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

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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