Lecture 5 Flashcards
How is uncertainty modelled? What is it?
It is modelled through a state space, S. it is a set of states where only one is true but it is uncertain which
Special name for states under uncertainty and why?
States of nature since the decision maker has no influence on which of the states is true
What are events?
Sunsets of the state space
What is E^c?
The complementary event to E (the negation of E)
How are outcomes denoted?
Either Greek letters or x’s
How is each state denoted?
Each state of S is denoted s1, s2….
See rest of notation in notes
Now
What are prospects?
Prospects designate state-contingent outcomes - they are courses of action for the decision maker
Define decision under uncertainty?
In decision under uncertainty, S is a finite or infinite state space, and R is the outcome set. Prospects map states to outcomes. The domain of preference is the set of all prospects. >= is a preference relation on the set of prospects
What is a certainty equivalent?
A certainty equivalent of a prospect x is an outcome α such that α~x
Explain first way to make decisions under uncertainty?
By choosing probabilities for uncertain events (subjective) and then maximising EV wrt those probabilities
What is an EV maximiser? And what if not?
If your preference towards all decisions can be represented by EV we say you are an EV maximiser
If not, we say EV doesn’t hold for you
With EV maximin method what are the subjective parameters?
The subjective probabilities - they characterise the decision maker
See
Slides 19-24
See
Top of 2nd side of notes on steps 1-3
What is the axiom of additivity?
[x>=y -> x+z>=y+z] for all prospects x, y and z
Define arbitrage/a Dutch book?
When preferences x(j)>=y(j) for j=1,…,m, such that the preferred prospects x(j), when COMBINED, always (for each state S) yield less than the non-preferred prospects y(j)
See notes
Define risk neutral?
When a party is neither risk averse or risk seeking. Tf given 2 events with equal EVs, they will be indifferent even if one is much riskier!
Fundamental point in finance?
Idea that no arbitrage implies an as-if risk neutral (ie. EV) evaluation of financial assets
Define de finetti’s theorem?
The person’s preferences follow the preference relation >= as a weak order; for each prospect there is a CE, and additivity, transitivity and monotonicity all hold
What does de finetti’s theorem indicate?
That the EV maximiser will not be arbitraged
Read
In finance, arbitrage means that you can combine a number of portfolios in such a way that you always (in every state of the world) make a profit. In such a case, you can make money from the market without contributing anything, which you will of course do as much as possible. In practice, arbitrage opportunities disappear within fractions of seconds because they are immediately exploited by the largest and quickest market participants, i.e. investment banks that use automated computer programs. A fundamental result in finance is that no-arbitrage implies an as-if risk neutral (= EV) evaluation of financial assets. The same result was discovered before by de Finetti for individual choice, where arbitrage is called a Dutch book.
A person maximises EV is the same as saying?
The person’s preferences follow the preference relation ≽as a weak order; for each prospect there exists a certainty equivalent; and additivity, transitivityand monotonicity are satisfied.
ie. De Finetti’s Theorem!
What does a dutch book show?
It shows that if someone is not an EV maximiser (eg. they follow a maximin rule) then arbitrage can be made against them. This arbitrage is what a Dutch Book shows!
