Derivatives Primer

What is a Derivative?
A derivative is a contract between two parties for the sale or purchase
of an underlying asset at a fixed price, at the maturity of the contract.
Categories of
Derivatives
Forward
Commitments
Contingent
Claims
A forward commitment is a legally
binding promise to perform some action
in the future.
A contingent claim is a claim (to a
payoff) that depends on a particular
event.
• Forward Contracts
• Futures contracts
• Swaps
• Options
• Credit based derivatives

Some other categorization of Derivatives:
Exchange traded :
• These are traded on
registered exchanges.
The contracts are
standardized and
backed by a
clearinghouse
Over-the-counter:
• Custom instruments
traded/created by
dealers in a market with
no central location. They
are largely unregulated
markets, and each
contract is with a
counterparty, which
exposes the owner of a
derivative to default risk
(when the counterparty
does not honor their
commitment).
• Eg: Forwards, Swaps
Based on Mode of Trading Based on Manner of settlement
Cash Settled :
• The net difference
between the positions
of the buyer and
seller is simply
exchanged.
Physically Settled
• The underlying asset
is exchanged at the
agreed price as per
the derivative
contract.

Forward Contract
What is it ?
• In a forward
contract, one party
agrees to buy and
the counterparty to
sell a physical or
financial asset at a
specific price
(forward price) on a
specific date in the
future.
Why is it used ?
• A forward contract
can be used to
reduce or eliminate
uncertainty about
the future price of
an asset (hedge) or
to speculate on
movements in asset
prices.
How is it
traded?
• The contracts are
traded over the
counter (i.e. outside
the stock exchanges,
directly between the
two parties) and are
customized
according to the
needs of the parties.
Hence, they have an
inherent counter-
party risk.

The party to the forward contract who agrees to buy the financial or
physical asset has a LONG FORWARD POSITION and is called the LONG
The party to the forward contract who agrees to sell or deliver the
asset has a SHORT FORWARD POSITION and is called the SHORT
Neither party makes
payment on initiation
of contract
If the price of the asset :
Increases Decreases
Value of the long position will increase &
that of the short position will decrease
Value of the long position will decrease & that of the
short position will increase
Eg: ‘X’ enters into a forward to buy an asset after 3
months @ Rs. 100. If the asset’s price after say 2
months is Rs. 110 then ‘X’ stands to benefit as he can
get the asset for Rs. 100
Eg: ‘Y’ enters into a forward to buy an asset after 3
months @ Rs. 100. If the asset’s price after say 1
months is Rs. 95 then ‘X’ stands to lose as he will have
to buy that asset for Rs. 100 under the contract.

Upon Maturity if :
Market Price > Forward Price Market Price < Forward Price
Long will receive from short an amount equivalent to :
Market Price of Underlying (less) Forward price
Short will receive from Long an amount equivalent to :
Forward price (less) Market Price of Underlying

Valuing Forwards
The valuation of derivative securities is based on a no-arbitrage condition.
What is Arbitrage?
Arbitrage refers to a transaction in which an investor purchases one asset or portfolio of
assets at one price and simultaneously sells an asset or portfolio of assets that has the same
future payoffs, regardless of future events, at a higher price, realizing a risk-free gain on the
transaction.

With derivative securities, the risk of the derivative is entirely based on the risk of the
underlying asset.
So, we can construct a portfolio consisting of the underlying asset and a derivative based on
it that has no uncertainty about its value at some future date (i.e., a hedged portfolio).
Because the future payoff is certain, we can calculate the present value of the portfolio as
the future payoff discounted at the risk-free rate. This will be the current value of the
portfolio under the no-arbitrage condition, which will force the return on a risk-free
(hedged) portfolio to the risk-free rate.
How is Arbitrage model used for pricing
forwards?

Simplifying the Valuation !
Let us say a person “Z” buys a stock @ Rs. 100 (i.e. invests Rs. 100). He then immediately enters into a Short forward
position (commits to sell in future) @ Rs. 110 after 1 year (This will have no immediate cash flow)
“Z” thereby locks in a profit of Rs. 10 as soon as he enters in these transactions. It is a riskless profit & hence should
return only risk-free rate.
We can use the above to arrive at a forward price given that we know the Risk free rate and Spot Price of asset.
That is to say : Rf = (FoT – So) / So
where, So : Spot Price ; FoT : Forward Price of Contract expiring after Time T
; Rf : Risk Free rate
Fo = S0 ( 1 + Rf)

During a forward contract’s life, at time t < T, its value is the spot price of the asset
minus the present value of the forward price:
Vt(T) = St – F0(T) / {(1 + Rf)^(T–t)}

