Market making formulas

Market structure: the book, the order types and who pays whom

Queue-weighted microprice

P_{\mathrm{micro}}=\frac{P_bQ_a+P_aQ_b}{Q_a+Q_b}

Weights the bid by ask size and the ask by bid size; it leans toward the thinner side of the book.

Remember

Price–time priority makes queue position an asset worth protecting.

Theoretical value, width and skew

Turning uncertainty into a width

\text{half-width} \approx z \cdot \sigma_{\text{your estimate}}

Treat your own estimate as a distribution. A market roughly one standard deviation either side of your theo is a sensible default; when asked explicitly for a confidence interval, use the z-score for the level requested.

Width grows with the square root of the pieces

\sigma_{\text{sum}} = \sigma\sqrt{n}\ \ (\text{independent pieces})

A market on a sum of $n$ independent pieces is $\sqrt n$ times as wide as a market on one piece, not $n$ times: errors partly cancel. Ten dice need a market about $\sqrt{10} \approx 3.2$ times as wide as one die, around a theo ten times as large.

Remember

Answer three questions in order: what is it worth, how sure am I, what am I holding.

Reading the fill: adverse selection

Kyle’s lambda

\Delta p = \lambda\,(x + u), \qquad \lambda = \frac{\sigma_v}{2\sigma_u}

In Kyle’s model a market maker sets the price change proportional to net order flow. The slope rises with how much there is to know ( $\sigma_v$ ) and falls with how much noise trading hides it ( $\sigma_u$ ). A deep market is one with plenty of uninformed flow to camouflage the informed.

Remember

A fill is information: you are selected against by whoever most disagrees with your price.

Inventory: carrying risk, shedding it, and the reservation price

The Avellaneda–Stoikov spread

\delta^a + \delta^b = \gamma\sigma^2(T - t) + \frac{2}{\gamma}\ln\!\left(1 + \frac{\gamma}{k}\right)

The total optimal spread has two parts: compensation for carrying inventory risk until the end of the horizon, and a term set by how quickly the chance of a fill falls as you quote further away ( $k$ ). The quotes are centred on the reservation price, not the mid.

Remember

Your position, updated after every single trade.

The trading game: quoting, requoting, and the questions at the end

Key rules

On a known distribution, width comes from the standard deviation and a wide market is indefensible.
On an unknown quantity, decompose out loud first and quote around the estimate.
Separate the two reasons to requote: inventory always, theo only against informed flow.

Bet sizing, Kelly and risk of ruin

The Kelly criterion

f^{*} = \frac{bp - q}{b} = \frac{\text{edge}}{\text{odds}}

The fraction of your bankroll that maximises the long-run growth rate of a repeated bet.

Kelly for any payoff

f^* = \arg\max_f\ \mathbb{E}\big[\ln(1 + fX)\big]

For a bet whose return per unit staked is the random variable $X$ , Kelly maximises expected log wealth. With a win of $b$ or a total loss, it reduces to the familiar edge-over-odds. For small, frequent bets it approximates $\mu/\sigma^2$ .

Risk of ruin with fixed bets

\Pr(\text{ruin}) = \left(\frac{q}{p}\right)^{B}

Betting one unit at a time with win probability $p > \tfrac{1}{2}$ at even money, against an opponent with unlimited capital, a bankroll of $B$ units is eventually lost with this probability. Edge shrinks the base; bankroll shrinks it exponentially.

Drawdowns under Kelly

\Pr(\text{wealth ever falls to } x\,W_0) = x^{2/c - 1}

For a continuously rebalanced bet at a fraction $c$ of the Kelly stake, the probability of ever falling to a fraction $x$ of starting wealth. At full Kelly ( $c = 1$ ) the chance of ever halving is one half; at half Kelly it is one eighth.

Remember

Kelly stakes the edge divided by the odds; with even money that is just the edge.