6.2.2 The Market-Maker's Problem

Basic Prerequisites: 01 · From Zero and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

This page states the market-maker’s optimization problem precisely and derives its first object - the reservation (indifference) price. The practical objective: before solving for quotes, understand what the dealer is maximizing, why exponential utility is the right choice, and how the arrival rates of orders enter the problem. Everything is the AS 2008 setup (§2) restated with the derivations filled in.

The model has three ingredients:

1. A price process - the mid-price $S_t$ follows arithmetic Brownian motion $d{S}_u=\sigma d{W}_u$. It is the dealer’s marking price, not a price at which he can trade costlessly.
2. A utility objective - the dealer maximizes the expected utility of terminal wealth $X_T+q_T{S}_T$ under constant absolute risk aversion (CARA): $u(z)=-\exp (-\gamma z)$.
3. An arrival-rate mechanism - his limit orders fill as Poisson processes whose intensities decrease with distance from the mid: $\lambda (\delta )=A{e}^{-k\delta }$.

Why exponential utility? It is the unique (up to affine) utility with no wealth effect: the optimal strategy does not change when you scale wealth, so the reservation price is independent of initial cash $x$. That is what makes the whole problem separable - the same reason AS can define clean indifference prices.

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2. Mathematical Ground Truth & Derivations

2.1 The frozen-inventory value function

Consider first an inactive dealer who holds $q$ shares to time $T$ and cannot trade. His wealth is $X_T+q{S}_T=x+q{S}_T$ (no interest). Since $S_T=s+\sigma (W_T-W_t)\sim \mathcal{N}(s,{\sigma }^2\tau )$ with $\tau =T-t$,

$$v(x,s,q,t)={\mathbb{E}}_t\left[-\exp \left(-\gamma (x+q{S}_T)\right)\right]=-\exp (-\gamma x)\exp (-\gamma qs)\exp \left(\frac{{\gamma }^2{q}^2{\sigma }^2\tau }{2}\right).(\text{AS }2.3)$$

The derivation is Gaussian-integral bookkeeping: $\mathbb{E}[-\exp (-\gamma q{S}_T)]=-\exp (-\gamma qs+\frac{1}{2}{\gamma }^2{q}^2{\sigma }^2\tau )$ because the moment generating function of $\mathcal{N}(s,{\sigma }^2\tau )$ is $\exp (\gamma qs+\frac{1}{2}{\gamma }^2{q}^2{\sigma }^2\tau )$ evaluated at the right sign.

2.2 Reservation (indifference) prices (AS Definition 1)

The reservation bid $r^b$ is the price making the dealer indifferent between his current portfolio and his portfolio plus one share:

$$v(x-r^b,s,q+1,t)=v(x,s,q,t).(2.4)$$

The reservation ask $r^a$ solves $v(x+r^a,s,q-1,t)=v(x,s,q,t)$ (2.5). Substituting the closed-form $v$ and cancelling (both sides share $-\exp (-\gamma x)$, and the indifference makes the price independent of $x$):

$$r^a(s,q,t)=s+(1-2q)\frac{\gamma {\sigma }^2\tau }{2},r^b(s,q,t)=s+(-1-2q)\frac{\gamma {\sigma }^2\tau }{2}.(2.6\text{–}2.7)$$

Their average is the reservation price

$$\boxed{r(s,q,t)=\frac{r^a+r^b}{2}=s-q\gamma {\sigma }^2(T-t)}$$

Interpretation. Long $(q>0)$ ⇒ $r<s$ (wants out). Short $(q<0)$ ⇒ $r>s$ (wants in). The wedge $\gamma {\sigma }^2\tau$ is the price of one unit of inventory risk; the skew is $q$ times that.

2.3 Order arrivals and the intensity law

AS assume market buy orders lift the dealer’s ask at Poisson rate ${\lambda }^a({\delta }^a)$ and sell orders hit his bid at rate ${\lambda }^b({\delta }^b)$, both decreasing in the distance from the mid. Aggregating the empirical market-order size distribution ($f_Q(x)\propto x^{-1-\alpha }$) with the logarithmic price-impact law ($\Delta p\propto \ln Q$) gives

$$\lambda (\delta )=\Lambda \mathbb{P}(\Delta p>\delta )=A{e}^{-k\delta },A=\Lambda /\alpha ,k=\alpha K.(2.11)$$

($A$ = baseline arrival intensity; $k$ = how fast fills die off as you quote further out - the “liquidity density” of the book.)

