Class Business


Upcoming Groupwork
Course Evaluations
Profit Profiles for Long Calls
Payoff
Long Call
ST – X – C
0
C
X
Spot Price (ST)
Profit Profiles for Short Calls
Payoff
Spot Price (ST)
C
0
X
ST – X – C
Short Call
Call Options - Zero Sum Game
Payoff
0
Long Call
X
Spot Price (ST)
Short Call
Profit Profiles for Long Puts
Payoffs
X – ST – P
0
X
P
Spot Price (ST)
Long Put
Profit Profiles for Short Puts
Payoffs
Short Put
X
0
P – (X – ST)
P
Spot Price (ST)
Put Options - Zero Sum Game
Payoffs
Short Put
0
X
Spot Price (ST)
Long Put
Market and
Exercise Price Relationships
In the Money - exercise of the option would be
profitable
Call: market price>exercise price (St > X)
Put: exercise price>market price (St < X)
Out of the Money - exercise of the option would not
be profitable
Call: market price<exercise price (St < X)
Put: exercise price<market price(St > X)
At the Money - exercise price and asset price are
equal (St = X)
Bull Spread Using Calls
Profit
ST
X1
Positon: Long 1 call at X1
Short 1 call at X2
X2
Straddle Combination
Profit
X
Position: Long 1 call at X
Long 1 put at X
ST
Option Strategies involving stock
Protective Put
Long Stock
Long Put
Profit
X
ST
) All you are doing is buying calls
Option Strategies involving stock
Covered Call
Long Stock
Short Call
Profit
X
) All you are doing is writing puts
ST
Chapter 15: Pricing Options
Put-Call Parity



Consider the following portfolio:
– Going LONG one European call option,
– Going SHORT one European put option
– LENDING the present value of the
exercise price PV(X) = X/(1+r)T at the
interest rate r.
What are the payoffs of this portfolio?
Notation:
– c = price of call
– p = price of put
Put-Call Parity
Action
Buy a call
Write a put
Buy bond with
face value = X
Today
(cost)
At maturity (cash flows: what you get)
ST < X
ST > X
ST=X
c
0
ST - X
0
-p
-(X – ST)
0
0
X/(1+r)T
X
X
X (ST)
c – p + X/(1+r)T
ST
ST
ST
Total cost
Put-Call Parity



The future cash flows of this transaction are identical
to the value of the stock ST.
You can get the same cash flows by buying the stock.
No arbitrage implies that the price of the stock equals
the price of the portfolio:
S0 = c – p + PV(X)
or
S0 + p = c + PV(X)
(S0 = current price of stock)
Example: Put-Call Parity

Suppose you are a trader on the floor of the CBOE,
you notice the following:
– Price of IBM stock = 100
– Call(X=110,T=1) price = 12.95
– Put(X=110,T=1) price = 7.45
– Risk-free rate is 5%
– Note: T is time to maturity

Is there an arbitrage opportunity?
How would you take advantage of it?

Example: Put-Call Parity

Step 1: Write down PCP equation
S0 + p = c + PV(X)

Step 2: Plug in numbers on each side
– S0 + p =100+7.45=107.45
– c + PV(X)= 12.95+110/(1.05)=117.71

Step 3: Go long the cheap side and short the expensive side
– Long: stock and put
– Short: call and bond with FV=X
Example: Put-Call Parity
Action
Today
(cost)
At maturity (flows: what you get)
ST < X
short a call
short PV(X)
Long put
-12.95
-104.76
0
ST > X
-(ST-110)
ST =X
0
-110
-110
-110
+7.45
(110-ST)
0
0
+100.00
ST
ST
ST
0
0
Long Stock
Net
-10.26
0
Binomial Option Pricing:
Call Option on Dell


The current price is S0 = $60.
After six months, the stock price will either grow to
$66 or fall to $54.
– Pick what ever probabilities you want.

