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# Deflationary sAMM Model

The Speculative AMM's volatility pricing model adopts two conditions:
1. 1.
appreciation of the token
2. 2.
risk-free interest rate.
Thus, in theory, a deflation token over the Ethereum network can assume that neither of these two conditions will change.
Therefore, the sAMM can simplify the volatility pricing model by averaging the volatility effect. To achieve a deflationary sAMM model, the formula changes to:
$P_t C_t = { N_t S_t A_t \left({1 - \alpha \over r - \mu^P}\right)^{1 \over \alpha} }$
And the average μ of the volatility as the following:
$\left({1 - \alpha \over r - \mu^P}\right)^{1 \over \alpha} \implies { 1 \over K }$
The value of
$K$
should be reasonably maintained as a constant, say
$K = 1$
. Thus, the deflationary sAMM pricing model becomes:
$K = { N_t S_t A_t \over P_t C_t } = 1$

## Price Tick

The sAMM utilities the above formula's inverse function to represent the order's quantity at price
$P_t$
. For example, to stake amount
$M_a$
of DEXG tokens and
$M_b$
collateral tokens to the exchange Pool at time
$( t+1 )$
, the price of the deflation token can be determined as:
${P_{t+1}}^{-1} = { C_t + M_a \over N_t + M_b }$
In the example, collateral tokens are required prior to pooling DEXG tokens. The Speculative AMM can use this model to reduce price volatility and ensure a stabilized exchange rate. Thus, the DEXToken Protocol can use the Token Swap mechanism to exchange tokens. More details about the token swap function can be viewed in the DEXToken Protocol Whitepaper.
The underlying algorithmic implementation of the smart contract can be found at the early staging commit algo-1 on Github.