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Price Impact by Size

Premia pools disincentivize highly disruptive trades through size price impact.
After every transaction, the pool price level updates from
CtCt+1C_t \rarr C_{t+1}
, depending on the size and the direction of the trade. This results in either an increased or decreased price for the next buyer/underwriter. There are no obvious reasons to disincentivize larger blocks of provided liquidity (on the LP side), however, whale-buying behavior needs to be accounted for.
Suppose a whale is waiting on the sidelines for the C-level to fall below their perceived market equilibrium, just to scoop up 50% of the pool's liquidity. Such a trade would cause a significant pool price level disruption; this disruption needs to be accounted for in the price charged to the whale. If the starting value is
, and the ending value (post whale trade) becomes
, s.t.
Ct+1>CtC_{t+1} > C_{t}
, what price impact penalty should be imposed on the whale?
In discrete form, the whale would end up paying
BS(Vi)(Ct+Ct+1)2BS(V_i)* \frac{(C_t+C_{t+1})}{2}
, however the differential form is slightly more accurate:
or more intuitively - the normalized step size, relative to the free capital in the pool.
Putting it all together, using
Ct=Ct adjusted for slippageC^*_t=C_t \text{ adjusted for slippage}
as a potential future trade-specific steepness modifier, we get a final pricing function:
Pt(Vi;Ct)=BS(Vi)Cts.t.Ct=Ctxt0exαx(10xt)P_{t}(V_i;C_t)=BS(V_i)*C^*_t \\ s.t.\hspace{0.25cm}C^*_t=C_t*\int^0_{x_t}e^{-x} \alpha_x*(\frac{1}{0-x_t})
This ensures large traders have no advantage over smaller traders.