Economics 434 Theory of Financial Markets

Economics 434Theory of Financial Markets Professor Edwin T Burton Economics Department The University of Virginia

Finite State Approach • Usually we assume an infinite number of states of the world (uncountably infinite, in fact) • Instead, image that we can identify and list each possible state of the world that may occur next period: e.g. recession, slow growth, boom times, etc. • Then, imagine that securities are defined by what they pay in each possible state of the world

Finite States of the World States: S1 S2 S3 Prob of States: π1π2π3 Securities: B1 B2 B3

Dividend Payoffs States 1 2 3 Securities 1 D11 D12 D13 2 D21 D22 D23 3 D31 D32 D33

Security Prices P1 P2 P3

A Portfolio (ϴ1, ϴ2, ϴ3) V = Value of a portfolio: ϴ1P1 +ϴ2P2 + ϴ3P3 {In matrix notation, this is simply ϴP, where ϴis a row vector and P is a three by one column vector}

What about security returns? For state 1 occurs: Security 1 will pay D11 Security 2 will pay D21 Security 3 will pay D31 A portfolio with quantities of securities: (ϴ1, ϴ2, ϴ3) Pays ϴ1D11 + ϴ2D21 + ϴ3D31 if state 1 occurs

For each state, the ϴ portfolio returns are given as: • Pays ϴ1D11 + ϴ2D21 + ϴ3D31 if state 1 occurs • Pays ϴ1D12+ ϴ2D22+ ϴ3D32if state 2 occurs • Pays ϴ1D13+ ϴ2D23+ ϴ3D33if state 3 occurs In matrix notation: Returns = ϴD where ϴis a row vector And D is a three by three matrix

A Portfolio that Always Makes Money is a ϴ such that • ϴ1D11 + ϴ2D21 + ϴ3D31>= 0 • ϴ1D12+ ϴ2D22+ ϴ3D32 >= 0 • ϴ1D13+ ϴ2D23+ ϴ3D33 >= 0 • And, at least one of the above is > 0 (not = 0) • {In matrix notation, this is the situation when ϴD > 0, where ϴ is a one by three row vector, D is a three by three matrix, and the product of the two is a one by three row vector}

Arbitrage exists when there is a ϴ such that The value of the portfolio is less than or equal to zero and the portfolio never loses money and always makes a profit in at least one state. If ϴP ≤ 0 then ϴD > 0 Or If ϴP < 0 then ϴD ≥ 0

Assume No Arbitrage Opportunities Exist If ϴD > 0 which means that the portfolio never loses money and makes money in at least one state of the world Then ϴP > 0, meaning that the portfolio must involve a positive investment This means that you cannot make a profit without putting up money.

So, what can we conclude if there are no arbitrage opportunities? • First, we can conclude that there is a set of prices: Ψ1, Ψ2, Ψ3 (all ≥ 0) such that: P1 = Ψ1D11 + Ψ2D12+ Ψ3D13 P2= Ψ1D21+ Ψ2D22+ Ψ3D23 P3= Ψ1D31+ Ψ2D32+ Ψ3D33 These Ψi’s are state prices (the marginal cost of obtaining additional dollar in state i

Now convert state prices to probabilities: Define: Ψ = Ψ1+ Ψ2+ Ψ3 Define πi = Ψi / Ψ Then π1+ π2+π3 = 1 These can be thought of as probabilities of state one, two and three occurring. They are not the true probabilities, though. They are known as “risk-adjusted” probabilities.

Now, using π’s, we can rewrite: P1 = Ψ1D11 + Ψ2D12+ Ψ3D13 P2= Ψ1D21+ Ψ2D22+ Ψ3D23 P3= Ψ1D31+ Ψ2D32+ Ψ3D33 as P1 =Ψ{π1D11+ π2D12+ π3D13} P2 = Ψ{π1D21+ π2D22+ π3D23} P3 = Ψ{π1D31+ π2D32+ π3D33} Note that the prices seem equal to “expected value” of dividends where probabilities are the πi’s (except for the role of Ψ)

How to interpret Ψ? • Suppose there is some ϴR with the property that: • ϴ1RD11+ ϴ2RD21+ ϴ3RD31 = 1 • ϴ1RD12+ ϴ2RD22+ ϴ3RD32 = 1 • ϴ1RD13+ ϴ2RD23+ ϴ3RD33 = 1 • Which means ϴRis a riskfree portfolio (asset) • Then: • Ψ = ϴ1Rp1+ ϴ2Rp2+ ϴ3Rp3 • Today’s value of the riskless porfolio • Then any security’s price = discounted expected return, where the • discount is Ψ and the probabilities are π’s (risk adjusted probabilities)

Second Mid Term Thursday, Nov 8 • Will Cover • Markowitz • Tobin • CAPM • Beginnings of State Price Approach • Will Not Cover • State prices • Risk adjusted probabilities • No class on Tuesday, Nov 6th(election day)

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Economics 434 Theory of Financial Markets