Basics of Option

HUANG Dashan

Lee Kong Chian School of Business

Term 2 2023/24

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European Call Option

A European call option gives the buyer the right, but not the obligation,
to buy a given amount of a given asset at a given price on a given date.
The call seller1 has the obligation to sell if exercised by the buyer
- underlying asset: S0
- exercise (or strike) price: K
- expiration (or maturity) date: T
- contract size (e.g., 100 shares for a stock call)
- option price or premium: c0

1
The seller of an option is also called the writer of the option.
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An Example

Suppose today is January 20. A March 20 European call 125 on IBM is traded
at $11.45.
Today you pay c0 = $11.45 to buy a right that allows you to buy a share
of IBM stock on March 20 at price K = $125.
On March 20, you have two choices.
1 If ST > 125, you exercise your option by buying one share from the seller of the
call at price 125. Then you sell the one share to the spot market and, therefore,
your cash flow will be CFT = ST − 125 > 0.
2 If ST ≤ 125, you give up your right, and your cash flow will be CFT = 0.
Hence, your cash flow can be written as CFT = max(ST − 125, 0).
It is you, the buyer, to determine to exercise the call or not.

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European Put Option

A European put option gives the buyer the right, but not the obligation,
to sell a given amount of a given asset at a given price on a given date.
The seller of a put option has the obligation to buy if exercised by the
buyer.
What is the critical difference between an option contract and a futures
(forward) contract?

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An Example

A March 20 European put 125 on IBM is traded at $2.24.

Today you pay p0 = $2.24 to buy a right that allows you to sell a share of
IBM stock on March 20 at price K = $125.
On March 20, you have two choices.
1 If ST < 125, you exercise the option by selling one share to the seller of the put at
price 125, and your cash flow will be CFT = 125 − ST > 0.
2 If ST ≥ 125, you don’t exercise the option (give up your right), and your cash
flow will be CFT = 0.
Hence, your cash flow can be written as CFT = max(125 − ST , 0).
It is you, the buyer, to determine to exercise the put or not.

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American Options

An American call option gives the buyer the right, but not the obligation,
to buy a given amount of a given asset at a given price on or before a
given date.
Example: A March 20 American call 200 on Facebook is trading at $17.10.
An American put option gives the buyer the right, but not the obligation,
to sell a given amount of a given asset at a given price on or before a
given date.
Example: An March 20 American put 60 on CISCO is traded at $12.60.

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European vs. American Option

An American option can be exercised on or before the expiration date,

while a European option can only be exercised on the expiration date.
Most of the exchange-traded options are American options.
European options are easier to analyze, and some of the properties of
American options are frequently deduced from European options.
With other things equal, is an American option worth more or less than a
European option?

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Margin

No margin required for buying an option, but cannot borrow to buy either.
For selling naked options,2 margin is required. The amount depends on
the composition of the position.

2
A naked option is a trading position where the seller of an option contract does not own any, or
enough, of the underlying security to act as protection against adverse price movements.
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Payoff and Profit

Payoff/cash flow of a long position is CF0 = −price0 at 0 and CFT at T .

The profit of a position is equal to the aggregate cash flow across time,

profit = CFT + CF0 = CFT − price0 (for buyer)

= −(CFT + CF0 ) = −CFT + price0 (for seller)
1 For a stock buyer, profit = ST − S0 (suppose no dividend payment);
2 For a stock short seller, profit = −ST + S0 .

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European Call Option Payoff

Let ST be the spot price of the underlying asset on the maturity date and K
be the strike price of the option. Then
Payoff to a call buyer cT = max(ST − K , 0)
Payoff to a call seller − cT = − max(ST − K , 0)
An example: A March 20 IBM European call 125 is traded at $11.45.

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European Put Option Payoff

For a put option, following the same argument as in the call option case, we
can show that
Payoff to a put buyer pT = max(K − ST , 0)
Payoff to a put seller − pT = − max(K − ST , 0)
An example: A March 20 IBM Put 125 is traded at $2.24.

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European Option Profit

Let c0 be the call price. Then

Profit to a call buyer = cT − c0 = max(ST − K , 0) − c0

Profit to a call seller = −cT + c0 = − max(ST − K , 0) + c0

Let p0 be the put price. Then

Profit to a put buyer = pT − p0 = max(K − ST , 0) − p0

Profit to a put seller = −pT + p0 = − max(K − ST , 0) + p0

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Profit Diagram for A Call

For the IBM example, c0 = 11.45, K = 125.

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Profit Diagram for A Put

For the IBM example, p0 = 2.24, K = 125.

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Some Terminologies

The exercise value is ST − K for a call and K − ST for a put.

