1. Financial Market Theory
Thursday, October 19, 2017
Professor Edwin T Burton

2. Finite State Version of MPT
October 19, 2017

3. Asset choice in a two period economy
Suppose that the world only has two periods; there is only one more period after today
Suppose we want to buy assets now (in this period) that will do well by the end of this upcoming single period
What should we own?
October 19, 2017

4. possible states in a two period economy
What can happen?
We can simplify and just think about these three possibilities
State 1 – Gets better
Economy S2
Now
State 2 – Gets worse S3
State 3 – Muddles along
October 19, 2017

5. Three possible states and three available assets
Three states can occur – Good, bad, and mediocre (S1, S2, S3)
What are the available assets? X1, X2, X3
How will each asset perform in each state?
October 19, 2017

6. The Definition of a “Real-World” Security
Given the states of the world: s1, s2, s3
A security is defined by its payoff in dollars in each state of the world
p1,i is the payoff for security i in state one
p2,i is the payoff for security i in state two
p3,i is the payoff for security i in state three
October 19, 2017
September 1, 2015

7. X1 X2 X3 X4 X5 s1 s2 s3 Definition of Securities p1,1 p1,1 p1,2 p1,3
October 19, 2017
September 1, 2015

8. What would constitute a riskless asset?
Assume that owning one unit of Xr will return exactly 1 dollar regardless of state
Return doesn’t have to be 1; could be anything. Easier to simply assume 1 unit of return in each state
Xr is the “riskless asset”
Return $1
State 1 – Economy gets better X1 $1
State 2 – Economy gets worse $1
State 3 – Economy muddles along
October 19, 2017

9. What Does a Security Cost Today?
P1 times Ɵ1 is what it costs to buy a quantity Ɵ1 of security one at price P1.
Or simply: P1 Ɵ1
Similarly for 2, 3, etc.
P1. is always a positive number, but what about Ɵ1. That might be negative
You may have sold security one
Long sale if you already owned it, but could be a short sale
October 19, 2017
September 3, 2015

10. So, What Does a Portfolio of Securities Cost?
A portfolio is three numbers in a world of three securities: Ɵ1, Ɵ2, Ɵ3 where the Ɵ’s are the amounts purchased or sold of securities one, two and three
Ɵ1P1 + Ɵ2P2 + Ɵ3P3
This could be positive or negative
October 19, 2017
September 3, 2015

11. What does this security pay
What does this security pay? (these can be negative as well as positive)
In state one:
Ɵ1p1,1 + Ɵ2p1,2 + Ɵ3p1,3
In state two:
Ɵ1p2,1 + Ɵ2p2,2 + Ɵ3p2,3
In state three:
Ɵ1p3,1 + Ɵ2p3,2 + Ɵ3p3,3
October 19, 2017

12. No Arbitrage Means P1φ1 + P2 φ2 + P3 φ3 ≤ 0 (Budget) Implies
The following three conditions are not all true:
p1,1φ1 + p1,2 φ2 + p1,3 φ3 ≥ 0
P2,1φ1 + p2,2 φ2 + p2,3 φ3 ≥ 0
P3,1φ1 + p3,2 φ2 + p3,3 φ3 ≥ 0
If the Budget holds exactly (equals zero), then at least one of the three conditions must be strictly < 0.
October 19, 2017

13. Fundamental Theorem of Finance
The Assumption of No Arbitrage is True
If and only if
There exist positive state prices (one for each state) that represent the price of a security that has a return of one dollar in that state and zero for all other states
October 19, 2017

14. Diversification in a “Finite State” World
Most assets perform well in good state –that’s the definition of a “good state”
Most assets do terribly in the bad state – that’s the definition of a “bad state”
Diversification in the sense of protection against downside losses – finding assets that pay off in bad states
October 19, 2017

15. State Prices
A state price is the price of a security that pays one unit in that state and zero in all other states
q1, q2, q3 are the state prices for states 1, 2, 3
q3 > q2 > q1
October 19, 2017

16. Again: How can you use “state prices?”
To price any security
Price of a security j equals:
Pj = (pj,1 * q1) + (pj,2 * q2) + (pj,3 * q3)
This pricing formula is true if and only if the no-arbitrage assumptions is true
Price of risk-free asset q = q1 + q2 + q3
October 19, 2017

17. Analyzing the risk free rate
Buy the risk free asset, paying q
Invest it
Next period, you will have q (1+r)
We know that equals one
q (1+r) =1
So q = 1/(1+r)
October 19, 2017

18. Risk Adjusted Probabilities
Pj = (pj,1 * q1) + (pj,2 * q2) + (pj,3 * q3)
Define πi = qi/q
These πi ‘s can be interpreted as probabilities since π1 + π2 + π3 = 1
Substituting in
Pj = q { (pj,1 * π1) + (pj,2 * π2) + (pj,3 * π3) }
October 19, 2017

19. Pj = q { (pj,1 * π1) + (pj,2 * π2) + (pj,3 * π3) } But q = 1/(1+r)
price equals discounted expected value!
October 19, 2017