Lecture 1: Time-State Claims
Economics of Finance
School of Economics, UNSW
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What is finance?
’Finance concentrates on exchanges in which money of one type or another is likely to appear on both sides of a trade.’
– . deals with payment now, payment in the future and uncertainty.
• Key factors: Time & Uncertainty.
Example: an apple tree
Good weather
Two time periods: t = 0, t = 1, spring- no apples and fall-uncertain apples.
Bad weather
Why do we care?
Types of questions relevant for finance:
• How much apple does an apple tree worth?→ Pricing
• How do we optimise our future apple stream? → Portfolio Problem
Why is it important?
Catastrophic financial consequences when we get it wrong!
Why economics of finance?
Economics is a science of people and market. Some questions:
• Why do people behave differently in financial market?
• Does the asset price reflects efficient allocation?
• How does financial market implement the risk sharing role? • How does trading financial asset improves social welfare?
• Does completing the financial market improves efficiency?
Financial contract: an Arrow-Debreu paradigm
and introduced the concept of (state-) contingent contract:
’A contract for the transfer of a commodity [specifying], in addition to its physical properties, its location and date, an event on the occurrence of which the transfer is conditional.’
, Theory of Value, The Cowles Foundation Monograph, 1959.
In short, a financial contract is a Time-state claim.
A simple environment
Key elements: Discrete time & discrete states Two time periods:
• Time 0 – today
• Time1-ayearfromnow
Two possible states of the world: • G: good weather
• B: bad weather
These states of the world are:
• mutually exclusive (no state that is both good and bad) • exhaustive (one and only one of the states will occur)
An all-apple economy
Suppose the only commodity traded in this economy is apple • No money per se;
• Apple is the unit of account (numeraire)
Why apples?
• Consumable (it’s good);
• Countable, and perfectly divisible (it is measurable); • Non-storable (timing matters!).
State-contingent production
The only type of productive investment is: APPLE TREE The tree will produce:
• 63 apples if the weather is good • 48 apples if the weather is bad
An apple tree
Good weather
Bad weather
Elementary claims
There are two elementary time-state claims:
• One apple at time 1 if the weather is GOOD
• One apple at time 1 if the weather is BAD We will refer to these claims as:
• GA – ’Good weather apples’, • BA – ’Bad weather apples’
Similarly, we will refer to a present apple as ’PA’.
Atomic security
We interchangeably refer to a claim as a security. A security is a certificate of the following form:
I, , promise to deliver to the bearer of
this certificate one apple at the end of year 1 if
and only if the weather during the year has been
• Implicitly, we assume that a credit agency has established that the security is AAA, i.e. default free.
• Atomic security is an atomic time-state claim (also known as basic Arrow-Debreu security, ’primitive’ security)
There exists a group of dealers stand ready to trade atomic claims.
G Dealer is willing to trade:
• 0.285 PA for 1.0 GA or • 1.0 GA for 0.285 PA or • any multiple of those.
Good weather apple
Good weather
0.285 apples
Bad weather
Further to Dealer G, Dealer B is willing to trade:
• 0.665 PA for 1.0 BA or • 1.0 BA for 0.665 PA or • any multiple of these
Atomic security: Bad weather apple
Good weather
0.665 apples
Bad weather
Complete Market
We have two explicit markets: PA GA and PA BA.
• Notice we can make combinations of any arbitrary number
GA and BA to construct any portfolio we desire
• We call this complete market
• So far all trades involves PA payment. What about other possible trades?
Other Types of Trade
Consider the following contract:
• Party A promises to pay Party B: 6 apples if the weather is good
• Party B promises to pay Party A: 3 apples if the weather is bad
• Neither party pays the other anything today (on signing) Such a contract is called a swap.
Swap is the third possible type of trade in our world: GA BA.
Perspective of Party A
Good weather
Bad weather
Perspective of Party B
Good weather
Bad weather
Put it all together
• Dealer G trades 0.285 PA for 1.0 GA or vice versa; • Dealer B trades 0.665 PA for 1.0 BA or vice versa; • Party A trades 6 GA for 3 BA or vice versa;
Looks like we can make some profit out there. How?
An arbitrage provides a positive net payoff in at least one time and state and no negative net payoff in any time and state.
The payment matrix
• each row represents a transaction;
• each column represents a time-state combination;
Party A Dealer B Dealer G Net
Arbitrage strategy
We now construct a set of transactions which implements an arbitrage.
Step 1: Go to Party A, and sign a contract swapping 6GA with 3BA;
Party A Dealer B Dealer G Net
This creates a position of −6GA and 3BA on your balance sheet.
Step 2: Go to Dealer B, and sell 3BA to her. In return, receive a credit of
3 × 0.665 = 1.995P A.
0 3 −6 3×0.665=1.995 −3 0
Party A Dealer B Dealer G Net
This transaction close out the position of BA.
Step 3: Go to Dealer G, and buy 6GA from her. Pay 6 × 0.285 = 1.710P A.
Party A Dealer B Dealer G Net
0 3×0.665=1.995 −6 × 0.285 = −1.710
3 −6 −3 0 0 6
This transaction close out the position of GA.
To finish off, summarize the transactions:
Party A Dealer B Dealer G Net
0 3×0.665=1.995 −6 × 0.285 = −1.710 0.285
3 −6 −3 0 0 6 0 0
Our strategy is creating a profit without any lost in any state. By definition, this is an arbitrage.
Several ways to arbitrage
Step 1: Go to Party A, sign a contract swapping 6GA with 3BA Step 2: Go to Dealer B, and sell 3BA to her. In return, receive a credit of
3 × 0.665 = 1.995P A.
Step 3: Go to Dealer G, and use all the of 1.995P A you
received to buy GAs from her
1.995P A/0.285 = 7GA.
Party A Dealer B Dealer G
0 3×0.665=1.995 −7 × 0.285 = −1.995
3 −6 3 0 0 7
This is still an arbitrage as there is a net profit in GA. It will be realised only if GA happens.
Arbitrage free environment
• When an arbitrage opportunity arises, traders will exploit it and cause the terms of trade to adjust until no arbitrage is possible.
• We call this arbitrage free environment.
The Law of One Price (LOP)
Definition: (LOP) In an arbitrage-free economy with no transactions costs, any given time-state claim will sell for the same price, no matter how obtained. This holds for any ’package’ of time-state claims.
• In the ‘real world’ transactions costs are usually present;
• The lack of arbitrage opportunities only insures that prices for a given set of time-state claims will fall within a band narrow enough to preclude generating a positive profit net of transactions costs out of trading.
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