Avellaneda-Stoikov Optimal Quotes with CARA Utility

Options Pricing · Hard · Free problem

A market maker quotes bid and ask prices around a midprice $S_t$ that follows a driftless Brownian motion $dS_t = \sigma \, dW_t$. The market maker has CARA (exponential) utility $U(x) = -e^{-\gamma x}$, where $\gamma > 0$ is the risk-aversion parameter. The market maker sets bid and ask offsets $\delta^b, \delta^a > 0$ (so the bid is $S_t - \delta^b$ and the ask is $S_t + \delta^a$).

Buy market orders (hitting the ask) arrive as a Poisson process with intensity $\lambda^a(\delta^a) = A e^{-k \delta^a}$, and sell market orders (hitting the bid) arrive as a Poisson process with intensity $\lambda^b(\delta^b) = A e^{-k \delta^b}$, where $A, k > 0$ are constants.

Over a short horizon $[t, T]$ with $\tau = T - t$ small:

  1. Approximate the value function to first order in $\tau$ and derive the optimal offsets $\delta^{a,*}$ and $\delta^{b,*}$ as functions of current inventory $q$, risk aversion $\gamma$, volatility $\sigma$, order-flow parameter $k$, and time remaining $\tau$.
  1. State any small-$\tau$ approximations you use.
  1. Interpret how the spread asymmetry $\delta^{a,*} - \delta^{b,*}$ depends on inventory $q$.

Hints

  1. Start by writing down the CARA value function and noting that exponential utility lets you separate cash from inventory risk. What is the market maker's reservation price -- the effective midprice adjusted for inventory?
  2. The reservation price is $r = S - q \gamma \sigma^2 \tau$ to first order in $\tau$. Now think of the optimal quoting problem as a monopolist pricing against Poisson demand with exponentially decaying intensity -- optimize the expected utility gain from each side independently.
  3. For each side, maximize $A e^{-k \delta} (1 - e^{-\gamma \delta})$ over $\delta$. The first-order condition gives $\delta = \frac{1}{\gamma} \ln(1 + \gamma/k)$ for the symmetric component. The inventory skew enters through the reservation price shift.

Worked Solution

How to Think About It: This is the core Avellaneda-Stoikov (2008) market-making model. The market maker faces a tension: quoting tighter spreads attracts more flow and earns more spread, but holding inventory exposes you to adverse midprice moves. CARA utility penalizes variance in terminal wealth, so the optimal quotes must balance spread capture against inventory risk. The key insight is that inventory skews your quotes -- if you are long, you want to sell more aggressively (tighter ask) and buy less aggressively (wider bid). This is exactly how real market makers manage inventory.

Quick Estimate: For a risk-neutral market maker ($\gamma = 0$), the optimal spread is symmetric and equals

/k$ on each side (the classic monopolist markup against elastic demand). With risk aversion, the total spread widens by approximately $\gamma \sigma^2 \tau$ to compensate for inventory risk, and the midpoint of the quotes shifts by $-q \gamma \sigma^2 \tau$ away from the midprice (leaning against inventory). So if $q > 0$ (long), the reservation price drops below the midprice, tightening the ask and widening the bid.

Approach: We use the Hamilton-Jacobi-Bellman equation for the value function under CARA utility, expand to first order in $\tau$, and optimize the offsets.

Formal Solution:

Let $W$ be the market maker's cash and $q$ the inventory. Terminal utility is $U(W + q S_T) = -\exp(-\gamma(W + q S_T))$. Define the value function:

$V(t, S, W, q) = \sup_{\delta^a, \delta^b} \mathbb{E}_t\bigl[-e^{-\gamma(W + q S_T)}\bigr]$

Since $S_T = S + \sigma(W_T - W_t)$ under driftless BM, the terminal wealth $W + q S_T$ is Gaussian conditional on no fills. Using the CARA structure, we can write:

$V = -\exp\bigl(-\gamma(W + q S)\bigr) \cdot \theta(t, q)$

where $\theta(T, q) = \exp\bigl(\tfrac{1}{2} \gamma^2 q^2 \sigma^2 \cdot 0\bigr) = 1$ at terminal time, and $\theta$ absorbs the inventory-risk penalty.

For the no-fill evolution over $[t, t+dt]$, the midprice risk contributes:

$\mathbb{E}\bigl[e^{-\gamma q (S_T - S_t)} \mid \text{no fills}\bigr] = \exp\bigl(\tfrac{1}{2} \gamma^2 q^2 \sigma^2 \tau\bigr)$

This gives the reservation price (the indifference midprice for the market maker):

$r(t, q) = S - q \gamma \sigma^2 \tau$

This is the price at which the market maker is indifferent to holding inventory $q$ -- it shifts below the midprice when long and above when short.

