Value Flow and Price Impact

Multiswap quotes are built from value flow.

For one reserve leg:

Sigma = da * s / (a + kappa * da)

where:

a  = reserve amount
s  = scale
da = reserve change

Normalize by scale:

sigma = Sigma / s

Define reserve-relative trade size:

r = da / a

Then:

sigma(r) = r / (1 + kappa * r)

Single-leg price impact

Let pre-trade price in scale terms be:

P = s / a

Let post-trade price implied by value flow be:

P' = Sigma / da

Then:

P' = (s * sigma) / (a * r)
P' = P * sigma / r

So:

P' / P = sigma / r

Using the normalized expression:

P' / P = 1 / (1 + kappa * r)

This is the compact price-impact identity.

Price impact is governed by the product:

kappa * r

When kappa * r is small, single-leg execution remains close to the current price.

With:

rt = 0
kappa = abs(r)^n

we get:

P' / P = 1 / (1 + r * abs(r)^n)

For positive inflow and stableness = 10:

P' / P = 1 / (1 + r^11)

This eleventh-order term explains why the balanced benchmark creates such strong low-impact depth.

For a swap, unnormalized value flow is conserved across legs:

sum Sigma_i = 0

Because:

Sigma_i = s_i * sigma_i

normalized value flow must be multiplied by scale before summing.