1976 the Arbitrage Theory of Capital Asset Pricing by ufe19594

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									                  The Return Form of Arbitrage
                            Pricing




Primbs, MS&E345                                  1
Pricing Theory:


                                           Linear function form
              Return form
                                              (risk neutral)
                (pdes)




                            Optimization

Primbs, MS&E345                                                   2
Pricing Theory:
                                                  Returns and Factor Models

              Return form
                                                     Profits and Losses
                (pdes)

                                                        Absence of Arbitrage
     Relationships between returns of assets

  Generalizes the Black-Scholes-Merton argument         Market Price of Risk


                                                     Multiple Factors

                                                  Futures




Primbs, MS&E345                                                                3
    Modeling Returns:

                                    P1
    You put something in.


                                         You get something out
                  P0
                         1 period


                          P  P0
                        r 1
                             P0

Primbs, MS&E345                                                  4
           The time period can be:   one year

                                     one month

                                     one week

                                     one day

                                     one second

                                     one millionth of a second

                                     an instantaneous dt.



Primbs, MS&E345                                                  5
   Example: an instantaneous dt:

            Assume we model a stock price as a geometric brownian
            motion.
                               dS  Sdt  Sdz
                                                         St+dt



                                        St       dt

                  What is the return?
                         St  dt  St
                      r                     
                                               dSt
                                                       dt  dz
                               St              St
Primbs, MS&E345                                                     6
        This is an example of a factor model:


                         dS
                      r     dt  dz
                         S

                                a  bf

          Where:    a  dt
                                          known at beginning of period
                    b 
                    f  dz                unknown factor



Primbs, MS&E345                                                          7
        This is an example of a factor model:


                            dS
                         r     dt  dz
                            S

                                    a  bf

     Note that  and  can depend on S at time t (the beginning
     of the time period).
                     dSt
                  r       ( St , t )dt   ( St , t )dz
                      St


Primbs, MS&E345                                                   8
  The modeling paradigm:

     We describe the return of a security over a time
     period dt as a factor model:

                  The factors:
                       •Time: dt (this is for convenience)
                       •Random factors: dz1, dz2, ...

     These random factors can be increments of Brownian Motion,
     Poisson Processes, or in general, whatever you want!!!

     I will write dt, but the time period could be of any length!!!



Primbs, MS&E345                                                       9
Pricing Theory:
                                                  Returns and Factor Models

              Return form
                                                     Profits and Losses
                (pdes)

                                                        Absence of Arbitrage
     Relationships between returns of assets

  Generalizes the Black-Scholes-Merton argument         Market Price of Risk


                                                     Multiple Factors

                                                  Futures




Primbs, MS&E345                                                                10
   Profits and Losses
             Consider an asset, S, with the following return
                           S            r  dt  dz

Invest x dollars at initial time             St+dt

That is, I purchase
 x                                                   Now each share is worth
    shares                     St
 St                                    dt                     St+dt


How much money did I make over dt?
                  x                x         x
 Profit:             S t  dt  x  St  dt  St  dS x  rx
                  St               St        St     S
          shares price     initial amount
Primbs, MS&E345                                                                11
Profit/Loss from a Portfolio

    x1            S1             r1  1dt   1dz        We are given the returns on
    x2            S2         r2   2 dt   2 dz         assets which all depend on a
                                                          common factor, dz.
    x3            S3         r3   3dt   3dz

Let xi be the dollar amount of money invested in asset i:

 Cost:         x1  x2  x3  1T x
 Profit/Loss:  r1 x1  r2 x2  r3 x3
               (1dt  1dz) x1  (2 dt   2 dz) x2  (3 dt   3 dz) x3
               (1 x1  2 x2  3 x3 )dt  (1 x1   2 x2   3 x3 )dz
                        (  T x)dt  ( T x)dz
                           1             1           1         x1 
          where        1  1           2         2    x   x2 
                                                                       
                                                      
                           1
                                         3 
                                                        3 
                                                                     x3 
                                                                       
Primbs, MS&E345                                                                          12
Pricing Theory:
                                                  Returns and Factor Models

              Return form
                                                     Profits and Losses
                (pdes)

                                                        Absence of Arbitrage
     Relationships between returns of assets

  Generalizes the Black-Scholes-Merton argument         Market Price of Risk


                                                     Multiple Factors

                                                  Futures




Primbs, MS&E345                                                                13
 What would an arbitrage portfolio be?



    An arbitrage is a riskless profit which requires no investment.


