# Modified Exchange Rate
First define *σ*(*t*) to be a piecewise sigmoid function of time (*t*) (*detailed in* [*Sigm*](https://providence-labs.gitbook.io/providence-labs/how-providence-works/markets/exchange-rate-adjustment/sigma)[*a* ](https://providence-labs.gitbook.io/providence-labs/how-providence-works/markets/exchange-rate-adjustment/sigma)*section*):
• Let *Nj* denote the number of outcome *j* tokens minted to the swapper.
• Let *Ni* denote the number of outcome *i* tokens burned by the swapper.
• Let *Eij* denote the exchange rate between *i* and *j* outcome tokens.
• Let *Si* denote the total supply of outcome *i* tokens.
• Let Sj denote the total supply of outcome i tokens.
A term *P* is given by the reciprocal of the current unmodified exchange rate (*Si / Sj*).
And define a term *a*:
A function *f*(*t*) can be derived:
Combining this, the exchange rate between outcome *i* and *j* (*Eij*) is given:
Note that the reciprocal relation is given by Eji = σ(t) · p/f(t).
Figure 2: Visualization of the exchange rates Eij(t) and Eji(t) over time, based on σ(t) and f(t), assuming constant odds p and Te (expiry) of 10.
A small swap fee as a percentage of the outcome tokens burned will be collected and automatically liquidated into USDC post-settlement.
