Chapter 17 Option Pricing
Framework

Background




One-period analysis




Put-call parity,
Arbitrage bound,
American call option
Black-Scholes Formula


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Definition and payoff
Some features about option
strategies
Price using discount factor
Derive Black-Scholes differential equation
Asset Pricing
2
Background (1)







Option;
Call/Put; C  Ct , CT P  Pt , PT
Strike Price X
Expiration Date T
Underlying Asset S  St , ST
European/ American Option
Payoff/Profit
Call payoff  CT  Max(ST  X ,0)
Put payoff  PT  Max( X  ST , 0)
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Background (2)
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Background (3)

Some Interesting Features of Options

High Beta (High Leverage)
-

Shaping Distribution of Returns:
-

Trading
Hedging
OTM Put + Stock
But Short OTM Put Option and Long Index



Return Distribution Extremely Non-normal
The Chance of Beating the Index for one or even five
years is extremely high, but face the catastrophe risk
So what kind of investments can and cannot be made
is written in the portfolio management contracts.
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Asset Pricing
5
Background (4)

Strategies



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By combining options of various strikes, you can buy
and sell any piece of the return distribution.
A complete set of option is equivalent to complete
markets.
Forming payoff that depends on the terminal stock
price in any way
Asset Pricing
6
One-period analysis

The law of one price


No arbitrage






existence of a discount factor
existence of positive discount factor
How to pricing option p  E  mx 
Put-Call Parity
Arbitrage Bounds
Discount Factors and Arbitrage Bounds
Early Exercise
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Asset Pricing
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Put-call parity

In the book of John C. Hull,
C  Xe rt  max( ST , X )
P  S  max  ST , X 
Strategies


(1) hold a call, write a put ,same strike price
(2) hold stock, borrow strike price X
CT  PT  ST  X
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PT  CT  ST  X
Asset Pricing
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Put-call parity
PT  CT  ST  X

According to the Law of One Price,
applying E  m to both sides for any m,
We get
P C S  X /Rf
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Asset Pricing
9
Arbitrage bounds


PT  CT  ST  X
Portfolio A dominates portfolio B
A  B, m  0
 E  mA  E  mB 
Arbitrage portfolio
(1)CT  0  C  0
(2)CT  ST  X  C  S  X / R f
(3)CT  ST  C  S
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Asset Pricing
10
Arbitrage bounds
(1)CT  0  C  0
(2)CT  ST  X  C  S  X / R f
C
Call value
Today
(3)CT  ST  C  S
Call value
in here
X/Rf
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S
Stock value
today
Asset Pricing
11
Discount factors and arbitrage bounds
This presentation of arbitrage bound is unsettling for two reasons,
First, you many worry that you will not be clever enough to dream up
dominating portfolios in more complex circumstances.
Second, you may worry that we have not dream up all of the arbitrage
portfolios in this circumstance.
Ct  E  mt ,T , xtc  , where, xtc  max  ST  X , 0 
m0
max Ct  Et (mxTc ), s.t.
St  Et (mST )
1  Et (mR f )
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Asset Pricing
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Discount factors and arbitrage bounds
m( s )  0
max Ct    ( s)m( s) xTc ( s),s.t.
m s 
s
St    ( s )m( s ) ST ( s )
s
1    ( s ) m( s ) R f
s
• This is a linear program. In situations where you do not know the
answer, you can calculate arbitrage bounds.(Ritchken(1985))
• The discount factor method lets you construct the arbitrage bounds
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Asset Pricing
13
Early exercise?

