size, allowing for customization based on the size of the
exposure or the desired risk management strategy. Additionally, if
parties wish to exit or modify the swap contract before its maturity, it
can be reversed or unwound, providing flexibility in managing positions.
✓ Cost Efficiency: Transaction costs for swaps are relatively low compared
to other financial instruments. Swaps do not involve commission fees
or premiums, making them cost-effective for market participants. The
primary cost associated with swaps is typically legal fees incurred
during the negotiation and documentation process.
✓ Market Avoidance: Swaps enable transacting parties to obtain the
currency they require without directly participating in the foreign
exchange markets. This can be advantageous when there is uncertainty
or volatility in the foreign exchange market. By utilizing swaps, parties
can bypass the need to convert currencies in the open market, reducing
exposure to exchange rate fluctuations and potential transaction costs.
✓ Financial Restructuring: Swaps can be used for financial restructuring
purposes, particularly in cases where a company operates
internationally and generates revenues in foreign currencies while
having liabilities denominated in the home currency. A currency swap
can help align the company's liabilities with its revenue streams,
reducing the currency risk associated with the debt. This allows for
better management of foreign currency exposure and helps mitigate the
impact of exchange rate fluctuations on the company's financial
position.

Drawbacks of using currency swaps

✓ Counterparty Risk: Swap contracts expose parties to counterparty risk,
which is the risk of default by the other party. If one party fails to meet
its obligations, the other party may face financial losses or difficulties
in finding a replacement counterparty. It is important to carefully
assess the creditworthiness and reliability of the counterparty before
entering into a swap contract.
✓ Arrangement Fees: Swaps often involve arrangement fees payable to
third parties facilitating the transaction. These fees can add to the
overall cost of using swaps as hedging or risk management tools.

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 ✓ Difficulty in Finding Reliable Counterparty: For a swap to be executed
successfully, both parties involved must have complementary needs,
meaning they should want to exchange exact amounts of the currencies
or cash flows involved. Finding a reliable counterparty with matching
requirements can sometimes be challenging. The search for a suitable
counterparty can be time-consuming and may involve additional costs,
such as legal fees or due diligence expenses.
✓ Sovereign Risk: When engaging in swaps involving currencies of
different countries, there may be a risk of sovereign interference or
exchange controls. Governments may impose restrictions or regulations
that affect the ability to carry out the swap or the free movement of
funds

Options and their Valuation

An option is a contract that gives the holder a right, without any obligation,
to buy or sell an asset, at a pre-agreed price, on or before a specified period of
time. An option on a building might give the buyer the right to buy the building
for Shs 200 million on or any time before the end of one year. Options are a
unique type of financial contract because they give the buyer the right, but
not the obligation, to do something. The buyer uses the option only if it is
advantageous to do so, otherwise the option can be thrown away. The option
to buy an asset is known as a call option, and the option to sell an asset is
called a put option. The asset on which the put or call option is created is
referred to as the underlying asset
Here are some important definitions:
1. Exercising the option: The act of buying or selling the underlying asset
via the option contract.
2. Strike or exercise price: The fixed price in the option contract at which
the holder can buy or sell the underlying asset.
3. Expiration date: The maturity date of the option; after this date, the
option expires.
4. American and European options: An American option may be exercised
anytime up to the expiration date. A European option differs from an
American option in that it can be exercised only on the expiration date
5. Option premium: Options do not come free. They involve cost. The
option premium is the price that the holder of an option has to pay for
obtaining a call or a put option. The price will have to be paid, generally
in advance, whether or not the holder exercises his option. Once an
option buyer pays the premium, they own the option and he will also
never lose more money than the premium.