So far, we have assumed that there were no benefits of holding the asset and no costs of holding the
asset, other than opportunity cost of the funds to purchase the asset (the risk-free rate of interest). There
may be additional costs of owning an asset, such as storage and insurance costs.
There may also be benefits to holding the asset, both monetary and nonmonetary.
F0(T) = [S0 + PV0(cost) − PV0(benefit)](1 + Rf)T
Vt(T) = St + PVt(cost) – PVt(benefit) – F0(T) / (1 + Rf)T − t
The no-arbitrage forward price is lower to the extent the present value of any benefits is greater, and
higher to the extent the present value of any costs incurred over the life of the forward contract is
greater.
Value of the forward at any point in time t is :

Futures Contract
Futures contracts are similar to forward contracts in terms of payoffs and pricing, the only
difference is that forwards are private contracts and typically do not trade whereas futures are
exchange-traded and hence standardized.
A clearinghouse is the counterparty to all futures contracts. Forwards are contracts with the
originating counterparty and, therefore, have counterparty (credit) risk.
Margin is money that must be deposited by both the long and the short as a performance
guarantee prior to entering into a futures contract.
Each day, the margin balance in a futures account is adjusted for any gains and losses in the
value of the futures position based on the new settlement price, a process called the mark to
market or marking to market.

Options
An option contract gives its owner the right, but not the obligation, to either buy or sell an
underlying asset at a given price (the exercise price or strike price).
The option buyer can choose whether to exercise an option, whereas the seller is obligated to
perform if the buyer exercises the option.
The owner of a call option has the right to purchase the underlying asset at a specific price for
a specified time period.
The owner of a put option has the right to sell the underlying asset at a specific price for a
specified time period.
The buyer of the option pays an amount called “Premium” to the seller for entering into the
contract.

Position Contract Action Underlying Action
Long Call
Buy the contract by paying premium
Option to buy the underlying
Short Call
Sell/write the contract & receive
premium
Obligation to sell the underlying
Long Put Buy the contract by paying premium Option to sell the underlying
Short Put
Sell/write the contract & receive
premium
Obligation to buy the underlying

An option holder will exercise the option only if they stand to benefit from it. Such an
option is called an In-The-Money option.
Call Option
Moneyness Relationship to Stock
In-the-money Stock price > Strike Price
At-the-money Stock price = Strike Price
Out-of-the-money Stock price < Strike Price
Put Option
Moneyness Relationship to Stock
In-the-money Stock price < Strike Price
At-the-money Stock price = Strike Price
Out-of-the-
money
Stock price > Strike Price
An option holder will let the Out of the money option expire.

Intrinsic value of a call option = max[0, S – X]
Intrinsic value of a put option = max[0, X – S]
where,
S : Spot Price of Asset
X : Exercise Price of Asset

Consider a put option with a $5 premium. The buyer pays $5 to the writer.
When the price of the stock at expiration is greater than or equal to the $50 strike price, the put
has zero value. The buyer of the option has a loss of $5, and the writer of the option has a gain of
$5.
If the stock’s price falls below $50, the buyer of the put option starts to gain (breakeven will come
at $45, when the value of the stock equals the strike price less the option premium). However, as
the price of the stock moves downward, the seller of the option starts to lose (negative profits will
start at $45, when the value of the stock equals the strike price less the option premium).
Overview of how it works
Market Price Gain/(loss) for Long* Gain/(loss) for Short*
55 -5 5
50 -5 5
48 -3 3
45 0 0
40 5 -5
*ignoring the time value of money

OPTION VALUATION AND PUT CALL PARITY
The intrinsic value (or exercise value) of an option the maximum of zero and the
amount that the option is in the money
Prior to expiration, an option has time value in addition to any intrinsic value. The
time value of an option is the amount by which the option premium (price) exceeds
the intrinsic value and is sometimes called the speculative value of the option
Option premium = Intrinsic value + Time value
When an option reaches expiration, there is no time remaining and the time value is zero. This
means the value at expiration is either zero, if the option is at or out of the money, or its
intrinsic value, if it is in the money

Factors determining Option Value
• For call options, the higher the price of the underlying, the greater its intrinsic
value and the higher the value of the option
Price of the underlying asset :
• : A higher exercise price decreases the values of call options and a lower
exercise price increases the values of call options
Exercise price
• An increase in the risk-free rate will increase call option values, and a decrease
in the risk-free rate will decrease call option values
Risk-free rate of interest :
• An increase in the volatility of the price of the underlying asset increases the
values of both put and call options.
Volatility of the underlying :
• : Longer time to expiration effectively increases expected volatility and
increases the value of a call option.
Time to expiration
• : If there are benefits of holding the underlying asset (dividend or interest
payments on securities or a convenience yield on commodities), call values are
decreased and put values are increased
Costs and benefits of holding
the asset

