Derivative Securities
Forwards and Options
PERTEMUAN 19-20
381 Computational Finance
Imperial College
London
Computational Finance 1/47
Topics Covered

Derivatives:
Forward Contracts, Options
Valuation
Option
techniques
Pricing Models
Binomial Option Pricing
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Introduction to Derivatives

security
whose payoff is explicitly tied to value or price of other financial security
that determines value of derivative is called underlying security
derivatives
arise when individuals or companies wish to buy asset or commodity in
advance to insure against adverse market movements;
effective tools for hedging risks – designed to enable market participants to
eliminate risk.
business dealing with a good faces risk associated with price fluctuations.
control that risk through use of derivative securities.
Example:

farmer can fix price for crop even before planting, eliminating price risk
an exporter can fix a foreign exchange rate even before beginning to
manufacture product, eliminating foreign exchange risk.

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Example 1: Derivatives

A forward contract to purchase 2000 pounds of sugar at 12 cents
per pound in 6 weeks.
The contract is a derivative security because its value is derived from
the price of sugar.


No reference to payoff - contract only guarantees purchase of sugar.

The payoff is implied and determined by the price of sugar in 6 weeks.
If price of sugar was 13 cents per pound, then contract would have a
value of 1 cent per pound,

Strategy: the owner of contract could

buy sugar at 12 cents according to the contract

then sell that sugar in the sugar market at 13 cents.
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Example 2: Derivatives
Assume that a contract gives one the right, but not the obligation to
purchase 100 shares of GM stock for $60 per share in exactly 3 months.

This is an option to buy GM.
Payoff of option will be determined in 3 months by the price of GM
stock at that time.


If GM is selling then for $70, the option will be worth $1000
The owner of option could at that time
purchase 100 shares of GM for $60 per share according to
option contract,


immediately sell those shares for $70 each
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Forward Contracts
Forward
contract is specified by a legal document, the terms of which bind
two parties involved to a specific transaction in the future.
on a priced asset is a financial instrument, since it has an intrinsic value
determined by the market for underlying asset
on a commodity is a contract to purchase or sell a specific amount of
commodity at specific time in future at a specific price agreed upon
today
Contract
is between two parties, buyer and seller.
buyer (long ): obligated to take delivery of asset & pay agreed-upon
price at maturity
seller (short): obligated to deliver asset & accept agreed-upon price at
maturity
Claims
are settled at defined future date; both parties must carry out their
side of agreement at that time.
Forward
price applies at delivery, negotiated so that initial payment is zero.
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Commodity Forwards

owner of commodities has to maintain their value,

requires storage (wheat, gold), feeding (live hogs), or
security (gold)

cost is called cost of carry

expressed as an annual percentage rate q

It is treated as a negative dividend.

the valuation formula for commodity forwards is
obtained as
F  S 0 e ( r  q )T
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Options
Holder
of forward contract is obliged to trade at maturity of contract
Unless
the position is closed before maturity, the holder must take possession of
the commodity, currency or whatever is the subject of the contract, regardless of
whether the price of the underlying asset has risen or fallen.
An
option gives holder a right to trade in the future at a previously agreed price
but takes away the obligations. If stock falls, we do not have to buy it after all.
An
option is a privilege sold by one party to another that offers the buyer the right
to buy or sell a security at an agreed-upon price during a certain period of time or
on a specific date.
Option holder has the right to chose to purchase a stock at a set-price within a certain period
Option writer has the obligation to fulfil the choice of the holder:

deliver the asset (for call option ) OR buy the asset (for put option )

receives the premium
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Vanilla Options: Call and Put
Call
option – right to buy particular asset for an agreed amount at specified time in future
Put
option – right to sell a particular asset for an agreed amount at a specified time in future
Example: Consider a call option on IBM stock which gives the holder the right to buy IBM stock
for an amount of $25 in one month. Today's stock price is $24.5.
amount
date
$25 which we can pay for stock is called exercise or strike price
on which we must exercise our option, if we decide to, is called expiry or expiration date
stock
(IBM ) on which option is based is known as underlying asset
premium
Let’s
is the amount paid for the contract initially
see what may happen over the next month until expiry!
Case 1: Suppose that nothing happens – stock price remains at $24.5. What do we do at expiry?
- exercise the option, handing over $25 to receive the stock.
- !!!! This is not a sensible decision since the stock is only worth $24.5.
- not exercise option or if really wanted the stock we would buy it in the stock
market for the $24.5.
Case 2: What happens if the stock price rises to $29?
- exercise the option, paying $25 for a stock, worth $29, and get a profit of $4
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Types of Options
Vanilla
Options – simplest ones
Call and Put

European Options – exercise only at expiry

American Options – exercise at any time before expiry

Asian Options – payoff depend on average price of
underlying asset over a certain period of time
Bermudan options – exercise on specific days,
periods


Exotic Options –more complex cash flow structures
Barrier, Digital, Lookback so on
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Options Valuation

procedure for assigning a market value to an option
market value of an asset is the value for which it could be sold in the
market today.