The annual risk-free interest rate is 1%.
– Assume yield curve is flat

What is the value of a call option with a strike price of
$65 that expires after 6 months?
Binomial Option Pricing:
Call Option on Dell

Find value of a corresponding call option with X=65:
Stock
Price Tree
H

Option Price Tree
66
60
1
?
L
54
0
Binomial Option Pricing:
Call Option on Dell

Claim: we can use the stock along with a risk-free bond
to replicate the option

Replicating portfolio:
– Position of D shares of the stock
•
•
–
If D is positive, that means you “own” the stock
If D is negative, that means you are “short” the stock
Position of $B in bonds (B=present value)
•
•
If B is positive, that means you “own” the bond
If B is negative, that means you are “short” the bond
Binomial Option Pricing:
Call Option on Dell



Strategy: If we know that holding D shares of stock and
$B in bonds will replicate the payoffs of the option, then
we know the cost of the option is DS0 + B
Example: Suppose the stock is currently $60, and we find
that holding 1 share of stock and shorting $55 in bonds
will give us the exact same payoffs as the option (in either
state).
60-55 = 5
Then we know the price of the option is ________.
Binomial Option Pricing:
Call Option on Dell

We want to find D and B such that
D66  B(1.01)1/ 2  1
D54  B(1.01)1/ 2  0



D66 and D54 are the payoffs from holding D shares of the stock
B(1.01)1/2 is the payoff from holding $B of the bond
Mathematically possible
–
Two equations and two unknowns
Binomial Option Pricing:
Call Option on Dell
D66  B(1.01)1/ 2  1
D54  B(1.01)1/ 2  0

Shortcut to finding D:
D

CH  CL
1 0
1


S H  S L 66  54 12
Subscripts:
– H – the state in which the stock price is high
– L – the state in which the stock price is low
Binomial Option Pricing:
Call Option on Dell

Once we know D, it is easy to find B
1
1/ 2
54

B
(1.01)
0
 
 12 

1  54 
B 
 4.48
1/ 2 
12  (1.01) 
So if we
– buy 1/12 shares of stock
– Short $4.48 of the bond
– Then we have a portfolio that replicates the option
Binomial Option Pricing:
Call Option on Dell

Do we know how to price the replicating portfolio? Yes:

We know the price of the stock is $60
– 1/12 shares of the stock will cost $5

When we short $4.49 of the bond
– we get $4.48

Total cost of replicating portfolio is
– 5.00 - 4.48 = 0.52

This is the price of the option. Done.
Binomial Option Pricing:
Put Option On Dell

Find value of a corresponding put option with X=65:
Stock
Price Tree

Option Price Tree
66
60
0
?
54
11
Binomial Option Pricing:
Put Option on Dell

We want to find D and B such that
D66  B(1.01)1/ 2  0
D54  B(1.01)1/ 2  11



D66 and D54 are the payoffs from holding D shares of the stock
B(1.01)1/2 is the payoff from holding B shares of the bond
Mathematically possible
–
Two equations and two unknowns
Binomial Option Pricing:
Put Option on Dell
D66  B(1.01)1/ 2  0
D54  B(1.01)1/ 2  11

Shortcut to finding D:
D

PH  PL 0  11 11


S H  S L 66  54 12
Subscripts:
– H – the state in which the stock price is high
– L – the state in which the stock price is low
Binomial Option Pricing:
Put Option on Dell

Once we know D, it is easy to find B
 11 
1/ 2
66

B
(1.01)
0


 12 

11  66 
B 
 60.20
1/ 2 
12  (1.01) 
So if we
– short 11/12 shares of stock
– buy $60.20 of the bond
– Then we have a portfolio that replicates the option
Binomial Option Pricing:
Put Option on Dell

Do we know how to price the replicating portfolio? Yes:

The price of the stock is $60
– When we short 11/12 shares of the stock we
will get $55.00

To buy $60.20 of the bond
– This will cost $60.20

Total cost of replicating portfolio is
– 60.20 - 55.00 = 5.20

This is the price of the option. Done.
Insights on Option Pricing

The value of a derivative
– Does not depend on the investor’s risk-preferences.
– Does not depend on the investor’s assessments of
the probability of low and high returns.
– To value any derivative, just find a replicating portfolio.
– The procedures outlined above apply to any
derivative with any payoff function