# Asset Prices

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```							     Lectures on Probability and
Statistics
• 1) Asset Prices
Binomial Random Variable Random Walks
• 2) Insurance
Jensen’s Inequality and Weak Law of Large
Numbers
• 3) Prediction and Trust
Conditional Distribution and Bayes’ Law

• 4) Hypothesis Testing
Asset Prices

Binomial Distribution
Expectation
Random Walks
Asset Prices
• A security whose price today (January 1)
is S0 = \$50
• The interest rate (for borrowing and
lending) is i = 10%
• On December 31 the security will sell for
either S'1 = \$66 with probability p = 0.75 or
S''1 = \$22 with probability q = 0.25
• Notice (\$66/1.1)0.75 + (\$22/1.1)0.25 = \$50
Call Option
• What is the value, today, of an option to
buy the security in one year for \$50?
• With p = 0.75 there is a gain of \$16 whose
present value is \$16/1.1 = \$14.56
• With probability q = 0.25 the present value
of the option is 0
• The option value is
V0 = 0.75*14.56 + 0*0.25 = \$10.92
Asset Prices
• An asset with a current market price of
S0=\$50
• Interest rate i = 10%
• Two possible states from one year to the
next:
Price increase of 32% with
probability p = 0.75
Price decline of 56% with
probability q = 0.25
Asset Price

\$50(1.32/1.1)=\$60

\$50                              E(ST) = 0.75(\$60) + 0.25(\$20) = \$50

\$50(0.44/1.1)=\$20

ST is the terminal value, in this case T = 1
Asset Price

\$50(1.32)2/(1.1)2 = \$72

\$50(1.32/1.1)
\$50(1.32)(0.44)/(1.1)2 = \$24
\$50
\$50(0.44)(1.32)/(1.1)2 = \$24
\$50(0.44/1.1)

\$50(0.44)2/(1.1)2 = \$8
Asset Prices

Path    Value # Up   Prob     PxV

Up-Up    \$72   2     0.5625   40.5

Up-Dn    \$24   1     0.1875    4.5

Dn-Up    \$24   1     0.1875    4.5

Dn-Dn    \$8    0     0.0625    0.5
The number of k element subsets
in an n element set

n!     n
 
k
k!(n  k)!  
Counting
• Index set – the set which includes the
number of the time period t = {1,2,…,T}
that represent the transitions from 0 to 1, 1
to 2 … T-1 to T.
• In each of the transitional periods equity
prices can either go Up or Down
• Suppose we know there are  ups, how
many different orderings of the transitional
periods can there be?
Counting
• Suppose T = 2 and  = 0, there is only one
subset of the index set, namely {1,2}, that
index the Downs.  2 

   

0

• Suppose T = 2 and  = 1, there are two
 2
subsets of the index set, namely {1} and {2}   
1
  

• Suppose T = 2 and  = 2, there is only one
subset of the index set, namely {1,2}.  2 

   

 2
Counting
• The number of  element subsets (the
number of ways  Ups can happen in T
transitions) is

T       T!
 
   !(T  )!
 
Combinatorial
n n
     1
0 n
   
n  n 
 
k n  k
        
 n  1  n   n 

 k    k    k  1
            
                  
Binomial Theorem

n   n  n  k n k
( a  b )    a b
k
k 0  
The number of subsets of an n-
element set = 2n
n
The number of k - element subsets:  
k
 
n
n
The total number of subsets:   k
k 0  

Let a  b  1 and the number of subsetsis
n
 n  k n k
  k a b  (a  b)  2
 
k 0  
n    n
Asset Prices
• Path Probabilities

 2 2 0
P ( 2 UP )    p q
 2
 
 2 1 1
P (1UP )    p q
1
 
 2 0 2
P (0 UP )    p q
0
 
Asset Prices
• S0 is the initial market value of the asset
• ST(t) is the market price at some terminal
date T if there have been t UP years
• r is 1 plus the market rate of interest (the
discount rate
• u is the percentage increase in market
value in an up year and d is the
percentage decrease in a down year
• a = 1 + u and            b=1+d
Asset Prices
• ST(t) is the market price at some terminal
date T if there have been t UP years

• ST(t) = (a/r)t(b/r)T-tS0

• ST(t) = atbT-tS0/(r)T
Asset Prices
• ST(t) = atbT-tS0/(r)T
T
E(ST )   p q  t   Tt     t
(a b   Tt
)S0 /(r )     T

t 0
T
E(ST )   (pa ) (qb)t          Tt
S0 /(r )   T

t 0

• Binomial Theorem
E (ST )  (pa  qb) S0 /(r )
T         T
Asset Prices
• With Risk Neutrality The Equilibrium
Market Price of an Asset Must Equal Its
Expected Value

• S0 = E(ST) = (ap + bq)TS0/(r)T

• Market Equilibrium r = (ap + bq)
Option Price
• A Call Option to Purchase Asset at time T
for a price of K

• At T the value, VT, of the option is ST – K if
K is no greater than ST and 0 if it is greater

• Suppose there must be  UP years for ST
to exceed K
Option Prices

• VT = P(ST>K){ E(ST |ST>K) – K/(1+r)T}
Option Prices

T T
t Tt
P(ST  K)  P( UP  )    p q
t
t   

```
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