9.3 Arrow-Debreu Securities Model and Mathematical Framework

Arrow-Debreu securities are used in economics to model a complete market, where risks can be segmented and priced according to different future states. Each security corresponds to a state of nature and pays out when that state is realized.

Mathematically, the model can be expressed as:

$$x_1 + x_2 + \dots + x_n = 1$$

Where:

• x_1, x_2, \dots, x_n are the prices of Arrow-Debreu securities for states 1 through n.

• The sum of all state-contingent securities equals 1, representing the full market structure.

For example, in a two-period model, where the market can have High, Medium, or Low states, Arrow-Debreu securities would correspond to assets purchased at prices

$$P_H, P_M, and P_L$$

for each state, with returns based on which state actually occurs.

Mathematically, we can write:

$$P_H \cdot x_H + P_M \cdot x_M + P_L \cdot x_L = 1$$

Where:

• x_H, x_M, x_L represent the quantities of each Arrow-Debreu security for High, Medium, and Low states, respectively.

• The prices P_H, P_M, P_L reflect the risk-neutral probabilities of each state occurring.

In a multi-state market, this model can be scaled to account for infinite scenarios, making it highly adaptable to various asset classes and market conditions.

The Arrow Token (ARR) mirrors this concept by acting as a dynamic risk management tool, adjusting based on the risks associated with different states in the decentralized finance world. This means that ARR can provide state-contingent coverage for DeFi protocols, NFTs, and real-world assets, adapting to real-time market conditions and ensuring accurate risk pricing.