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 6. In-the-money: A put or a call option is said to in-the-money when it is
advantageous for the investor to exercise it. In the case of in-the-money
call options, the exercise price is less than the current value of the
underlying asset, while in the case of the in-the-money put options, the
exercise price is higher than the current value of the underlying asset
7. Out-of-the-money: A put or a call option is out-of-the-money if it is not
advantageous for the investor to exercise it. In the case of the out-of-
the-money call options, the exercise price is higher than the current
value of the underlying asset, while in the case of the out-of-the-money
put options, the exercise price is lower than the current value of the
underlying asset.
8. At-the-money: When the holder of a put or a call option does not lose or
gain whether or not he exercises his option, the option is said to be at-
the-money. In the case of the out-of-the-money options, the exercise
price is equal to the current value of the underlying asset

Call Option

A call option on a share (or any asset) is a right but not obligation to buy the
share (underlying asset) at an agreed exercise (strike) price. Suppose that the
current share price of share is Shs 1,300. You expect that price in a 3-month
period will go up to Shs 1,500. But you do fear that the price may fall below
Shs 1,200. To reduce the chance of your risk and at the same time to have an
opportunity of making profit, instead of buying the share, you can buy a 3-
month call option on share at an agreed exercise price (E) of, say, Shs 1,250.
Will you exercise your option if the price of the share is Shs 1,500 in three
months?
You will exercise your option since you get a share worth Shs 1,500 by paying
an exercise price of Shs 1,250. You will gain Shs 250; that is, the pay-off or
the value of your call option at an expiration of Shs 250. Your call option is
in-the-money at maturity.
What will you do if the price of the share is Shs 1,200 when the call option on
share expires? Obviously, you will not exercise the option. You gain nothing.
Your call option is worthless, and out-of-the-money at expiration. You may
notice that the value of your call option can never be less than zero.
Thus, you should exercise call option when share price at expiration exceeds
exercise price. On the other hand, you should not exercise the call option
when the share price at expiration is less or equal to the exercise price.
The value of the call option at expiration is:
Value of call option at expiration = Maximum [Share price – Exercise price, 0]

11
The expression above indicates that the value of a call option at expiration is
the maximum of the share price minus the exercise price or zero. The call
option holder’s opportunity to make profits is unlimited. It depends on what
the actual market price of the underlying share is when the option is
exercised. The greater the market value of the underlying asset, the larger is
the value (pay-off) of the option
Figure showing the pay-off or value of a call option.
It may be observed from Figure below that the call buyer’s potential pay-off
becomes unlimited, once the price of the share (the underlying asset) goes
beyond the exercise price. If the share price is on or below the exercise price,
the call buyer will not exercise his option. Thus, his pay-off will be zero, since
the option is worth nothing
Value of call option

(bad outcomes) (good outcomes)

It may also be observed from Figure above that the possible outcomes can be
divided into two parts: one, above the exercise price and other, below the
exercise price. The outcomes above the exercise price are said to be in-the-
money and are beneficial to the option holder but not the outcomes below the
exercise price. It is the exercise price that divides the good and bad outcomes
How is the seller (or the writer) of a call option affected when the value of the
underlying asset changes? Figure below shows his position as a mirror image
of the call buyer’s position. The call the buyers gain is called seller’s loss. The
seller of the call option will not incur any loss when the price of the share (the
underlying asset) is less than the exercise price since the buyer will not
exercise his option. However, if the share price rises and goes beyond the
exercise price, the potential loss of the call seller is very high.

12
Figure showing a pay-off of call option writer (seller)
In-the-money Out-of-the-monEy
(good outcomes ) (bad outcomes) Value of share/underly ing asset
Value of call option

Unlimited potential loss

Exercise price

Call premium
A call buyer exercises his right only when the outcomes are favorable to him.
The seller of a call option, being the owner of the asset, gives away the good
outcomes in favor of the option buyer. The buyer of a call option must,
therefore, pay an up-front price, called call premium, to the call seller to buy
the option. The call premium is a cost to the option buyer and a gain to the
call seller. What is the net pay-off of the buyer and the seller of a call option
when the call premium (that the buyer has to pay to the seller) is involved?
Illustration of call option pay-off
The share of Pakasa Ltd is selling for Shs 10,400. An investor buys a 3 months
call option at a premium of Shs 500. The exercise price is Shs 10,500. What
is the investor’s pay-off if the share price is Shs 10,000, or Shs 10,500, or Shs
11,000, or Shs 11,500, or Shs 12,000 at the time the option is exercised?
solutions
The investor will exercise his option for any price above the exercise price Shs
10,500. Since the exercise price is Shs 10,500 and the investor (the buyer)
has to pay a premium of Shs 500, his pay-off will be zero when the share price
rises to Shs 11,000. Thus, Shs 11,000 is a break-even price (i.e., the exercise
price plus the call premium) for him. The exercise price, Shs 10,500, separates
the good outcomes from the bad outcomes. The seller of the call option (the
asset owner) is being paid the call premium, Shs 500, for giving up the good
outcomes in favour of the buyer of the call option.

