Economics 434

Financial Markets Professor Burton

University of VirginiaFall 2015

September 8, 2015

Today

Tomorrow

s1

s3

s2

And, we may not have any idea what the probabilities of s1, s2, s3 may be!!September 8, 2015

So, Again, What are “State Prices”

• A state price, qi, is the price of a security that returns $ 1 in state i

• The rate of return of the ith state price security would be:(1 – qi) divided by qi

September 8, 2015

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 has a return of one dollar in that state and zero for all other states

September 8, 2015

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

September 8, 2015

The Risk Free Security• Imagine a portfolio that consisted only of one of each of the state price

securities: Q1, Q2, Q3 with prices q1, q2, q3. (Call this portfolio, Q, which consists of one unit of each state prices security).

• That portfolio, Q, would return exactly $ 1 regardless of which state occurred – that means that portfolio would be the riskless asset.

• Price of Q, the riskless asset = q = q1 + q2 + q3

• Risk free rate = r = so that q =

September 8, 2015

Interpreting the risk free rate

• What is the value of $ 1 tomorrow?• What would you have to invest today to be

absolutely certain to receive $ 1 tomorrow?• $ X (1 + r) = $ 1 which says: “if I invest $ x

today and earn the risk free rate, I will have $ 1 tomorrow.

• Thus $ X =

September 8, 2015

Create pseudo-probabilities (risk adjusted probabilities)

• Define πi = (recall that qi is the ith state price and q is the sum of all state prices)

• Then: πi > 0 for all I π1 + π2 + π3 = 1 This looks like probabilities for each state!

In fact, these πi ‘s are called “risk-adjusted probabilities”

September 8, 2015

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

September 8, 2015

The Pricing of Security j

Pj = (pj,1 * q1) + (pj,2 * q2) + (pj,3 * q3)

Now substitute πk = Pj = Since q = ; Pj = = price equals discounted expected value!

September 8, 2015

September 8, 2015