                  In the current setting, a portfolio that

                           (1) costs nothing

                           (2) has no risk

                           (3) but makes a profit


Primbs, MS&E345                                                       14
 Profit/Loss from a Portfolio

 Cost:         x1  x2  x3  1T x
 Profit/Loss:  r1 x1  r2 x2  r3 x3
                   ( 1dt   1dz) x1  (  2 dt   2 dz) x2  (  3dt  dz) x3
                   ( 1 x1   2 x2   3 x3 )dt  ( 1 x1   2 x2  x3 )dz
                   (  T x)dt  ( T x)dz


 Consider the following portfolio:

     No Cost:       1T x  0
                                                No Profit/Loss: T x  0
     No Risk:       Tx 0

                         Otherwise, Arbitrage!!!
Primbs, MS&E345                                                                      15
                  A necessary condition for No Arbitrage

     No Cost:        1T x  0
                                           No Profit/Loss: T x  0
     No Risk:        Tx 0




                    What are the implications of this?




Primbs, MS&E345                                                       16
                                           1T 
 Let’s write the condition as follows:     T   x  0  T x  0
                                           
 This truthfulness of this implication is equivalent to saying that
  is a linear combination of 1 and .

    Why?
         Assume:     10  1  1  

                               1T 
           Then:    T x  T  T  x  0
                               


Primbs, MS&E345                                                       17
                                               1T 
 Let’s write the condition as follows:         T   x  0  T x  0
                                               
 This truthfulness of this implication is equivalent to saying that
  is a linear combination of 1 and .

    Another Argument:
                           1T 
                  Let   A T
                           
                  then x  N (A) and         N (A)

                  but N ( A)   R( AT )

                  hence   R ( AT ) or      AT   1  

Primbs, MS&E345                                                         18
                    A necessary condition for No Arbitrage

     No Cost:           1T x  0
                                                No Profit/Loss: T x  0
     No Risk:           Tx 0




                  There exists a  such that:        1  




Primbs, MS&E345                                                            19
Pricing Theory:
                                                  Returns and Factor Models

              Return form
                                                     Profits and Losses
                (pdes)

                                                        Absence of Arbitrage
     Relationships between returns of assets

  Generalizes the Black-Scholes-Merton argument         Market Price of Risk


                                                     Multiple Factors

                                                  Futures




Primbs, MS&E345                                                                20
 Our return was given by: r  dt  dz

     We have derived a relationship between:

                               1  

                    Expected Returns and Risk
  (Actually, this is nothing more that Ross’ 1976 Arbitrage Pricing Theory.)


                     10  1  1  
 The ’s tell you how much expected return you are rewarded for taking risk.

           They are generally referred to as the “Market Price of Risk”.


                   Let’s look at them in a little more depth...

Primbs, MS&E345                                                                21
 Consequences of the Return Form of AOA.

        A Risk Free Asset:
                                      the risk free rate
                     r0  r f dt

        Then, from AOA:
                                  0 
                1    1             0 is the risk free rate!
                                   1 

                                               We can think of 0 as the
                  r f  1 0  0            “market price of time”.




Primbs, MS&E345                                                            22
 Consequences of the Return Form of AOA.

        A Risk Free Asset:                 Two other assets:

                       r0  r f dt           r1  1dt   1dz
                                            r2   2 dt   2 dz
        Then, from AOA:
       (1) r f  0
       (2) 1  0   11  r f   11
            2  0   2 1  rf   2 1
      Solve for 1:               1  rf  2  rf
                             1         
                                    1      2

                  We refer to 1 as the “market price of risk” for dz.

Primbs, MS&E345                                                          23
   The Market Price of Risk

                               1  rf  2  rf
                          1         
                                 1      2


        (1) It looks like the Sharpe ratio.


        (2) The market price of risk is associated with the factor dz.


        (3) All securities that only depend on the random factor dz
            will have the same market price of risk
            (instantaneous Sharpe ratio).



Primbs, MS&E345                                                          24
   The Market Price of Risk

                                  1  rf  2  rf
                             1         
                                    1      2

     An important Note:
           The market price of risk is often a function of St and t
           because it depends on  and  which are often
           functions of St and t.
                                                               St+dt
            The market price of risk is
            known here, just like  and .
                                                    St
                                                            dt
           I won’t always make this dependence explicit, but you should keep
           it in mind!


Primbs, MS&E345                                                                25
  Connections with CAPM:

     CAPM says:                                rf   (rM  rf )


     We just derived that:                     r f  


     Hence, we should have that:              (rM  rf )  

Example:           If returns are uncorrelated with the market:

                       0                 0
               Sometimes used to determine market price of risk.