If exercise value is positive, then we say that the option is in the money. If
it is negative, then the option is out of the money. If it is zero, then the
option is at the money.
Call Put
In ST > K ST < K
At ST = K ST = K
Out ST < K ST > K
Investors also use S0 to describe if an option is in, at, or out of the money.

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Notations

c0 : European call price

C0 : American call price
p0 : European put price
P0 : American put price
I : the present value of the dividends paid by the underlying asset during
the remaining life of an option
S0 , ST : the spot price of the underlying asset
T : the time to maturity
r : the interest rate

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Strike Price Condition
Let c0 (K1 ) and C0 (K1 ) be the European and American call prices at time 0
with strike K1 .
Suppose K1 < K2 .
1 European c0 (K1 ) > c0 (K2 ) and American C0 (K1 ) > C0 (K2 ). In words,

lower strike call has higher price.

2 European p0 (K1 ) < p0 (K2 ) and American P0 (K1 ) < P0 (K2 ). In words,

higher strike put has higher price.

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What if c0 (K1 ) = 1 and c0 (K2 ) = 2 with K1 < K 2?

Since c0 (K1 ) and c0 (K2 ) violate the strike price condition, there must exist
an arbitrage opportunity.
However, we do not know which one is mispriced. It is possible that both
are overpriced or underpriced, and also possible that one is overpriced and
the other one is underpriced. So we talk about their relative mispricing.
If c0 (K1 ) is fairly priced, c0 (K2 ) is overpriced. In contrast, if c0 (K2 ) is
fairly priced, c0 (K1 ) is underpriced.
Example
Suppose that StarHub is overvalued by $0.5 and SingTel is overvalued by $1. In
the long run when new information comes out, StarHub’s price will drop by $0.5
whereas SingTel’s price will drop by $1.
Today, if you think StarHub is fairly priced, then SingTel would be overvalued by
$0.5. Instead, if you think SingTel is fairly priced, then StarHub would be
undervalued by $0.5.
If you buy StarHub and sell SingTel today, you will make a profit of $0.5 when
both prices converge to their fundamentals.
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What if c0 (K1 ) = 1 and c0 (K2 ) = 2 with K1 < K 2? Cont’d

By using “buy low sell high”, you should buy c0 (K1 ) and sell c0 (K2 ).
1 At time 0, CF0 = −price0 = −(c0 (K1 ) − c0 (K2 )) = −(1 − 2) = 1.
2 At time T ,

CFT = cT (K1 ) − cT (K2 ) = max(ST − K1 , 0) − max(ST − K2 , 0)


 0 − 0 = 0, ST ≤ K1 ;
= (ST − K1 ) − 0 = ST − K1 , K1 < ST ≤ K2 ;
(ST − K1 ) − (ST − K2 ) = K2 − K1 , ST > K2 .


This is an arbitrage because your CF0 = 1 > 0 and CFT ≥ 0. You will
make an at least $1 profit for sure.3

3
We do not require the aggregate CF at 0 equal to 0, because money changes hands at 0.
Making it equal to 0 is easy but redundant.
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What if p0 (K1 ) = 2 and p0 (K2 ) = 1 with K1 < K 2?

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An Example

Suppose the current price of a non-dividend payment stock is $40. Consider

the following American call prices:

K Oct. Nov. Dec.

35 1 6 8
40 2 5 7
45 4 3 5

Which two quotes violate the strike price condition?

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Put-Call Parity only apply to European options

Since Portfolios A and B have the same terminal cash flow, they should have
the same price today.
c0 + Ke −rT = p0 + S0 .
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Put-Call Parity only apply to European options

Let c0 be the price of a European call with strike K and maturity T , and
let p0 be the price of the otherwise identical European put. Let I be the
present value of the income from holding the underlying asset during the
life of the option.
Consider the following two portfolios:
A: Long one European call, costing c0 .
B: Long one European put option, long one share of the stock, and borrow an
amount of cash Ke −rT for time to maturity T and an amount of cash I for time
to maturity t (at which the stock pays dividend),4 costing p0 + S0 − Ke −rT − I .

CF0 = −price0 = −(p0 + S0 − Ke −rT − I ),

CFt = D − Ie rt = 0,
CFT = pT + ST − K .
4
This loan is to make the aggregate cash flow zero at t, as the call’s cash flow is zero.
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Put-Call Parity, Cont’d

Cash flows at T of these two portfolios

Portfolio ST ≤ K ST > K
A 0 ST − K
B (K − ST ) + ST − K = 0 ST − K

Therefore the put-call parity:

c0 = p0 + S0 − Ke −rT − I

Portfolio B is also called a synthetic call. How to form a synthetic put?

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