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Table showing the call option holder’s pay-off at expiration
Shs Shs Shs Shs Shs
Share price 10,000 10,500 11,000 11,500 12,000
Investor's
inflows: Sale of
shares ---- --- 11,000 11,500 12,000
Investor's
outflows:
Exercise option --- --- 10,500 10,500 10,500
Call premium 500 500 500 500 500
Net pay-off -500 -500 0 500 1,000

What is the pay-off of the seller of the call option? The position of the call
option seller will be opposite to that of the buyer as shown in Table above. If
the buyer (investor) exercises his option, the seller will lose. His (seller’s)
potential loss is very high, and his profit is limited to Shs 500 (the call
premium). If Figure above is turned up side down, the call seller’s position can
be depicted graphically in table below
Table showing the call option seller’s pay-off at expiration
Shs Shs Shs Shs Shs
Share price 10,000 10,500 11,000 11,500 12,000
Seller's inflows:
Exercise option ---- --- 10,500 10,500 10,500
Call premium 500 500 500 500 500
Seller's outflows:
Share price --- --- 11,000 11,500 12,000
Net pay-off/profit 500 500 0 -500 -1,000

Figure showing the pay-off of the call option buyer

10
Exercise price
Pay-off

5 Unlimited profit potential

0 Share price
10,000 10,500 11,000 11,500 12,000
-5
Premium Break-even price
-10
Limited loss area

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Pay-off of call option seller

10
Pay-off

5 Break-even price

0 Share price
10,000 10,500 11,000 11,500 12,000
-5
Exercise price Unlimited loss
-10

Put Option

A put option is a contract that gives the holder a right but not an obligation
to sell a specified share (or any other asset) at an agreed exercise price, on or
before a given maturity period.
Suppose you expect price of share to fall in the near future. Therefore, you
buy a 3-month put option at an exercise price of Shs 5,000. The current
market price of a share is Shs 4,800. If the price actually falls to Shs 3,500
after three months, you will exercise your option. You will buy the share for
Shs 3,500 from the market and deliver it to the put-option seller (writer) to
receive Shs 5,000. Your gain is Shs 1,500, ignoring the put option premium,
transaction costs and taxes.
You will forgo your put option if the share price rises above the exercise price;
the put option is worthless for you and its value for you is zero. A put buyer
gains when the share price falls below the exercise price. Ignoring the cost of
buying the put option (called put premium), his loss will be zero when the
share price rises above the exercise price since he will not exercise his option.
Thus, you should exercise the put option when Exercise price is greater than
share price at expiration. On the other hand, you should not exercise the put
option when the Exercise price is less or equal to share price at expiration
The value or pay-off of a put option at expiration will be: Value of put option
at expiration = Maximum [Exercise price – Share price at expiration, 0]
Pay-off diagram for the put option buyer
The figure shows that the value of the put option for the holder depends on
the value of the underlying asset. The value of the put option is zero when it
is out-of-the-money. You may observe from diagram that the potential profit
of the put option buyer is limited, since share price cannot fall below zero.
The exercise price is again the dividing point between the good and bad

15
outcomes. The put option buyer’s gain is the seller’s loss. The seller insures
the buyer from the bad outcomes.
Value of put option

Limited profit

Exercise price

Value of share/ underly ing asset

In-the-money Out-of-the-money
(good outcomes) (bad outcomes)

Pay-off diagram for the put option seller

Figure below shows the pay-off of the seller of a put option. It should be clear
from Figure 7.6 that the potential loss of the put-option seller is limited to the
exercise price. Since the buyer has to pay a premium to the seller for
purchasing a put option, the potential profit of the buyer and the potential
loss of the seller will reduce by the amount of premium.

