Asset Prices

							     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   