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# Algorithmic functions

As outlined in Speculative AMM (sAMM) the Volatility Effect on the Adoption and Valuation of Tokenomics research paper provided the mathematics which is adopted in the deployed sAMM models. At the core of each sAMM model, such as the Deflationary sAMM, is the 'Universal Pricing Model'.
Conception - UPM builds the core of the sAMM

## Universal Pricing Model (UPM)

The UPM is an algorithmic formula derived from the academic research paper to determine the token price with a set of specific on-chain factors. The sAMM will pull these data-points directly from the Ethereum blockchain.
• $N_t$
: Total amount of collateral tokens in the pool
• $S_t$
: Aggregated transaction needs
• $A_t$
: Platform productivity
• $C_t$
: Amount of tokens in circulation, including locked tokens
The model proposed by Jollen Chen et al describes the token valuation algorithm that the DEXToken Protocol uses to provide token liquidity. Utilising the mentioned data-points the UPM will use the following formula to assess the token price
$P_t$
.
$P_t= { N_t S_t A_t \over C_t }$

## Volatility Effect on the Token Price

Adding the expected token appreciation
$u^P$
and risk-free interest rate
$r$
to this model, and under the condition of full adoption, setting to a constant between 0 and 1, then:
$P_t= { N_t S_t A_t \over C_t } \left({1 - \alpha \over r - \mu^P}\right)^{1 \over \alpha}$
The Speculative AMM uses this volatility pricing model to determine the mintable tokens at time
$t$
.