One Period Model

Pre Black-Scholes/Merton

Samuelson, …

Equity bias. Bonds…

Let T be the set of all possible trading times and I be the set of market instruments.

Let T(i) \subseteq T be the trading times for instrument i\in I.

See mp-units

The type of the amount is dividend, interest, principal, fee, tax, …
The kind of the type is the instrument.

The same instrument may trade on multiple exchanges, over the counter, through a broker/dealer, on a dark market, etc. This is the type of the instrument.

Instruments

Stock. Has an issuer. One or more IPO’s that determine the float.

An amount is an integral quantity of the minimum trading size of an instrument and a kind.
An amount also has a kind, such as dividend, interest, or principal payment.

A holding is an amount and owner that decides to buy or sell the holding.

A position is a (multi)set of holdings.

We assume holdings are divisible. The holding (a, o) can be split into $(a’, o),

A real-valued functions takes points in a set to the real numbers.

A real-valued measure takes subsets of a set to the real numbers, does not count things twice, and the measure of nothing is 0.

B(S), B(S)^*\cong ba(S).

Exercise. If \mu(A\cup B) = \mu(A) + \mu(B) - \mu(A\cap B) then \nu(A) = \mu(A) - \mu(\emptyset) is a measure.

Every instrument determines a type. Every type has a kind.
E.g., a $ can be a dividend, interest, principal,…

How to connect real-world vs risk-neutral measures.

The concept of no arbitrage. Efficient markets.

Probabilities change over time.

\Omega is all possible outcomes …

Risk-neutral vs real-world measures.

Samuelson - deflated prices are martingales (under risk-neutral measure) What about real world measures???

Futures quotes are martingales.

Ross Recovery.

Neo-economics.

Markowitz - stock prices (efficient markets)