15.401 Recitation
5: Options
Learning Objectives

R i off Concepts
Review
C
t
o Payoff profile
o Put‐call parityy
o Valuation of options
o Binomial tree

Examples
o Payoff replication
o Arboreal Corporation
2010 / Yichuan Liu
2
Review: elements of a call/put option

Type:
o Call: holder has the right but not the obligation to buy
o Put: holder has the right but not the obligation to sell

Q
Quantity
tit off the
th underlying
d l i asset:
t
o Usually one share of stock with current price S
Strike/exercise price (K)
 Expiration date (T)
 Style:

o European: can only be exercised at T
o American: can be exercised at any time between 0 and T.
2010 / Yichuan Liu
3
Review: payoff profile
Call
Put
Payoff
Long
Payoff
45°
Asset Price
K
45°
Asset Price
Asset Price
K
45°
K
Asset Price
Short
K
45°
Payoff
2010 / Yichuan Liu
Payoff
4
Review: payoff profile
 The payoff of a portfolio of options is the sum of

payoffs of the individual components:
1 put
Straddle
1 call
+
=
K1
K1
1 call @ K1
‐2 call @ K1
+
K1
2010 / Yichuan Liu
K1
+
K2
Butterfly spread
1 call @ K3
=
K3
K1 K2 K3
5
Review: put‐call parity

Two portfolios with identical payoffs
Portfolio 1
1 call @ K
Bond w/ FV=K
+
=
K
K
K
Portfo
olio 2
1 put @ K
+
K
2010 / Yichuan Liu
1 stock
=
K
6
Review: put‐call parity
 No arbitrage implies that the two portfolios must

have the same cost:
C
C
 PV K   P  S
K
 P  S
C
T
1  r 
This is the put‐call parity.
 Note: the call and put must have the same exercise

exercise
price (K).

2010 / Yichuan Liu
7
Pricce of options
Review: value of an option
Price of underlying asset
2010 / Yichuan Liu
Value of call
Value of put
price ((K))
Strike p
Decrease
Increase
Price of underlying
asset (S)
Increase
Decrease
Volatility of the
underlying asset (σ)
Increase
Increase
Maturity (T)
Increase
Increase
Interest rate (r)
Increase
Decrease
8
Review: binomial tree
 Idea: if there are only two states of the world next

period, we can price options given the underlying
asset and a risk‐free asset (“bond”) by replication:
Underlying Asset
p
Su
S
S
Call
B
Sd
Cu
C
B/(1+r)
1‐p
2010 / Yichuan Liu
Bond
B
Cd
9
Review: binomial tree

Replication:
A shares of underlying asset
Bond (FV=B)
Total
Replication
CF at t = 0
CF at t=1
(“up” state)
CF at t=1
(“down” state)
‐A x S
A x Su
A x Sd
‐ B/(1+r)
B
B
‐A
A x S ‐ B/(1+r)
A x Su + B
A x Sd + B
= ‐C
= Cu
= Cd
o A = ( C u – C d ) / ( Su – Sd )
o B = C u – A x Su
o C = A x S + B/(1+r)
2010 / Yichuan Liu
10
Review: binomial tree
 Equivalently, we can solve for the risk

risk‐neutral
neutral
probability, q:
 Then,

qSu  1 q S d
q
S
S
1r
qCu  1 q Cd
C
1 r
Note: q is not related to the state probability p. In
fact,, p is not used in the pricing
g of C.
2010 / Yichuan Liu
11
Example 1: payoff replication

How would you replicate the following payoff profile
using only call and put options?
a)
10
15
25
30
b)
8
8
2010 / Yichuan Liu
12
16
20
12
Example 1: payoff replication

Answer:
a) Long 1 call (K=10)
Short 1 call (K=15)
Sh t 1 callll (K
Short
(K=25)
)
Long 1 call (K=30)
b) Long 1 put (K=8)
Short 1 call (K=8)
Long 2 calls (K=12)
Short 1 call (K=20)
(K 20)
2010 / Yichuan Liu
13
Example 2: Arboreal Corporation
 Arboreal Corporations stock price is currently $102.

At the end of 3 months it will be either $120 or $90.
The 3‐month spot rate is 2%. What is the value of a
3‐month European call option with a strike price of
$110?
Call
Stock
$120
$102
C
$90
2010 / Yichuan Liu
$10
$0
14
Example 2: Arboreal Corporation

The call can be replicated with:
o Long 1/3 stock: costs $34
o Short bond with FV=30: costs ‐$30/(1+2%) = ‐$29.41

Th price
The
i off th
the callll mustt b
be
C  34  29.41  $4.59

Alternatively, we can solve for the risk
Alternatively
risk‐neutral
neutral
probability: 120q  901 q 
1 2%

 102  q  0.468
The price of the call is then
2010 / Yichuan Liu
100.468  01 0.468
 $4.59
C
$4 59
1 2%
15
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15.401 Finance Theory I
Fall 2008
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