Financial Market
Derivatives Market  Vanilla Options  Call option
Financial Market
Derivatives Market  Vanilla Options  Call option
Participans:
The deal:
For a specific period of time
HOLDER WRITER
HOLDER will have a RIGHT to buy
some underling security for a price specified today.
WRITER will have an OBLIGATION to deliver
the underlying security.
Options features:
Underlying security = what security is going to be traded.
Multiplier (N) = the quantity of the underlying asset.
Expiration date (Maturity) (T) = the day when the options expiry.
Strike (Exercise price) (K) = price at which the underlying transaction will occur upon exercise.
CALL option = holder gets the right to buy = when the maturity comes, holder has the right
to demand from option s writer to sell him particular security at price specified in contract.
Financial Market
Derivatives Market  Vanilla Options  Call option
Today Maturity
Holder can (but doesn t have to) finalize
Holder gets the right for a
transaction.
future transaction.
Writer is obliged to finalize transaction
but only if holder calls for that.
Example 1:
A is signing a contract with B specifying that in a one year from now he will have the right to
buy 1 stock of X company for a 10$ (the price specified in the contract).
Today After 1 year
A can buy 1 stock of X company.
A gets the right to buying 1 stock of
B must sell that 1 stock, if A will call for that.
X company from B but after 1 year.
Financial Market
Derivatives Market  Vanilla Options  Call option
Will CALL s holder execute his rights ??
Data from example 1: Underlying security = Stock X
N = 1
T = 1 year
K = 10 $
Let s assume that stock X price after T is going to be:
CALL s holder has the right to buy stock X for 10 $ (stock X is listed with higher
a) ST = 20 $
price at this moment)  Holder executes his right and in the same moment
b) ST = 15 $
sells stock X at the market for a higher price.
c) ST = 8 $
CALL s holder will not execute his right while he can buy the same stock at the
market much cheaper. Hence CALL option is worthless.
d) ST = 5 $
CALL s holder will not execute his right while he can buy the same stock at the
e) ST = 10 $
market at the same price. Hence CALL option is worthless.
Financial Market
Derivatives Market  Vanilla Options  Call option
CALL option HOLDER will execute his right when:
ST > K
What will be CALL s holder payoff (if Strike = 10 $):
a) ST = 20 $ CALL s holder executes his right: he buys stock X for 10 $ from writer.
At the same moment, he sells this security for:
b) ST = 15 $
a) 20 $ b) 15 $
Payoff: a) 10 $ b) 5 $
c) ST = 8 $ CALL s holder will not execute his right, while he can buy the same stock from
market for:
d) ST = 5 $
c) 8 $ d) 5 $
Payoff: c) 0 $ d) 0 $
e) ST = 10 $ CALL s holder will not execute his right, while he can buy the same stock from
market for the same price.
e) 0 $
Payoff:
Financial Market
Derivatives Market  Vanilla Options  Call option
CALL s holder payoff function:
In general: Data from example 1 (K = 10 $):
Payoff
Payoff
10 $
5 $
ST ST
0
0
5 $ 10 $ 15 $ 20 $
K
Payoff function for CALL s holder =
MAX(ST  K , 0)
Financial Market
Derivatives Market  Vanilla Options  Call option
CALL s writer payoff function:
Data form example 1: Underlying security = Stock X
Underlying security = Stock X
N = 1
N = 1
T = 1 year
T = 1 year
K = 10 $
K = 10 $
Let s assume that stock X price after T is going to be:
Writer will be asked by the holder to sell the security. Hence, writer will buy
a) ST = 20 $
stock X from the market (paying a) 20 $ b) 15 $) and will sell this stock
b) ST = 15 $
immediately to holder (asking 10 $).
Payoff: a) -10 $ b) -5 $
c) ST = 8 $
d) ST = 5 $
Writer won t be called to sell the stock X, while holder will not execute his right.
e) ST = 10 $
Payoff: c) 0 $ d) 0 $ e) 0 $
Financial Market
Derivatives Market  Vanilla Options  Call option
CALL s writer payoff function:
In general: Data from example 1 (K = 10 $):
Payoff
Payoff
ST ST
0
0
5 $ 10 $ 15 $ 20 $
K
- 5 $
- 10 $
Payoff function for CALL s writer =
- MAX(ST  K , 0)
Financial Market
Derivatives Market  Vanilla Options  Call option
Holder ßð Ä…ð Writer
Payoff
Holder: has nothing to lose  can make a big profit.
Writer: he is risking a lot - got nothing to gain.
HOLDER
This is definitely unfair.
