Tick Rounding Cost in Market Making

$a_c = P(V = 1 \mid \text{buy}) = \frac{p(1-q) + p \cdot q/2}{p(1-q) + p \cdot q/2 + (1-p) \cdot q/2}$

Simplifying the numerator: $p(1 - q/2)$. The denominator: $p(1 - q/2) + (1-p) \cdot q/2 = p - pq/2 + q/2 - pq/2 = p + q/2 - pq$.

$a_c = \frac{p(1 - q/2)}{p + q/2 - pq} = \frac{p(2 - q)}{2p + q - 2pq} = \frac{p(2-q)}{2p + q(1-2p)}$

Similarly, a sell arrives from an informed trader when $V = 0$, or from noise:

$b_c = P(V = 1 \mid \text{sell}) = \frac{p \cdot q/2}{(1-p)(1-q) + (1-p) \cdot q/2 + p \cdot q/2}$

Numerator: $pq/2$. Denominator: the total sell probability is $(1-p)(1-q) + q/2$, since sells come from informed traders when $V=0$ (probability $(1-p)(1-q)$) or from noise traders (probability $q/2$). Of these, $V=1$ sells come only from noise when $V=1$, with probability $p \cdot q/2$. Thus:

$b_c = \frac{pq/2}{(1-p)(1-q) + q/2} = \frac{pq}{2(1-p)(1-q) + q} = \frac{pq}{2 - 2p - 2q + 2pq + q} = \frac{pq}{2 - 2p - q + 2pq}$

Part 2: Rounding loss per fill.

The rounded ask is $a = \lceil a_c / \delta \rceil \cdot \delta$. The rounding loss on the ask side (from the perspective of the market maker as a cost of not being at fair value) is:

$\ell_a = a - a_c$

This is the amount by which the posted ask exceeds the fair conditional value. Note $0 \leq \ell_a < \delta$. This is actually a rounding *gain* for the maker on the ask side (she sells above fair value). The rounding loss on the bid side is:

$\ell_b = b_c - b$

where $b = \lfloor b_c / \delta \rfloor \cdot \delta$, and $0 \leq \ell_b < \delta$. Again, this is a gain for the maker (she buys below fair value).

If we define rounding loss from the market's perspective (cost to liquidity takers), then per fill the ask-side loss is $\ell_a = a - a_c$ and bid-side loss is $\ell_b = b_c - b$.

Part 3: Expected per-trade rounding loss averaged over sides.

Let $\pi_a$ be the probability a trade is a buy (fill on the ask) and $\pi_b = 1 - \pi_a$ the probability it is a sell. The expected per-trade rounding loss is:

$\bar{\ell} = \pi_a \cdot \ell_a + \pi_b \cdot \ell_b$

where $\pi_a = p(1 - q/2) + (1-p) \cdot q/2$ and $\pi_b = (1-p)(1-q/2) + p \cdot q/2$ from the arrival probabilities.

If $p = 1/2$ (symmetric prior), then $\pi_a = \pi_b = 1/2$ and $\bar{\ell} = (\ell_a + \ell_b)/2$.

Part 4: Upper bounds.

Since $\ell_a = a - a_c$ where $a = \lceil a_c/\delta \rceil \cdot \delta$, we always have:

$0 \leq \ell_a < \delta, \quad 0 \leq \ell_b < \delta$

So the per-side rounding loss is strictly bounded by one tick:

$\ell_a < \delta, \quad \ell_b < \delta$

The expected per-trade rounding loss is bounded by:

$\bar{\ell} < \delta$

A tighter bound uses the fill probabilities. Let $f_a$ and $f_b$ be the fill probabilities on each side. The expected rounding cost to the market maker (the cost of not being at the exact continuous optimum, accounting for the change in fill rate) is bounded by:

$\text{Rounding cost} \leq \frac{\delta}{2}(f_a + f_b)$

This follows because rounding shifts each quote by less than $\delta$, and the first-order effect on expected profit is the shift times the fill probability.

Answer: The continuous quotes are $a_c = \frac{p(2-q)}{2p + q(1-2p)}$ and $b_c = \frac{pq}{2(1-p)(1-q) + q}$. Rounding losses per fill are $\ell_a = \lceil a_c/\delta \rceil \cdot \delta - a_c$ and $\ell_b = b_c - \lfloor b_c/\delta \rfloor \cdot \delta$, each bounded above by $\delta$. The expected per-trade loss averaged over sides is $\bar{\ell} = \pi_a \ell_a + \pi_b \ell_b < \delta$, with the tighter bound $\bar{\ell} \leq \frac{\delta}{2}(f_a + f_b)$ in terms of local fill probabilities.