# Chapter 2. Two-State Economy

As an empirical domain, finance is aimed at specific answers, such as an appropriate value for a given security, or an optimal number of its shares to hold.

Darrell Duffie (1988)

The notion of arbitrage is crucial to the modern theory of Finance.

Delbaen and Schachermayer (2006)

The analysis in this chapter is based on the most simple model economy that is still rich enough to introduce many important notions and concepts of finance: an economy with two relevant points in time and two uncertain future states only. It also allows us to present some important results in the field, like the Fundamental Theorems of Asset Pricing, that are discussed in this chapter.1

The simple model chosen is a means to simplify the formal introduction of the sometimes rather abstract mathematical concepts and financial ideas by avoiding as many technicalities as possible. Once these ideas are fleshed out and well understood, the transfer to more realistic financial models usually proves seamless.

This chapter covers mainly the following central topics from finance, mathematics, and Python programming:

Finance Mathematics Python

Time

Natural numbers $double-struck upper N$

`int`, `type`

Money (currency)

Real numbers $double-struck upper R$

`float`

Cash flow

Tuple

`tuple`, `list`

Return, interest

Real numbers $double-struck upper R$

`abs`

(Net) present value

Function

`def`, `return`

Uncertainty

Vector space $double-struck upper R squared$

`NumPy`, `ndarray`, `np.array`

Financial asset

Process

`ndarray`, `tuple`

Risk

Probability, state space, power set, mapping ...

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