In-the- money Out-of- the-monEy

Value of share/ underly ing asset
Value of put option

Exercise price

Limited loss

Factors determining the option value

The value of an option depends on the following factors:
1. Underlying Asset Price: The price of the underlying asset has a direct
impact on the value of an option. For call options, as the underlying
asset price increases, the value of the call option typically increases,
assuming other factors remain constant. Conversely, for put options,
as the underlying asset price increases, the value of the put option
usually decreases.
2. Exercise price (Strike Price): The strike price is the predetermined price
at which the underlying asset can be bought or sold (in the case of call
and put options, respectively). For call options, a lower strike price

16
 relative to the underlying asset price increases the intrinsic value,
making the call option more valuable and vice versa. For put options,
a higher strike price relative to the underlying asset price increases
the intrinsic value and the put option's value.
3. Volatility: Volatility refers to the degree of price fluctuations in the
underlying asset. Higher volatility generally increases the value of
options because it implies a greater potential for large price swings,
which can lead to higher profits for option holders. Increased volatility
raises the probability of the option ending up in-the-money (for both call
and put options) and thus contributes to their value.
4. Time to expiration: The time remaining until the option's expiration
affects its value. Generally, as the time to expiration decreases, the value
of the option decreases, assuming other factors remain constant. This is
because options have a limited lifespan, and the potential for favorable
price movements diminishes as the expiration approaches.
5. Risk-free Interest rates: Options depend on the interest rate that applies
to the period that matches the length of time of the option’s expiration.
For a call option, the present value of the exercise price will reduce if the
interest rate is high for a given time to expiration. This consequently
increases the value of a call option and the reverse is also true. For put
option, the value of put options tends to decrease as the risk-free interest
increases (due to lower opportunity costs).

Black–Scholes Model for Option Valuation

The Black-Scholes-Merton (BSM) model is a pricing model for financial

instruments. It is used for the valuation of stock options. The BSM model is
used to determine the fair prices of stock options based on five variables:
volatility, underlying stock price, Exercise price, time to expiration, and risk-
free rate. It is based on the principle of hedging and focuses on eliminating
risks associated with the volatility of underlying assets and stock options.
Assumptions The BSM model is based on the following assumptions:
• There are no transaction costs or taxes: The model assumes that there
are no transaction costs or taxes associated with buying or selling the
option or the underlying asset.
• The option can only be exercised at expiration: The model assumes that
the option can only be exercised at the expiration date, which is known
as a European-style option. This assumption simplifies the calculation
of the option price because it eliminates the need to consider the
possibility of early exercise.
• Random walk: The stock market is a highly volatile one, and hence, a
state of random walk is assumed as the market direction can never truly

17
 be predicted.
• The risk-free interest rate is constant and known: The model assumes
that the risk-free interest rate is constant over the life of the option and
is known with certainty. This assumption allows the model to discount
future cash flows to their present value at the risk-free rate.
• Normal distribution: Stock returns are normally distributed. It implies
that the volatility of the underlying asset is constant over the life of the
option and is known with certainty.
• The underlying asset price follows a log-normal distribution: The model
assumes that the price of the underlying asset follows a log-normal
distribution, which means that the logarithm of the price follows a
normal distribution

According Black-Scholes model, the value of call and put options are given as
follows:
Value of call option: C0 = S0 N(d1) – K e-r*T N(d2)
Value of put option: P0 = K e-r*T N(– d2) – S0 N(– d1)
𝐼𝑛 (𝑆/𝐾)+ (𝑟 +𝜎 2/2)𝑇
Where; d1 =
𝜎√𝑇

d2 = d1 - 𝜎√𝑇
where;
C0 = the current value of call option
P0 = the current value of put option
S0 = the current market value of the share
K = the exercise price
E = 2.7183, the exponential constant
R = the risk-free rate of interest
T = the time to expiration (in years)
N( ) = the cumulative normal probability density function
Ln = the natural logarithm
Σ = standard deviation (volatility of the underlying stock
σ2 = variance of the continuously compounded annual
return on the share.
The Black–Scholes model has two features. First, the parameters of the model,
except the share price volatility, are contained in the agreement between the
option buyer and seller. Second, in spite of its unrealistic assumptions, the
model is able to predict the true price of option reasonably well.

