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Complete Markets

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Definitions

Event State of the world State Contingent Claim (State Claim)

Payoff Vector Market is a payoff vector

Exchange dollars today for state-contingent bundle of dollars tomorrow

Markets are complete If we can arrange a portfolio with any payoff vector

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Uncertainty

Market complete? Interest rate? Probability of War?

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exampleIf I know what pure securities pay TOMORROW ($1 in only one state - e.g. "u" or "d") and I know their prices TODAY (p_u and p_d) then I can figure out the price TODAY of any security generating payoffs (cash flows) TOMORROW.

In the example you refer to, we work `backwards'. We know the price TODAY (V_1 = 1) of a security that pays (TOMORROW) 1.5 in the "u" state and 0.5 in the "d" state AND we know the price TODAY of the risk-free bond (b = 1) that pays 1 in BOTH states TOMORROW (that's why it is risk-free - it doesn't matter which state prevails) - note that since b = 1 TODAY, the risk-free rate of interest is 0. Knowing these 2 prices allows us to compute the prices of the pure securities TODAY: p_u = 0.5 and p_d = 0.5. Now we can determine the price TODAY of ANY other security in this world - e.g.: a security that pays (TOMORROW) 0.5 in "u" and 0 in "d," must have a price of 0.25 TODAY ...

TODAY, in this example, simply means some time before the state (here "u" or "d") is revealed at some later time (perhaps only an instant later) - here called TOMORROW.

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Financial Decision Making

Market prices determine value Competitive markets One-sided markets

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Time Value of Money

$1 today is worth more than $1 tomorrow Interest rate is the exchange rate across time $1 in your pocket is worth more than $1 promised

Which is worth more than $1 expected Which is worth more than $1 hoped for

Risk-free rates PV NPV NPV + Borrowing or Lending

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Time Value of Money

Interest rate is the exchange rate across time

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Time Value of Money

PV, NPV

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Time Value of Money

NPV + Borrowing and Lending

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Arbitrage

Arbitrage Certain profit by exploiting different pricing for the

same asset Law of one price

An asset has the same price in all exchanges No-arbitrage and security pricing

Bond $1000, 1 year, 5% What if over-priced or under-priced? Determine interest rates from bond prices

Other securities

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Separation Principle

Security transactions in a normal market do not create nor destroy value

This allows us to only focus on the NPV of the project And not worry about the financing choice

Example: Cost today: $10M Benefit in 1 year: $12M Risk-free rate: 10% Ability to issue $5.5M security today Does the issuance matter?

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Portfolio Valuation

Value additivity Price of a portfolio is the sum of the prices of individual

securities A firm is a portfolio of projects

The value of the firm is the sum of the values of all projects

Maximizing NPV for each decision maximizes the value of the firm

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Price of Risk

$1 in your pocket is worth more than $1 promised Which is worth more than $1 expected Which is worth more than $1 hoped for

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Risk Premium

Expected return Risk premium No-arbitrage pricing of a risky security

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Risk Premiums

Depends on risk Riskier securities command higher risk premium

Risk is relative to the overall market Risk premium can be negative

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Risk Premiums

Risk premium depends on risk:

rs = rf + (risk premium for investment s)

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Arbitrage and Transaction Costs

Two types of costs: Commissions Bid-ask spreads

No arbitrage conditions hold “up to transaction costs”

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Financial System

Financial market Security

Bond Stock Option Mutual fund Exchange-traded fund Hedge fund Private equity fund

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Asymmetric Information

Adverse selection Moral hazard Financial intermediaries Free markets

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Money

Barter system Money is a medium of exchange

Commodity money Fiat mondey

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Money supply

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Money supply

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Money Supply

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The Euro

What happens if a country imports too much Currency devaluates

What if currency is fixed (Greece) Wages must fall or Output declines

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Bitcoin

http://bitcoin.org/en/