2.4 The Hamilton–Jacobi–Bellman equation (AS eq. 3.1–3.3)

With the dealer controlling ${\delta }^a,{\delta }^b$, wealth jumps as $d{X}_t=p^{a}d{N}_t^a-p^{b}d{N}_t^b$ and inventory is $q_t=N_t^b-N_t^a$. The value function

$$u(s,x,q,t)=\underset{{\delta }^a,{\delta }^b}{\max }{\mathbb{E}}_t\left[-\exp \left(-\gamma (X_T+q_T{S}_T)\right)\right]$$

solves the HJB equation

$$u_t+\frac{1}{2}{\sigma }^2{u}_{ss}+\underset{{\delta }^b}{\max }{\lambda }^b({\delta }^b)\left[u(s,x-s+{\delta }^b,q+1,t)-u(s,x,q,t)\right]+\underset{{\delta }^a}{\max }{\lambda }^a({\delta }^a)\left[u(s,x+s+{\delta }^a,q-1,t)-u(s,x,q,t)\right]=0,$$

with terminal condition $u(s,x,q,T)=-\exp (-\gamma (x+qs))$. The two “max” terms are simply the expected gain from a filled bid (inventory $+1$, cash $-p^b$) and a filled ask (inventory $-1$, cash $+p^a$), weighted by their arrival intensities. Because utility is exponential, the ansatz

$$u(s,x,q,t)=-\exp (-\gamma x)\exp (-\gamma \theta (s,q,t))(3.2)$$

reduces the problem to a scalar equation for $\theta$ (AS eq. 3.3), which is the object the next page solves. Applying Definition 1 to the ansatz gives the clean relations $r^b=\theta (s,q+1,t)-\theta (s,q,t)$ and $r^a=\theta (s,q,t)-\theta (s,q-1,t)$ - the reservation prices are first differences of $\theta$ in $q$.

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3. Computational Implementation - verifying the reservation price

We solve the indifference conditions (2.4)–(2.5) numerically by bisection on the exponential value function, and check they match the closed forms (2.6)–(2.7).

The numerically-solved indifference prices match eqs. (2.6)–(2.7) to machine precision, and their mean is the reservation price $r=s-q\gamma {\sigma }^2\tau$. At $q=+4$ the dealer values a share at $98.40, $1.60 below the $100 mid - he is long and wants out.

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4. Failure Modes & First-Principles Breakdowns

1. Choosing the wrong risk measure. Exponential (CARA) utility is what buys the wealth-independent reservation price. Swap in mean–variance and the clean separation survives approximately (AS appendix shows the analogous $R^a,R^b$ for geometric BM), but other risk measures change the skew non-linearly (Cartea & Jaimungal 2015). “Why exponential?” is a real modelling choice, not a detail.
2. The intensity law is an empirical bet. $\lambda (\delta )=A{e}^{-k\delta }$ is derived from a power-law order-size distribution plus a logarithmic impact law - assume a different impact law (e.g. $\Delta p\propto Q^{\beta }$) and the intensity becomes a power law $B{\delta }^{-\alpha /\beta }$, changing the optimal spread. The functional form is a modelling choice that must be checked against the book.
3. No drift, no autocorrelation, no information. The mid-price has no predictable component in the AS setup. Any alpha signal (a drift $\mu$, an OFI-based forecast) is outside the model, and treating $dS=\sigma dW$ as literal when the flow is informed is the root cause of the failure modes in 05 · Failure Modes.
4. Unbounded inventory. The frozen-inventory problem allows any $q$; the reservation skew grows linearly in $q$ but never forbids a position. Real systems impose caps - see Guéant’s bounded-inventory treatment in 06 · Advanced Extensions.

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5. References

• Avellaneda & Stoikov (2008), Quantitative Finance 8(3)
• Guéant, Lehalle & Fernandez-Tapia (2013), Dealing with the inventory risk, Math. & Financial Econ. 7(4)
• Ho & Stoll (1981), JFE 9(1)

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · The AS Model · Index Hub
• Theory: Stochastic Calculus & Itô · Limit Order Book Mechanics