Now the market maker sets offsets relative to their reservation price. Define $\tilde{\delta}^a = \delta^a + S - r = \delta^a + q \gamma \sigma^2 \tau$ and similarly for the bid. The optimization over each offset decouples (to first order in $\tau$). For the ask side, the market maker maximizes the expected utility gain from a fill arriving at rate $\lambda^a = A e^{-k \delta^a}$:

$\max_{\delta^a} \lambda^a(\delta^a) \cdot h(\delta^a)$

where $h(\delta^a)$ is the utility benefit of earning spread $\delta^a$ while reducing inventory risk. Under the CARA/small-$\tau$ expansion, this leads to:

$\max_{\delta^a} A e^{-k \delta^a} \cdot \bigl(1 - e^{-\gamma \delta^a}\bigr)$

Taking the first-order condition and solving exactly, the optimal offsets are:

$\delta^{a,*} = \frac{1}{\gamma} \ln\!\left(1 + \frac{\gamma}{k}\right) + \left(\frac{1}{2} - q\right) \gamma \sigma^2 \tau$

$\delta^{b,*} = \frac{1}{\gamma} \ln\!\left(1 + \frac{\gamma}{k}\right) + \left(\frac{1}{2} + q\right) \gamma \sigma^2 \tau$

Small-$\tau$ approximations used:

  • The value function is expanded to first order in $\tau = T - t$, ignoring $O(\tau^2)$ terms (probability of two or more fills).
  • The reservation price $r = S - q \gamma \sigma^2 \tau$ uses the first-order expansion of the exponential moment of the Gaussian midprice move.
  • The fill intensities are treated as constant over $[t, T]$ (justified when $\tau$ is small).

**Interpretation of $\delta^{a,*} - \delta^{b,*}$:**

$\delta^{a,*} - \delta^{b,*} = -2 q \gamma \sigma^2 \tau$

This is the inventory skew:

  • When $q > 0$ (long inventory): $\delta^{a,*} < \delta^{b,*}$, so the ask is tighter than the bid. The market maker leans toward selling to reduce long exposure.
  • When $q < 0$ (short inventory): $\delta^{a,*} > \delta^{b,*}$, so the bid is tighter. The market maker leans toward buying.
  • When $q = 0$: quotes are symmetric.
  • The skew is proportional to $\gamma$ (more risk-averse means more aggressive inventory management), $\sigma^2$ (more volatile assets require faster inventory reduction), and $\tau$ (more time remaining means more exposure to manage, though this is the first-order term).

The total spread is:

$\delta^{a,*} + \delta^{b,*} = \frac{2}{\gamma} \ln\!\left(1 + \frac{\gamma}{k}\right) + \gamma \sigma^2 \tau$

The first term $\frac{2}{\gamma} \ln(1 + \gamma/k)$ is the monopolist markup against elastic demand (approaches

/k$ as $\gamma \to 0$). The second term $\gamma \sigma^2 \tau$ is the risk premium for holding inventory over the remaining horizon.

Answer: The optimal offsets are $\delta^{a,*} = \frac{1}{\gamma} \ln(1 + \gamma/k) + (\frac{1}{2} - q)\gamma \sigma^2 \tau$ and $\delta^{b,*} = \frac{1}{\gamma} \ln(1 + \gamma/k) + (\frac{1}{2} + q)\gamma \sigma^2 \tau$. The spread asymmetry $\delta^{a,*} - \delta^{b,*} = -2q\gamma\sigma^2\tau$ shows the market maker skews quotes against inventory: long positions tighten the ask and widen the bid, and vice versa. The magnitude of the skew increases with risk aversion, volatility, and time remaining.

Intuition

The Avellaneda-Stoikov model captures the fundamental tension in market making: earning spread versus bearing inventory risk. The CARA utility framework makes this tension analytically tractable because exponential utility separates the "how much cash do I have" question from the "how much inventory risk am I carrying" question. The result is clean and intuitive -- your quotes center around a reservation price that is shifted from the true midprice by an amount proportional to your inventory, your risk aversion, and the remaining variance you face.

In practice, every electronic market maker implements some version of this inventory skew. When you are long, you want to get flat, so you make it cheap for people to buy from you (tight ask) and expensive for them to sell to you (wide bid). The model also shows why spreads widen with volatility and close to news events -- the $\gamma \sigma^2 \tau$ risk premium directly inflates the total spread. The

/k$ component reflects the elasticity of order flow: if flow is very price-sensitive (large $k$), the market maker cannot charge much markup. This interplay between adverse selection, inventory risk, and order-flow elasticity is the backbone of modern market microstructure.

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