ST
0
K This is why when the contract is signed writer
demands a single payment (Premium) from
Payoff
holder for giving him the right to buy some security in
the future for a price specified today.
K
0 ST
c  CALL option price
(Premium, which needs to be paid by the holder to the
WRITER
writer for obtaining the right for a future transaction).
Financial Market
Derivatives Market  Vanilla Options  Call option
Holder s payoff - premium = Holder s profit
What will be CALL s holder profit (K = 10 $, c = 5 $) assuming that the stock X price is going to be:
a) ST = 20 $
CALL s holder executes his right an gain: a) 10 $ b) 5 $. Unfortunately, he has
paid a 5 $ premium to a writer, hence the final profit is going to be:
b) ST = 15 $
Profit: a) 5 $ b) 0 $
c) ST = 8 $
CALL option is worthless, but holder has paid 5 $ premium, hence the profit is:
d) ST = 5 $
Profit: c) - 5 $ d) - 5 $
e) ST = 10 $ CALL option is worthless, but holder has paid 5 $ premium, hence the profit is:
Profit: e) - 5 $
Financial Market
Derivatives Market  Vanilla Options  Call option
Payoff function ßð Ä…ð Profit function
for CALL s holder:
Payoff function (K = 10 $): Profit function (K = 10 $, c = 5 $):
Payoff Profit
10 $ 10 $
5 $ 5 $
ST
0 0
ST
5 $ 10 $ 15 $ 20 $
5 $ 10 $ 15 $ 20 $
- 5 $ - 5 $
Financial Market
Derivatives Market  Vanilla Options  Call option
Payoff function and Profit function
$
for CALL s holder:
- c
- c
ST
0
K K + c
- c - c - c
- c
Profit function for CALL s holder =
MAX(ST  K , 0) - c
Financial Market
Derivatives Market  Vanilla Options  Call option
Writer s payoff + premium = Writer s profit
What will be CALL s writer profit (K = 10 $, c = 5 $) assuming that the stock X price is going to be:
Writer will be asked to sell stock X, hence writer will make: a) - 10 $ b) - 5 $.
a) ST = 20 $
On the other hand writer gain 5 $ premium for writing the contract. To sum it up,
b) ST = 15 $
writer s profit is equal to:
Profit: a) - 5 $ b) 0 $
c) ST = 8 $
CALL option is worthless. Writer s will be left with a premium:
d) ST = 5 $
Profit: c) 5 $ d) 5 $
e) ST = 10 $ CALL option is worthless. Writer s will be left with a premium:
e) 5 $
Profit:
Financial Market
Derivatives Market  Vanilla Options  Call option
Payoff function ßð Ä…ð Profit function
for CALL s writer:
Payoff function (K = 10 $): Profit function (K = 10 $, c = 5 $):
Payoff Profit
5 $ 5 $
ST ST
0 0
5 $ 10 $ 15 $ 20 $ 5 $ 10 $ 15 $ 20 $
- 5 $ - 5 $
- 10 $
- 10 $
Financial Market
Derivatives Market  Vanilla Options  Call option
Payoff function and Profit function
$
for CALL s writer:
Profit function
c
+ c + c
+ c
K + c
ST
0
Payoff function K
+ c
Profit function for CALL s writer = - MAX(ST  K , 0) + c
Financial Market
Derivatives Market  Vanilla Options  Call option
Payoff and Profit Functions
for CALL s holder and writer
Payoff Profit
H
O
L
D
ST ST
E
R
Payoff Profit
W
R
I
T
ST ST
E
R
Financial Market
Derivatives Market  Vanilla Options  Call option
CALL Premium (c)
Option premium Intrinsic value Time Value
= +
What will be an option payoff if it
What could happen in a future.
would expire right now.
Max ( S0  K ; 0 )
c  Max ( S0  K ; 0 )
Time value approaches zero as the expiration date nears.
Financial Market
Derivatives Market  Vanilla Options  Call option
CALL Premium (c)
CALL option premium mostly depends on:
S0  spot price (what is the current underlying security price).
K  Strike (execution price).
T  Time (How long this option is going to live).
When the other variables are constant then:
If S0 raises then c raises too. If S0 falls then c falls too.
If K raises then c falls.
c raises.
If K falls then
c raises too.
If T raises then If T falls then c falls too.
Financial Market
Derivatives Market  Vanilla Options  Call option
Terminology:
Based on relationship between S0 and K we can distinguish such options:
IN the money (ITM)
AT the money (ATM)
OUT of the money (OTM)
CALL option is: Payoff
ITM when S0 > K
0
K
ATM when S0 = K
S0
OTM when S0 < K
OTM
ITM
ATM