Financial Market Theory Thursday, November 1, 2018 Professor Edwin T Burton

Possible states in a two period economy What can happen? We can simplify and just think about these three possibilities S1 State 1 – Great Economy Economy S2 Now State 2 – Average Economy S3 State 3 – Financial & Economic Collapse November 1, 2018

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? November 1, 2018

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 November 1, 2018

X1 X2 X3 s1 s2 s3 Definition of Securities p1,1 p1,1 p1,2 p1,3 p2,3 November 1, 2018

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 November 1, 2018

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. November 1, 2018

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 November 1, 2018

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 November 1, 2018

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 November 1, 2018

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 November 1, 2018

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) November 1, 2018

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) } November 1, 2018

Pj = q { (pj,1 * π1) + (pj,2 * π2) + (pj,3 * π3) } But q = 1/(1+r) price equals discounted expected value! November 1, 2018

November 1, 2018