–
how much is the contract worth now, at expiry, before expiry?
–
no idea on stock price is between now & expiry but contract has value
at least there is no downside to owning option – contract gives you specific
rights but no obligations
–

value of contract before expiry depends on 2 things:
–
how high asset price is today – the higher asset today the higher we expect
the asset to be at expiry, more valuable we expect a call option
–
how long there is before expiry – the longer time to expiry, the more time
for the asset to rise or fall
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Payoff Diagram
 value of an option at expiry as function of underlying stock price

explains what happens at expiry, how much money option contract is worth
•right to buy asset at certain price within specific time
•buyers of calls hope that stock will increase before expiry
•buy and then sell amount of stock specified in contract
•right to sell asset at certain price within specific time
•buyers of puts hope that stock will decrease before expiry
•sell it at a price higher than its current market value
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Call Option Value at Expiry
Consider a call option with stock price ST and the exercise price E
at the expiry date T
Value of a call option is zero or the difference between the value of
the underlying and strike price, whichever is greater.

C  max ST  E,0
If ST  E holder can purchase a share more cheaply in market
than by exercising option

If ST  E holder receives one share from writer of the call option
for price of E

then make a profit of ST  E
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Put Option Value at Expiry
Consider a put option with stock price ST and the exercise
price E at expiry date T
Value
of a put option is zero or the difference between
strike price and value of the underlying, whichever is
greater.
P  max E  ST ,0
ST  E holder sells share to the writer of the put
option at price E and makes a profit of E  ST
If
If
ST  E holder prefers not to exercise the option
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Example
What are the payoffs of a call and put option at expiry if the exercise price
is £50 and the stock prices are £20, 40, 60, 80?
Stock price
Buy Call
Write Call
Buy Put
Write Put
20
Max(20-50,0) = 0
0
Max(50-20,0) = 30
-30
40
0
0
10
-10
60
10
-10
0
0
80
30
-30
0
0
Write an option  Sell an option
Buy call :
C  max ST  E ,0
Write call : - C   max ST  E ,0
Buy put :
P  max E  ST ,0
Write put : - P   max E  ST ,0
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Example
Suppose the price of IBM is $666 now. The cost of a 680 call option with
expiry in 3 months is $39. You expect the stock to rise between now and
expiry. How can you profit if your prediction is right?
Suppose that you
buy the stock for $666.
Assume that just before expiry, the stock has risen to $730.
Profit is $64 and the investment rises by
Suppose that you
730  666
100  9.6%
666
buy the call option for $39.
At expiry, you can exercise the call : pay $680 to receive something
worth $730. You have paid $39 and gain $50.
Profit is $11 per option. In percentage the profit is
value of asset at expiry - strike - cost of call
100
cost of call
730  680  39

100  28%
39
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Profit 
Put-Call Parity
Suppose
that you buy one European call option with strike price of E and you
write one European put option with the same strike. Both options expire at T
and today’s date is t.
At
T, payoff of portfolio of call and put options is sum of individual payoffs.
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Put-Call Parity at T
payoff of portfolio of call & put options
max ST  E ,0  max E  ST ,0  ST  E
C  P  ST  E
Type
Call
Option
Put
Option
Portfolio
Value
ST  E
ST  E
C  max ST  E, 0
0
ST  E
 P   max E  ST , 0
 ( E  ST )
0
ST  E
ST  E
Option Value
C-P
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Option Pricing Models
Approaches
to option pricing problem based on different
assumptions about market, dynamics of stock price behaviour
Theories
based on the arbitrage principle,
applied
The
when dynamics of underlying stock take certain forms
simplest of these theories is based on binomial model of
stock price fluctuations

widely used in practice since it is simple and easy to calculate

approximation to movement of real prices

generalizes one period “up-down” model to multi-period setting
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