 Primbs, MS&E345                                                     26
Pricing Theory:
                                                  Returns and Factor Models

              Return form
                                                     Profits and Losses
                (pdes)

                                                        Absence of Arbitrage
     Relationships between returns of assets

  Generalizes the Black-Scholes-Merton argument         Market Price of Risk


                                                     Multiple Factors

                                                  Futures




Primbs, MS&E345                                                                27
 An extension to multiple factors
                       r1  1dt   11dz1  ...   1m dzm        x1
                       r2   2 dt   21dz1  ...   2 m dzm     x2
                       rn   n dt   n1dz1  ...   nm dzm      xn

Vector notation:
  r  dt  Kdz                   1             11   1m   dz1 
                                                dz    
                                                              
                  where                                             
                                  n 
                                                 n1   nm 
                                                               dzm 
                                                                      
No Arbitrage

                  1 x0
                   T
                                     KT x  0                    T x  0
                   No Cost            No Risk                    No Profit


Primbs, MS&E345                                                              28
No Arbitrage

                  1 x0
                  T
                                K x0
                                  T
                                                   T x  0
                  No Cost       No Risk            No Profit




                   There exists  such that:     1 K 




Primbs, MS&E345                                                29
  Does any of this make sense?

                                      r1  1dt   11dz1  ...   13dz3
           Consider:
                                      r2   2 dt   21dz1  ...   23dz3



   Necessary condition for no arbitrage:

                       1  0   111   122   133
                        2  0   211   222   233

In general, these equations do not have a unique solution for the ’s!
     That is, there could be many ’s that satisfy the equations.

     Nevertheless, it is still a necessary condition for no arbitrage.

Primbs, MS&E345                                                               30
Pricing Theory:
                                                  Returns and Factor Models

              Return form
                                                     Profits and Losses
                (pdes)

                                                        Absence of Arbitrage
     Relationships between returns of assets

  Generalizes the Black-Scholes-Merton argument         Market Price of Risk


                                                     Multiple Factors

                                                  Futures




Primbs, MS&E345                                                                31
  An interesting special case: Futures Contracts

         When you enter into a futures contract, no money
         exchanges hands. So, initially, it costs nothing. This
         makes it a special case.

          Let, f be the futures price, and assume f follows:
                         df
                              dt   dz
                          f
          But, when I enter into this contract, I pay nothing


          How does this fit into our factor model framework?


Primbs, MS&E345                                                   32
   Profit/Loss from a Portfolio that contains Futures
                                                          We are given the returns on
      x1            S1          r1  1dt   1dz
                                                          assets which all depend on a
      x2            S2         r2   2 dt   2 dz       common factor, dz.
      x3             f         r3   3dt   3dz         Future Price


Let x1, x2 be the amount of money invested in assets 1 and 2:
But, x3 is the total future price. You agree to pay this at the delivery date, not now!


   Cost:         x1  x2  1 1 0x
   Profit/Loss:  r1 x1  r2 x2  r3 x3
                          ( 1dt   1dz) x1  (  2 dt   2 dz) x2  (  3dt  dz) x3
                          ( 1 x1   2 x2   3 x3 )dt  ( 1 x1   2 x2  x3 )dz
                          (  T x)dt  ( T x)dz

  Primbs, MS&E345                                                                           33
                     A necessary condition for No Arbitrage

     No Cost:             x1  x2  0
                                                        No Profit/Loss:
     No Risk:  1 x1   2 x2   3 x3  0
                                                  1 x1   2 x2   3 x3  0




                                                   1  1     1   
                  There exist 0,   1 such that:                0
                                                   2   1   2   
                                                     0           1 
                                                                3   
                                                   3 


Primbs, MS&E345                                                                 34
                  There exist 0, 1 such that:    1  1     1   
                                                                 0
                                                   2   1   2   
                                                     0           1 
                                                                3   
                                                   3 



                         Therefore, from the last equation:

                                           3
                                      1 
                                           3

             I don’t need the risk free rate! I just need to know the
               dynamics of a futures price and I can calculate the
                               market price of risk!

Primbs, MS&E345                                                               35
    Some Intuition:

           Risk free rate: The “market price of time”

                  No risk, only time.
                                                        T   t



           Futures price: The “market price of risk”

                  No time, only risk.


                                                        T       t




Primbs, MS&E345                                                     36
                              References


           Ross, S. A., “The arbitrage theory of Capital Asset Pricing”, Journal of
           Economic Theory, 13, 341-360, 1976.

           Black, F. and M. Scholes, “The pricing of options and
           corporate liabilities”, Journal of Political Economy, 81, 637-659, 1973.

           Merton, R. C., “Theory of rational option pricing”, Bell Journal of
           Economics and Management Science, 4, 141-183, 1973.




Primbs, MS&E345                                                                       37

								
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