18
Example
Mr. Mugisha is interested in writing a six-months call option on Pakasa Ltd’s
share. Pakasa share is currently selling for Shs 12,000. The volatility
(standard deviation) of the share returns is estimated as 67%. Mr. Mugisha
would like the exercise price to be Shs 12,000. The risk-free rate is assumed
to be 10%. How much premium should Mr. Mugisha charge for writing the
call option? What would be the corresponding value of the put option?
Proposed solutions
Value of call option: C0 = S0 N(d1) – K e-r*T N(d2)
𝐼𝑛 (𝑆/𝐾)+ (𝑟 +𝜎 2/2)𝑇
Where; d1 =
𝜎√𝑇

d2 = d1 - 𝜎√𝑇
12,000)+ (0.1 0.672
) 0.5
𝐼𝑛 (
+
d1 = 12,000 2

0.67 𝑥 √0.5

d1 =
𝐼𝑛 (1)+ 0.1622 = 0.34
0.4738

19
 d2 = 0.34 - (0.67 𝑥 √0.5)= -0.13

From the standard normal distribution table, we obtain N(d1) and N(d2) as
follows;
N(d1) = 0.6331
N(d2) = 0.4483
Value of call option: C0 = (12,000 x 0.6331) – (12,000 e-0.1*0.5 x 0.4483)
C0 = 7,597 – 5,117 = Shs 2,480
The value of corresponding put option can be computed as follows:
P0 = K e-r*T N(– d2) – S0 N(– d1)
P0 = 12,000 e-0.1*0.5 N(0.13 ) – 12,000 N(– 0.34)
P0 = 12,000 e-0.1*0.5 (0.5517) – 12,000 (0.3669)
P0 = 6,298 – 4,403 = Shs 1,895

Put-call parity
Put-call parity is a fundamental concept in option pricing theory that shows
the relationship between the prices of call and put options with the same
underlying asset, strike price, and expiration date. The theory is based on the
principle of no-arbitrage, which states that two securities with identical
payoffs must have the same price. The call parity theory is expressed as
follows:

C0 + K e-r*T = P0 + S0
Where;
• C0 is the price of a call option on an underlying asset
• K e-r*T is the present value of the strike price, K, discounted at the risk-
free rate over the expiration time
• P0 is the price of a put option on the same underlying asset, with the
same strike price and expiration date as the call option
• S0 is the current price of the underlying asset
The call parity theory suggests that the sum of the price of a call option and
the present value of its strike price is equal to the sum of the price of a put
option and the current price of the underlying asset. This relationship holds
true at any point in time before the expiration date of the options.
The intuition behind put-call parity theory is that owning a call option gives
the holder the right to buy the underlying asset at the strike price K. If the
holder exercises the option, they pay K to acquire the asset. However, they

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can also achieve the same result by selling a put option with the same strike
price K. The seller of the put option is obligated to buy the underlying asset
from the holder at the strike price K if the holder chooses to exercise the put
option. By selling the put option, the holder receives the premium and has
effectively locked in the purchase price of the underlying asset at K.

Example
Determine the value of the put option in the above example using put-call
parity formula
Using put-call parity, the value of the put option can be computed as follows;
P0 = C0 + K e-r*T - S0

P0 = 2,480 + 12,000 e-0.1*0.5 – 12,000

P0 = Shs 1,895 (as obtained using the Black-Scholes model)

Test yourself
1) Determine the price a European Call option, currently ATM with three
months maturity. The current price of the underlying stock price is Shs
12,400. The volatility for this stock is 25% annually, while the risk-free
rate is at 8%. Also calculate the price of the corresponding put option

2) You just purchased a European Put option. The strike price of the
option is Shs 15,000 and there is one year left until maturity. The
current price of the underlying stock price is Shs 12,500. The volatility
for this stock is 45% annually, while the risk-free rate is at 10%.
Calculate the value of put option under the Black-Scholes framework

3) Determine the price a European put option. The strike price of this
option is Shs 3,500, with the current stock price at Shs 3,250. The risk-
free rate is 9% and the option will expire in 18 months. The volatility for
this stock is 45% annually.
Limitations of the Black-Scholes-Merton Model

• Limited to the European style options: The Black-Scholes model is

designed for European-style options, which can only be exercised at
expiration. It does not accurately value American stock options which
are exercisable any time prior to the expiration date. It tends to
undervalue American options because it does not take into account this
time flexibility
• Risk-free interest rate assumption: The model assumes a constant risk-