Put-call parity is based on the no-arbitrage principle that portfolios with identical payoffs must sell for
the same price. A fiduciary call (composed of a European call option and a risk-free bond that will pay
X at expiration) and a protective put (composed of a share of stock and a long put) both have identical
payoffs at maturity regardless of value of the underlying asset at expiration.
Based on this fact and the law of one price, we can state that, for European options:
C + X / (1 + Rf) T = S + P
where,
C : Call premium
P : Put Premium
X : Strike Price
Rf : Risk free rate
T : Time to Expiration
That is, the value of a call at X and the present value of the exercise price must equal the current asset price
plus the value of a put or there would be an opportunity for profitable arbitrage

SWAPS
❑ Swaps are agreements to exchange a series of payments on periodic settlement dates
over a certain time period.
❑ At each settlement date, the two payments are netted so that only one (net) payment is
made. The party with the greater liability makes a payment to the other party.
❑ The length of the swap is termed the tenor of the swap, and the contract ends on the
termination date.
https://www.investopedia.com/articles/optioninvestor/07/swaps.asp

Popular types of SWAPS
Interest Rate Swaps
❖ In a plain vanilla interest rate swap, one party makes fixed-rate interest payments on a notional principal amount
specified in the swap in return for floating-rate payments from the other party.
Eg : Company A and Company B enter into a five-year swap with the following terms:
Company A pays Company B an amount equal to 6% per annum on a notional principal of
$20 million.
Company B pays Company A an amount equal to one-year LIBOR + 1% per annum on a
notional principal of $20 million
Company A will pay Company B $1,200,000 ($20,000,000 * 6%). On Dec. 31, 2006, one-
year LIBOR was 5.33%; therefore, Company B will pay Company A $1,266,000
i.e. ($20,000,000 * (5.33% + 1%))
(The payments will be netted off)
❖ In a basis swap, one set of floating-rate payments which are based on a reference rate is swapped for another
which is based on some other reference rate

Currency Swaps
A currency swap, sometimes referred to as a cross-currency swap, involves the exchange of interest—and
sometimes of principal—in one currency for the same in another currency
Example:
Company C, a U.S. firm, and Company D, a European firm, enter into a five-year currency swap for $50 million. Let's
assume the exchange rate at the time is $1.25 per euro (e.g., the dollar is worth 0.80 euro). First, the firms will
exchange principals.
Company C pays $50 million, and Company D pays 40 million euros. This satisfies each company's need for
funds denominated in another currency (which is the reason for the swap).
Suppose, the agreed-upon dollar-denominated interest rate is 8.25%, and the euro-denominated interest rate is
3.5%
As with interest rate swaps, the parties will actually net the payments against each other at the then-prevailing
exchange rate.

A credit default swap (CDS) is an insurance contract against default. A bondholder pays a series
of cash flows to a credit protection seller and receives a payment if the bond issuer defaults.
Credit-default Swap
Equity Swap
An equity swap is similar to an interest rate swap, but rather than one leg being the "fixed" side,
it is based on the return of an equity index.
For example, one party will pay the floating leg (typically linked to LIBOR) and receive the
returns on a pre-agreed-upon index of stocks relative to the notional amount of the contract.

Forward Rate Agreements
A forward rate agreement (FRA) is a derivative contract that has an interest rate, rather than
an asset price, as its underlying. An FRA permits an investor to lock in a certain interest rate
for borrowing or lending at some future date.
One party will pay the other party the difference (based on an agreed-upon notional contract
value) between the interest rate specified in the FRA and the market interest rate at contract
settlement.
Consider an FRA that will, in 30 days, pay the difference between 90-day LIBOR and the 90-day rate specified in the
FRA (the contract rate). A company that expects to borrow 90-day funds in 30 days will have higher interest costs if
90-day LIBOR increases over the next 30 days.
A long position in an FRA (to “borrow” at the fixed rate & lend at floating rate) will receive a payment at settlement
(based on floating rate reference rate) that will offset the increase in borrowing costs from the increase in 90-day
LIBOR.
Conversely, if 90-day LIBOR decreases over the next 30 days, the long position in the FRA will make a payment to the
short in the amount that the company’s borrowing costs have decreased relative to the FRA contract rate.
Firms can, therefore, reduce or eliminate the risk of (uncertainty about) future borrowing costs using an FRA.

Formula & calculation of FRA payments

Derivatives Primer

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