Market making and inventory control

Definition A market maker posts a bid price $b$ at which it will buy and an ask price $a$ at which it will sell, and trades one standardized unit with each counterparty that arrives. With cash $x$ and inventory $q$, a fill at the bid changes them to $x - b$ and $q + 1$, and a fill at the ask changes them to $x + a$ and $q - 1$. Marked at the midpoint $s$, its wealth is $x + q s$, and the quote distances are $\delta^b = s - b$ and $\delta^a = a - s$. Inventory risk is the exposure of marked wealth to midpoint moves while $q \ne 0$. In this lesson the midpoint follows $dS_t = \sigma\,dW_t$, where $W$ is a Brownian motion under the dealer’s physical measure $\mathbb{P}$, so it is an arithmetic Brownian motion with zero drift and constant volatility $\sigma$ in currency per unit per square root of time, and the interest rate is zero. If the inventory is held fixed until the horizon $T$, with $H = T - t$ remaining, marked wealth changes by $q(S_T - s)$, which has mean zero and variance

Tick sizes, fees, queue priority and adverse selection are left out of the model.

Definition The dealer has exponential utility $U(w) = -e^{-\gamma w}$ of terminal wealth, with constant absolute risk aversion $\gamma > 0$ in inverse currency units. The certainty equivalent of a random terminal wealth $\Pi$ with $\mathbb{E}^{\mathbb{P}}[e^{-\gamma \Pi}] < \infty$ is the sure amount $\operatorname{CE}$ with $U(\operatorname{CE}) = \mathbb{E}^{\mathbb{P}}[U(\Pi)]$, that is

In the frozen inventory model of the previous card, $S_T$ is normal with mean $s$ and variance $\sigma^2 H$, and the normal moment generating function gives

so for every real $q$, exactly,

The inventory penalty is the last term $\tfrac12 \gamma \sigma^2 H\, q^2$, the amount by which the certainty equivalent falls short of the marked wealth. For any normal $\Pi$ the same computation gives $\operatorname{CE} = \mathbb{E}^{\mathbb{P}}[\Pi] - \tfrac12 \gamma \operatorname{Var}(\Pi)$, and the zero drift of the midpoint is what makes the mean equal to $x + qs$.

Definition In the frozen inventory model, the reservation bid $r_b$ is the price at which buying one unit leaves the certainty equivalent unchanged, $\operatorname{CE}(x - r_b, q + 1) = \operatorname{CE}(x, q)$, and the reservation ask $r_a$ solves $\operatorname{CE}(x + r_a, q - 1) = \operatorname{CE}(x, q)$. Since $(q \pm 1)^2 - q^2 = \pm 2q + 1$,

The reservation price is their midpoint

and the gap $r_a - r_b = \gamma \sigma^2 H$ does not depend on $q$. Neither value depends on the cash $x$, a property of exponential utility. Because the certainty equivalent is linear in cash, buying at any $b < r_b$ raises it by $r_b - b$ and selling at any $a > r_a$ raises it by $a - r_a$. Positive inventory places both values below $s$ and negative inventory places both above. They are exact indifference values for a single trade after which the inventory stays frozen. A dealer who keeps quoting has dynamic indifference values from the full control problem, which differ from these in general.

Definition Given the quote distances $\delta^b$ and $\delta^a$, sales to the dealer at its bid and purchases from it at its ask arrive as two counting processes with intensities

evaluated at $\delta^b$ and at $\delta^a$, where $A > 0$ is measured in fills per unit of time and $k > 0$ in inverse currency units. The model takes the two counts to be independent, and while the distances stay fixed they are Poisson processes, as in Limit order books and queueing models, so a fill occurs in a short interval of length $dt$ with probability $\lambda\,dt + o(dt)$, and the expected inventory changes at the rate $\lambda(\delta^b) - \lambda(\delta^a)$. The intensity equals $A$ at zero distance and is multiplied by $e^{-k}$ for each additional currency unit of distance. The law is a model of passive quotes. A negative distance, an ask below the midpoint or a bid above it, lies outside the range it is meant for, and a quote that reaches the other side of the book would trade at once instead of waiting for fills.

Definition Suppose the dealer quotes continuously until $T$, fills arrive at the exponential intensities, and the dealer maximizes $\mathbb{E}^{\mathbb{P}}\bigl[U(x_T + q_T S_T)\bigr]$. With zero interest the cash is riskless, so the value function satisfies $u(x, q) = e^{-\gamma x} u(0, q)$, with $s$ and $t$ suppressed. Let the dynamic indifference ask $r_a^{\text{dyn}}$ solve $u(x + r_a^{\text{dyn}}, q - 1) = u(x, q)$. A sale at $a = r_a^{\text{dyn}} + c$ then changes the value by

where $-u(x, q) > 0$, and the dynamic programming equation of Dynamic allocation and stochastic control multiplies this gain by the intensity $A e^{-k\delta^a}$, with $\delta^a = c + r_a^{\text{dyn}} - s$. Apart from factors that do not depend on $c$, the ask is chosen to maximize

Its derivative

is positive below and negative above

so $c^*$ is the unique maximizer over all real $c$, and the optimal ask is $r_a^{\text{dyn}} + c^*$. In the same way, with the dynamic indifference bid defined by $u(x - r_b^{\text{dyn}}, q + 1) = u(x, q)$, the optimal bid is $r_b^{\text{dyn}} - c^*$. This step is exact given the dynamic indifference values. To get a closed form, replace them by the frozen inventory values $r_b$ and $r_a$, which gives the approximate quotes

centered at the reservation price. This replacement is an approximation and not a solution of the full HJB equation, because future fills change the value of the inventory that the frozen values hold fixed. When the formula gives a quote that is not allowed, for example one off the tick grid, one that crosses the book or one that would breach an inventory limit, the quote has to be found by maximizing over the allowed set.