Lesson 1: Time-State Claims
Economics of Finance
School of Economics, UNSW
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What is finance?
Finance 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
Financial contract: an Arrow-Debreu paradigm (theory)
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
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)
Dealers (opportunities to trade)
• 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.
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