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 free interest rate throughout the option's life. However, interest rates
can change over time, and the assumption of a constant rate may not
hold in practice. Changes in interest rates can affect the pricing of
options, particularly those with longer maturities.
• Constant volatility assumption: The model assumes that volatility
remains constant throughout the option's life. However, volatility is
known to be dynamic and can change over time, particularly in
response to market events or news. This assumption may lead to
inaccuracies in pricing options, especially for longer-dated options or
during periods of high market volatility.
• Assumption of a frictionless market: Trading generally comes with
transaction costs such as brokerage fees, commission, etc. However,
the Black Scholes model assumes a frictionless market, which means
that there are no transaction costs. It is hardly ever the reality in the
trading market.
• No dividends or cash flows: The Black-Scholes model assumes that the
underlying asset does not pay any dividends or generate cash flows
during the option's life. In reality, many assets, such as stocks, do pay
dividends, which can have a significant impact on option pricing.
Adjustments need to be made to the model to account for dividend
payments.
• Assumption of normal and log normal distribution: The Black-Scholes
model assumes that stock price movements follow a log-normal
distribution and stock returns follow a normal distribution, implying
symmetric returns and no significant skewness or fat tails. However,
empirical evidence suggests that asset price returns often exhibit
skewness and fat-tailed distributions, which the model does not fully
capture.

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APPLICATION OF OPTION PRICING THEORY IN INVESTMENT DECISIONS

Introduction
The NPV analysis discussed earlier does not recognize the value of investing
in a project that can more easily adapt to changes in a company’s environment
after the decision to invest has been made. This strategic flexibility is valuable,
and this value is often not captured by the traditional NPV analysis. This leads
to some potentially value-adding investments being rejected.
It is possible to picture this strategic flexibility as a series of choices or options
that are available to managers.
In this chapter we shall see how Black-Scholes model can be adapted to value
real choices or real options that an investment decision may possess
Another benefit of this approach is that risk and uncertainties are viewed as
opportunities, where the upside outcomes can be exploited and the downside
risk can be managed.
The value of real option can then be added to the traditional NPV to give a
revised and more accurate assessment of the value created by the project
Different types of real options
1) Option to expand
Making an investment now, in addition to the cash flows created by that
investment, sometimes create an opportunity for a company to expand in
future. An option to expand can result from being able to apply the new
technology or brand name to other projects if the initial project is successful.
It can simply relate to the option to following on from a successful initial
investment by further expansion in that same area.
To be considered an option to expand, the expansion opportunity would need
to link to the initial investment i.e. the company would not be able to expand
successfully if it had not made the investment in the initial project. An option
to expand involves choosing to spend more money to exploit the upside risk

2) Option to delay
Making an investment now can, in addition to the cash inflows created by the
investment, sometimes create an opportunity for the company to reduce
further spending where an investment can potentially be delayed in a way that
adds value to the project. For projects that have a series of clearly identifiable
stages, managers may be able consider the option to delay at each stage.

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 An option to delay will exist if the project is protected for example by a patent
or a license, so that the competitors are unable to exploit the investment
opportunity. For example, an option to wait may be considered to exist where
an investment starts by establishing a patent, because after securing this, a
company can then wait to see how the market conditions develop before
deciding on when to start commercial production of the product. An option to
delay involves choosing to defer the decision spend more money
3) Option to redeploy
An investment can sometimes contain an opportunity for a company to easily
redeploy assets to another use if the initial investment is not successful. An
option to redeploy involves changing the use of assets to create different
revenue stream, to manage the downside risk
4) Option to withdraw
An investment can sometimes contain a feature that gives a company a choice
or option to withdraw from the market. This would mean that the project
creates a right to selling a license or patent to a competitor or to a joint venture
partner. An option to withdraw involves choosing to raise money by
abandoning the project, to manage the downside risk

Application of Black-Scholes model to value real options

As discussed earlier, the Black-Scholes model is used to determine the option

value based on five variables, that is, volatility, underlying asset price,
Exercise price, time to expiration, and risk-free rate
Value of call option, C0 = S0 N(d1) – K e-r*T N(d2)
𝐼𝑛 (𝑆/𝐾)+ (𝑟 +𝜎 2/2)𝑇
Where; d1 =
𝜎√𝑇

d2 = d1 - 𝜎√𝑇
where;
S0 = Present value of cash flows after exercising the option
K = Cost of the investment
R = the risk-free rate of return
T = the time to expiry of the option in years
Σ = the standard deviation of the project
When you input the variables into the formula, you should note the following;
✓ S0 is the present value of cash flows generated after exercising the
option. So, if you are told the cost of exercising an option to expand in

