A functional approach to the price impact of stock trades and the implied true price

We propose a functional approach to estimate the instantaneous price impact of a trade and to infer an implied true price. Our model expresses price impact as an S-shaped function of signed volume. It has two parameters, one for price impact and one for liquidity depth. The latter measures the differential impact of small and large trades. The price impact is instantaneous, that is, it occurs at the instant of trade execution. Our specification also permits the price impact of buys and sells to be asymmetric. We compute an implied true price from our model, and we find that it is closer than the quote midpoint to the unobservable true price. Our empirical analysis also shows that the effective spread, which is computed using the midpoint, has an upward bias, and that the implicit transaction costs may be lower than previous estimates.