By applying the absence of arbitrage, we can never
exercise an American call option without dividends
before the expiration date.
payoff

CT  max  ST  X ,0  ST  X
price
C
 S  X / Rf
R f 1
C
SX
S-X is what you get if you exercise now. the value of the
call is greater than this value, because you can delay
paying the strike, and exercising early loses the option
value
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Asset Pricing
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Black-Scholes Formula (Standard Approach Review)
Portfolio Construction:
S   S t   S z
1: derivative
f

: share
S
 f 1  2 f 2 2 
    
 S  t
2
 t 2 S

  rt
 f
f 1  2 f 2 2 
f
f    S  

S

t

 S z

2

S

t
2

S

S


f
f 
S
S
f
  f 
S
S
 f 1  2 f 2 2 
f 



S

t

r
f

S  t



2
S 

 t 2 S

f
f 1 2 2  2 f
 rS
  S
 rf
2
t
S 2
S
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Black-Scholes Formula (Standard Approach Review)
Risk Neutral Pricing:

E  max V  X  , 0   V  X g V  dV
X
m s 2 / 2
ˆ
E max V  X  ,0  e
N  d1   XN  d2 
Where:
m  ln  E V    s 2 2
c  e rt Eˆ  max  ST  X , 0  
 e  rt  S0 e rt N  d1   XN  d 2  
 S0 N  d1   Xert N  d2 
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Black-Scholes Formula (Discount Factor)

Write a process for stock and bond, then use

 to price the option. the Black-Scholes formula
results,

(1) solve for the finite-horizon discount factor T /  0
and find the call option price by taking the
expectation C0  E0  T /  0 xTc 

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(2) find a differential equation for the call option and
solve it backward.
Asset Pricing
17
Black-Scholes Formula (Discount Factor)


The call option payoff is CT  max  ST  X ,0
The underlying stock follows
dS
  dt   dz
S


The is also a money market security that pays the real interest
rate rdt
In continuous time, all such discount factors are of the form:
d
 r
 rdt 
dz   w dw;


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Asset Pricing
E  dwdz   0
18
Method 1: price using discount factor
c  e rt Eˆ  max  ST  X ,0  

Use the discount factor to price the option
directly:
 T

T
C0  E0 
max  ST  X , 0   
max  ST  X , 0  dF  T , ST 
0
 0

Where ST and  T are solutions to
dS
  dt   dz
S
d
 r
 rdt 
dz   w dw;


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dS
  dt   dz
S

How to find analytical expressions for the
solutions of equations of the form (17.2)
dY
 Y dt   Y dz
Y
dY 1 1
1 2

2
d ln Y 

dY   Y   Y  dt   Y dz
2
Y 2Y
2 

1 2

0 d ln Y   Y  2  Y  0 dt   Y 0 dzt
T
T
T
1 2

ln YT  ln Y0   Y   Y  T   Y  zT  z0 
2 

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Asset Pricing
20
Applying the Solution to (17.2)
1 

ln YT  ln Y0   Y   Y2  T   Y  zT  z0 
2 

dS
  dt   dz
S
d
 r
 rdt 
dz   w dw;


We get:
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Ignoring the term of  w dw
And Proof Later
1 2

ln ST  ln S0       T   T 
2 

2

1  r  
 r
ln T  ln  0   r  
T
  T 

2   


Asset Pricing
21
Evaluate the call option by doing the integral
 T

C0  E0 
max  ST  X , 0  
 0


T
 
 ST  X  dF  T , ST 

0
ST  X
T   



0
ST  X
T  



0
ST  X



e
 S    X  dF  
T
ST dF   

ST  X
 1    r 2    r
T 
 r  
T
 2    



ST  X
X
T  



e
0
S0 e
XdF  
1 2

    T  T 
2 

 1    r 2 
 r
T 
 r  
T

 2   



f ( )d 
f   d 
ST  X
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22


C0  S0
e
2

1  2  r  
 r
  r   
T  ( 
) T



2







1
S0  e
2 ST  X
2

1  2  r  
 r
1
  r   
T  ( 
) T   2


2
  

2



1
X  e
2 ST  X

1

S0  e
2 ST  X
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f ( )d   X 
e
 1    r 2    r
T 
 r  
T
 2    