25 | H E D G I N G N O T E S B Y C P A G E O F F R E Y K A R U B A N G A
 three years’ time is say Shs 100m and that this follow-on project is
expected to give an NPV in three years’ time of Shs 20m. Then the value
of S0 is Shs 120m (100m+20m), discounted back by three years at the
cost of capital that was used for the project
✓ K is the cost of exercising the option. This is not discounted back to the
present value. This is because in the formula valuing a call option, E is
multiplied by e-rt which is a type of discount factor. So, if you are told
the cost of exercising an option to expand in three years’ time is say Shs
100m and that this follow-on project is expected to give an NPV in three
years’ time of Shs 20m, then the value of E is Shs 100m
✓ rf is the risk-free rate of return, this is not the same as the cost of capital
of the company
✓ t is the time to expiry of the option, not of the project. So if you
considering an option to expand in three years’ time, and this project
lasts for 5 years, then t=3 (because this is when the option to expand
must be exercised)
✓ s is the standard deviation; you may have to calculate this as the square
root of variance
valuing a call option
Test yourself 1
Project 1 has a net present value of (Shs 10m). It will also develop an expertise
so that the company would be ready to penetrate the European market with
an improved product in four years’ time. The expected cost of the investment
in four years’ time is Shs 600m.
Currently the European project is valued at zero NPV nut management
believes that economic condition in four years’ time may change and the NPV
could be positive
The standard deviation is 30%, risk-free rate is 4% and the cost of capital is
10%
Required: Evaluate the value of this option to expand
Test yourself 2
ABC Ltd is a Ugandan based company located in Kampala. The company is
considering opening a new store in Northern region to exploit the available
business opportunity in the region. The new investment is estimated to cost
Shs 120m and generate a present value of cash receipt of Shs 100m.
These figures would suggest that the investment should be rejected. However,
if the first store is opened, then the company would gain the option to open
the second store (an option to expand or follow-on). Suppose this would have
the following details:

26 | H E D G I N G N O T E S B Y C P A G E O F F R E Y K A R U B A N G A
 Time (t) 5 years’ time
Estimated cost Shs 200m
Present value of net cash receipts Shs 150m
Volatility of cash flows (σ) 28%
Risk-free rate of return 6%
Required: Evaluate the value of this option to expand
Note that the option to delay is also a call option and should be valued in the
same way as the option to expand
Valuing a put option
Test yourself 5
Company X is considering an investment in a joint venture to develop high
quality office blocks to be let out to corporate clients. This project has a 30-
year life and is expected to cost Shs 900m and to generate an NPV of Shs
100m for company X.
The project manager has argued that this understates the true value of the
project because the NPV of Shs 100m ignores the option to sell company X ‘s
share back to its partner for Shs 400m at any time after the first ten years of
the project.
The standard deviation of the project is 45% and the risk-free rate is 5% per
year
Required: Ascertain the value of this option
Test yourself 6
An online DVD and CD retailer is considering investing Shs 20m on improving
its customer information and online ordering systems. The expectation is that
this will enable the company to expand by extending its range of products. A
decision will be made in one year’s time, when the directors have had a chance
to analyze the customer behavior competitors’ businesses in more details, to
assess whether the expansion is worthwhile
Preliminary estimates of the expansion programme have found that an
investment of Shs 50m in one year’s time will generate net receipt with a
present value of Shs 40m in the years thereafter. The project’s cash flows are
expected to be quiet volatile, with a standard deviation of 40%. The current
risk-free rate of interest is 5%
Required:
Advise the firm whether the initial investment in updating the systems is
worthwhile

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Test yourself 7
A firm is considering a project that is expected to cost Shs 50m. The project,
on an average, will generate after-tax cash flows of Shs 7.5m per annum over
its estimated economic life of 15 years. The firm’s cost of capital is 15%, and
the risk-free rate is 8%. The firm thinks that the cash flows will fluctuate and
variance of the value of the cash flows will be 0.0676. As an alternative to
taking up the project now, it is thinking of delaying the project.
Required: Advise the firm on what they should do?

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