Economics 434 Financial Markets Professor Burton University of Virginia Fall 2015 September 3, 2015

Today Tomorrow s1s1 s3s3 s2s2 And, we may not have any idea what the probabilities of s 1, s 2, s 3 may be!! September 3, 2015

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 – p i, 1 is the payoff for security i in state one – p i, 2 is the payoff for security i in state two – p i, 3 is the payoff for security i in state three September 3, 2015

s1s1 s3s3 s2s2 Definition of Securities X1X1 X2X2 X3X3 X4X4 X5X5 P 1,1 P 1,2 P 1,1 P 1,3 P 2,1 P 2,2 P 2,3 P 3,1 P 3,2 P 3,3 P 4,1 P 5,1 P 4,2 P 5,2 P 4,3 P 5,3 September 3, 2015

What Does a Security Cost Today? P 1 times Ɵ 1 is what it costs to buy a quantity Ɵ 1 of security one at price P 1. Or simply: P 1 Ɵ 1 Similarly for 2, 3, etc. P 1. is always a positive number, but what about Ɵ 1. That might be negative You may have sold security one Long sale if you already owned it, but could be a short sale September 3, 2015

So, What Does a Portfolio of Securities Cost? A portfolio is three numbers in a world of three securities: Ɵ 1, Ɵ 2, Ɵ 3 where the Ɵ’s are the amounts purchased or sold of securities one, two and three Ɵ 1 P 1 + Ɵ 2 P 2 + Ɵ 3 P 3 This could be positive or negative September 3, 2015

What does this security pay? (these can be negative as well as positive) In state one: Ɵ 1 p 1,1 + Ɵ 2 p 2,1 + Ɵ 3 p 3,1 In state two: Ɵ 1 p 1,2 + Ɵ 2 p 2,2 + Ɵ 3 p 3,2 In state three: Ɵ 1 p 1,3 + Ɵ 2 p 2,3 + Ɵ 3 p 3,3 September 3, 2015

Now we know What a portfolio costs And What a portfolio pays (and when) September 3, 2015

So, what is an arbitrage? Two cases – A portfolio that costs nothing (or a negative amount) – Has a positive payoff in at least one state and no negative payoffs in all others – A portfolio costs a finite amount of money – Has an infinite payoff September 3, 2015

No Arbitrage Means P 1 φ 1 + P 2 φ 2 + P 3 φ 3 ≤ 0 Implies The following three conditions: – p 1,1 φ 1 + p 2,1 φ 2 + p 3,1 φ 3 ≥ 0 – p 1,2 φ 1 + p 2,2 φ 2 + p 3,2 φ 3 ≥ 0 – p 1,3 φ 1 + p 2,3 φ 2 + p 3,3 φ 3 ≥ 0 Where at least one of the last 3 inequalities is strict, otherwise the budget constraint is strict September 3, 2015

Fundamental Assumption of Modern Finance Assume that given the availability securities and current prices, there are no arbitrage opportunities (portfolios) available This assumption drives all of modern finance Equilbrium is not required (but equilibrium would imply that there are no arbitrage opportunites September 3, 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 3, 2015