Derivative Market
Futures
Forwards
Options
What is in today’s lecture?
Introduction to Derivative
Forward and Futures
Financial
Derivatives
Various aspects of forwards
Pricing of forward contracts
Options
Forward Prices and Spot Prices
How forward price are determined?
In other words, how to price future
contracts?
 See example on the next slide and draw
conclusion for yourself
 1.
2
 3.
 4.
Forward Prices and Spot Prices
Suppose that the spot price of gold is $1000 per
ounce and the risk-free interest rate for
investments lasting one year is 5% per annum.
What is a reasonable value for the one-year
forward price of gold?
• Suppose first that the one-year forward price is
$1300 per ounce. A trader can immediately take
the following actions:
• 1. Borrow $1000 at 5% for one year.
• 2. Buy one ounce of gold.
• 3. Enter into a short forward contract to sell the
gold for $1300 in one year
• The trader earns a riskless profit of?
•

The trader pays a total price for one
ounce of gold = 1000 + (1000x5%
interest) = $1050
The trader sell the gold for Rs. 1300
 His riskless profit is = 1300-1050 = $250


The example shows that $1300 was too
high a forward price
Pricing aPrices
forward
Forward
andcontract
Spot Prices
Forward price = So + CC
 Where So is the spot/current/cash price
today
 CC is the cost of carry ( in the previous
example CC is the financing cost /
interest paid for buying gold)
 F = $1000 + (1000x.05) = 1000(1+.05)
 = $1050

The case of continuous
compounding
Like in time value of money concept,
when continuous compounding is the
rT
assumption, the interest rate formula
ie
becomes:
 Where e = 2.71828
 Forward price for a non-dividend paying asset
is

F  Soe
rT
Example
Consider a four-month forward contract to buy a zerocoupon bond that will mature one year from today. The
current price of the bond is Rs.930. (This means that
the bond will have eight months to go when the
forward contract matures.) Assume that the rate of
interest (continuously compounded) is 6% per annum.
 T = 4/12 = .333
 r = 0.06, and So = 930. The forward price,


F  Soe
rT
 930 e
. 06 *. 333
 948 . 79
For dividend or interest paying
securities

Since the forward contract holder does not
receive dividend/interest on the underlying
asset, but the present price So reflects the
future income from the asset, the present value
of dividends/interest should be deducted from
So while calculating Forward price
F  ( S o  l )e

rT
l is the present value of future dividends/
interest
Example
Consider a 10-month forward contract on
Nishat Mills Ltd (NML) stock with a price of
Rs.50. Assume that the risk-free rate of interest
(continuously compounded) is 8% per annum
for all maturities. Also assume that dividends of
is 0.75 per share are expected after three
months, six months, and nine months.
 The present value of the dividends

 0 . 75 e
 0 . 08 * 3 / 12
 0 . 75 e
 0 . 08 * 6 / 12
 0 . 75 e
 0 . 08 * 9 / 12
 2 . 162
Example continued..

The forward price of the contract is
F  ( S o  l )e
rT
F  ( 50  2 . 162 ) e
0 . 08 *10 / 12
 Rs . 51 . 14
Assets with storage costs
Storage costs can be regarded as negative
income
 If U is the present value of all the storage
costs that will be incurred during the life
of a forward contract, then the forward
price is given by:

F  ( S o  U )e
rT
Example
Consider a one-year futures contract on
gold. Suppose that it costs $2 per ounce
per year to store gold, with the payment
being made at the end of the year. Assume
that the spot price is $450 and the riskfree rate is 7% per annum for all
maturities.
 This corresponds to r = 0.07, and S0 =
450, T=1
 0 . 07 *1
U  2e
 1 . 865

Example continued..
F  ( S o  U )e
rT
F  ( 450  1 . 865 ) e
0 . 07 *1
 484 . 63
Forward price with known yield



If an asset pays income that can be
expressed as a present yield on per annum
continuous basis, the forward price of the
asset can be determined as,
F  Soe
( r  q )T
q = average yield per annum the asset
underlying the forward contract
 Consider a six month forward contract on
an asset that is expected to provide a yield
of 3.96% p.a. with continuous compounding.
Then price of a six month contract is????
Valuing the forward Contracts
The value of a forward contract at the time
it is first entered into is zero.
 At a later stage, it may prove to have a
positive or negative value.
 we suppose
 K is the delivery price for a contract that
was negotiated some time ago, the delivery
 date is T years from today, and r is the T-year
risk-free interest rate. The variable Fo is
 the forward price that would be applicable if
we negotiated the contract today.

