Preliminaries

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NOTATION

Nomenclature for the various parameters and settings applicable to formulas in this document is provided:

1. Matrix Notation for Protocol State. Vectors are capitalised and vector and matrix elements use subscript, such as discrete time (t) and the any/all symbol (*).

2. Lambda Notation for Parameters (λ). Lambda is used to represent coefficients that are applied throughout our set of risk and pricing functions such that they may be calibrated. Throughout, references are often to a single, general, “lambda” coefficient in equations, however these are distinct in implementation.

3. Theta Notation for Configuration variables (θ). Theta is used to represent general protocol settings, such as the value of the Protocol Fee. Throughout this document we refer to a general “theta” coefficient in equations, however these are distinct in implementation.

4. Reference values (r) are setpoints (aka “targets”) that are used in the calculation of various protocol functions. These may be static or dynamic.

5. Measured values and inputs are generally designated as (⍺).

6. Set indices generally have reserved lettering:

   • u for referring to price risk unit vectors

   • v for referring to liability risk unit vectors

   • w for referring to liquidity risk unit vectors

   • i for referring to Takers

   • j for referring to Makers

   • t for referring to time

   • k general

7. Special Values:

   • Virtual Probability of Claim (β)

   • Yield Target (E) / epsilon (ɛ)

CONVENTIONS

• Assets and Capital are commonly referred to as ETH and USD respectively throughout for brevity.

• We refer to “USD” as the stablecoin currency for capital so as to isolate a specific implementation currency (i.e. USDT for Bumper v1.0 on Arbitrum).

• For the formulations contained in these docs, full lambda notation may also be omitted for brevity.