f   d 
ST  X
ST  X


 1    r 2    r
1
T 
 r  
T   2

 2   

2


1 
 r  
     
T
2 
  
1  (1/ 2) 2
f   
e
2
d
d
2

d   1 Xe rt
e

2
ST  X
Asset Pricing
1    r 

   
 T
2   

2
d
23
1
2

e
 (1/ 2)(    )2
d  1    a         a 
a
1 2

ln X  ln ST  ln S0       T   T 
2 

1 

ln X  ln S0      2  T
2 


 T

1 2

ln
X

ln
S



 T

0

2

      r  T
C0  S0  

 
 T





1 2

ln
X

ln
S



 T

0

2

   r 
 Xe  rt  


 T

  



1 2

ln
S
X

r

 T

0

2


 S0 
 T



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







T





1 2

ln
S
X

r

 T


0

2


  Xe  rT  

 T





Asset Pricing






24
Proof:
 w dw
not Affect
dS
  dt   dz
S
d
 r
 rdt 
dz   w dw;




C0  E0  T max  ST  X , 0  
 0


T   ,  
  
ST     X  dF  dF   

0
ST  X

 

T  ,  
0
ST  X


 e
ST dF   dF   

 
T  ,  
ST  X
 1    r 2 1 2    r
 r  
  T 
T   w T 
 2    2 w 



S0 e
0
e
d
XdF   dF  
1 2

    T  T 
2 

f   d f ( )d 
ST  X

X
 e
 1    r 2 1 2 
 r
 r  
  w T 
T   w T 

 2   2 



f   d f   d 
ST  X
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Asset Pricing
25


 S0    e
ST  X 
1
  w2 T  w T 
2


f   d   e

  12 w2T  w
 X   e
ST  X 
Where:
e
1
  w2 T  w T 
2
T
2

1
 r     r 
  r   2 
 T    
 T

2









f   d   e

 1    r 2 
 r
T 
 r  
T

 2   



f ( )d 
f   d 
1
1
  w2 T  w T    2
1
2
2
f   d 
e
d

2
2
1




T
 w 
1
2

e
d

2
This is the integral under the normal distribution, with mean of  w T
and, standard variance of 1,so the integral is 1.we multiply both sides
without any change.
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Asset Pricing
26
Method 2:derive Black-Scholes
Differential Equation

Guess that solution for the call option is a
function of stock price and time to expiration,
C=C(S,t). Use Ito’s lemma to find derivatives of
C(S,t)
1
dC  Ct dt  CS dS  CSS dS 2
2
1

2 2
  Ct  CS S   CSS S   dt  CS S dz
2


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Et  dpt   Et ( pt   pt )  0
Et  d (p)   0
d (p)  pd   dp  d dp
Et  d  /    rt
d
 r
  rdt 
dz


1


dC   Ct  CS S   CSS S 2 2  dt  CS S dz
2


t Ct  Et  t t Ct t  
d (C )  Cd   dC  d dC
0  Et  d C   CEt (d )  Et (dC )  Et (d dC )
1


0  Cr dt    Ct  CS S   CSS S 2 2  dt  Et  CS S dz 
2


 
  r 
1

2 2
 Et   rdt 
dz    Ct  CS S   CSS S   dt  CS S dz  

2



 
1


0  Cr dt   Ct  CS S   CSS S 2 2  dt  CS    r  S dt
2


1
0  rC  Ct  CS S   CSS S 2 2  CS    r  S
2
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1
0  rC  Ct  SrCS  CSS S 2 2
2
f
f 1 2 2  2 f
 rS
 S
 rf
t
S 2
S 2
• This is the Black-Scholes differential equation
for the option price
Ct  ST , T   max  ST  X ,0
C  S , t 
C  S , t  1 C 2  S , t  2 2

  rC  S , t   Sr

S
2
t
S
2 S
• This differential equation has an analytic solution, one
standard way to solve differential equation is to guess and
check, and by taking derivatives you can check that (17.7)
does satisfy (17.8).
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Thanks
Your suggestion is welcome!
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