Valuing the forward Contracts
f:Value of forward contract today
 At the beginning of the life of the forward
contract, the delivery price, K, is set equal
to the forward price, Fo, and the value of
the contract, f, is 0
 Here K remains constant but Fo keeps
changing due to changing So therefore the
value of a forward contract becomes
either positive or negative.

Valuing the forward Contracts
Value of a long forward contract is,
Value of a short forward contract is,
Example

A long forward contract on a non-dividend-paying
stock was entered into some time ago. It
currently has 6 months to maturity. The risk-free
rate of interest (with continuous compounding) is
10% per annum, the stock price is $25, and the
delivery price is $24. In this case,. So= 25, r =
0.10, T = 0.5, and K = 24. From the 6-month
forward price, Fo, is given by

the value of the forward contract is
Pricing an option
Pricing an option means “finding the
amount that you pay to have the
right/option”
 Or pricing an option is finding the “fair”
value of the agreement
 “fair” means the value that precludes any
arbitrage opportunity
 Let’s use an example to find the value of
call option at expiration

Intrinsic Values
If a call is in the money, it will have
positive intrinsic value.
 Intrinsic value is equal to the difference
between current price and exercise price
 Example: FFC current price is Rs.65, a call
option on FFC is Rs.60, the option is in
the money
 IV = 65-60 = 5

Pricing call option at expiration
Suppose you own a call option on one
share of PTCL stock with a strike price of
Rs. 30
 Suppose that at expiration one share of
PTCL trades at Rs. 35
 What should be the minimum price of
this call now?

Price of call option
Minimum price should be
 35 – 30 = 5 OR
 Market price – Exercise Price Or
 S – E which is equal to Intrinsic value
 In other words, price of call option can
never be lower than intrinsic value

The Black-Scholes Model for valuing
options

Black-Schole model uses the five variables
to value non-dividend paying call option
•
•
•
•
•
S = Current stock price
E = Exercise price of call
r = Annual risk-free rate of return,
continually compounded
δ2 =Variance (per year) of the
continuous return on the stock
t = Time (in years) to expiration date
Determinants of option prices

1. Exercise Price
An increase in the exercise price reduces the
value of the call
 In the previous example, if exercise price was
32 instead of 30, the value of the call option will
be
 35 – 32 = 3 instead of 5

Determinants of option prices
 Expiration
date : the higher the
maturity, the higher will be the premium
of both call and put option
 The reason is that the option holder has
more time to exercise the option
 The extra time increases the probability
of profit for option holder and loss for
option writer
Determinants of option prices
Stock Price: Other things being equal, the
higher the stock price, the more valuable the
call option will be.
 In our earlier example, if current market
price of PTCL share was Rs. 40, then option
value would have been
 40 – 30 = 10 in stead of 5
 If S>E -------high profit if call option is
exercised-------high premium for call options
 If S>E------- loss if put option is exercised----low premium for put options

Determinants of option prices
The Variability of the Underlying
Asset prices
 The greater the variability of the
underlying asset, the more valuable both
the call and put option will be
 Greater variability can subject the option
writer to higher losses, this is why option
writer will need higher premium for giving
such option

Determinants of option prices
•
•
•
•
•
The Interest Rate
Call prices are also a function of the level of
interest rates.
Buyers of calls do not pay the exercise price
until they exercise the option.
The ability to delay payment is more
valuable when interest rates are high and
less valuable when interest rates are low.
Thus, the value of a call is positively related
to interest rates.
Difference between futures and
option contracts

Initial investment on the contract are
different
◦ margin requirements in forward contracts
◦ Premium paid on the option contracts


In future contracts money is needed on daily
basis for marking to market the contract
whereas in options money is only needed
when the option is exercised
Profits from the future contracts are linear
whereas profits from the option contracts
are non linear