Optimal quoting: the Avellaneda–Stoikov model

Lesson 10 of 111 min read

"Inventory risk and quote skewing" argued that a market maker holding too much of an asset should lower both quotes, and that the spread should reflect the risk it carries. This lesson turns the argument into one rule. Avellaneda and Stoikov start from the utility function of "Risk aversion and the optimal complete portfolio", let the market maker maximise it, and obtain a reservation price, which sets the skew, and a spread, which sets the width. We derive the first from scratch and state the second with a sketch of where it comes from.

Set-up

The mid price follows the Brownian motion of "Random walks and Brownian motion" without drift: $dS_t = \sigma\,dW_t$ . The maker has cash $x$ and an inventory of $q$ shares (negative when short), and trades until a horizon $T$ . At the end it is worth $x + qS_T$ , and it values that outcome with exponential utility:

U = -e^{-\gamma\,(x + q S_T)},\qquad \gamma > 0.

The parameter $\gamma$ is the maker's risk aversion. The maker chooses its quotes as distances from the mid: a bid at $S - \delta^b$ and an ask at $S + \delta^a$ . A quote at distance $\delta$ is filled at the rate $\lambda(\delta) = A\,e^{-k\delta}$ per unit of time, as in "Market makers and liquidity": the closer the quote, the more often it trades. A larger $k$ means that fills thin out faster as the quote moves away.

The reservation price

Suppose the maker does not trade at all until $T$ . With $\tau = T - t$ time left, $S_T = S + \sigma\sqrt{\tau}\,Z$ with $Z$ standard normal, and since $E[e^{cZ}] = e^{c^2/2}$ ,

E\bigl[-e^{-\gamma(x + qS_T)}\bigr] = -e^{-\gamma(x + qS)}\;e^{\gamma^2 q^2\sigma^2\tau/2}.

The last factor is the penalty for the variance of the position. Now ask what price $r^b$ makes the maker indifferent to buying one more share. After the purchase, cash is $x - r^b$ and inventory is $q+1$ , so equal utility requires equal exponents:

-\gamma\,(x - r^b + (q+1)S) + \tfrac12\gamma^2 (q+1)^2\sigma^2\tau = -\gamma\,(x + qS) + \tfrac12\gamma^2 q^2\sigma^2\tau.

Since $(q+1)^2 - q^2 = 2q + 1$ , this solves to the indifference bid below. The same steps for selling one share, with cash $x + r^a$ and inventory $q - 1$ , give the indifference ask:

r^b = S - \frac{(1 + 2q)\,\gamma\sigma^2\tau}{2},\qquad r^a = S + \frac{(1 - 2q)\,\gamma\sigma^2\tau}{2}.

Their average is the reservation price:

r = S - q\,\gamma\,\sigma^2\,(T - t).

With no inventory the reservation price is the mid. A long maker ( $q > 0$ ) values the asset below the mid, because every extra share adds variance to a position it already has too much of; a short maker values it above. The shift is proportional to the inventory, the risk aversion, the variance and the time left.

The optimal spread

The reservation price tells the maker where to centre its quotes. The spread is a separate decision, because each side now trades off two things: a quote far from the mid earns more per fill but fills less often. Writing the maker's value function and applying dynamic programming gives the Hamilton–Jacobi–Bellman equation of the problem. We do not derive it here. Avellaneda and Stoikov show that, to first order, the optimal distances are $\delta^a = r^a - S + \tfrac1\gamma\ln(1 + \gamma/k)$ on the ask side and the mirror image on the bid side, and the total spread is

\delta^a + \delta^b = \gamma\,\sigma^2\,(T - t) + \frac{2}{\gamma}\ln\!\Bigl(1 + \frac{\gamma}{k}\Bigr).

This is the paper's approximation, not an exact solution, but it is accurate in the range where the model is used. Each term has a meaning.

The first term, $\gamma\sigma^2(T-t)$ , is the price of inventory risk: it grows with risk aversion, with variance and with the time left to hold a position, and it vanishes at the horizon.
The second term, $\tfrac{2}{\gamma}\ln(1 + \gamma/k)$ , comes from the fill rate. As $\gamma \to 0$ it tends to $2/k$ , which is the spread of a risk-neutral maker who just maximises the income $2\delta A e^{-k\delta}$ : the half-spread $1/k$ of "Adverse selection and informed trading" when every customer is noise.

Quotes

The maker centres the spread on the reservation price, not on the mid:

b = r - \tfrac12(\delta^a + \delta^b),\qquad a = r + \tfrac12(\delta^a + \delta^b).

A long maker therefore has both quotes below where a flat maker would put them: the ask is cheap and likely to sell, the bid is low and unlikely to buy, which is the skew of the previous lesson, now with a size. As the horizon approaches, $T - t \to 0$ , the inventory term in the reservation price and the first term in the spread both shrink: the reservation price returns to the mid and the spread to its fill-rate term.

Worked example

Quotes at the middle of the day

Take $\gamma = 0.1$ , $\sigma = 2$ , $k = 1.5$ , mid $S = 100$ , $T - t = 0.5$ and an inventory of $q = 6$ shares. The inventory shift is:
$\displaystyle q\gamma\sigma^2(T-t) = 6 \cdot 0.1 \cdot 4 \cdot 0.5$ 1.2
The reservation price is the mid less the shift:
$\displaystyle r = 100 - 1.2$ 98.8
The inventory-risk term of the spread is:
$\displaystyle \gamma\sigma^2(T-t) = 0.1 \cdot 4 \cdot 0.5$ 0.2
The fill-rate term of the spread is:
$\displaystyle \tfrac{2}{0.1}\ln(1 + 0.1/1.5) = 20 \cdot 0.064539$ 1.2908
The total spread is their sum:
$\displaystyle 0.2 + 1.2908$ 1.4908
The bid is half a spread below the reservation price:
$\displaystyle b = 98.8 - 0.7454$ 98.0546
The ask is half a spread above it, well below the 100.75 a flat maker would post:
$\displaystyle a = 98.8 + 0.7454$ 99.5454
At the close, $T - t = 0$ , the spread is only the fill-rate term. With $\gamma = 0.5$ it is:
$\displaystyle \tfrac{2}{0.5}\ln(1 + 0.5/1.5)$ 1.1507

The effect of risk aversion

It is tempting to say that a more risk-averse maker always quotes wider. The two terms say otherwise. Early in the day the first term dominates and it grows with $\gamma$ . Near the close it vanishes, and the second term, $\tfrac{2}{\gamma}\ln(1+\gamma/k)$ , is decreasing in $\gamma$ . At the close above, $\gamma = 0.1$ gives a spread of 1.2908 and $\gamma = 0.5$ gives 1.1507: the more risk-averse maker is tighter. A maker that fears inventory wants to trade, and near the end it has no time left to be hurt, so it quotes close to the mid to get fills.

The figure shows the quotes on a price column and the total spread against the time left for three values of $\gamma$ . Set the inventory to 0 and move the day clock: the reservation price sits on the mid and the quotes are symmetric. Now drag the inventory to $-10$ and the quotes shift up. At the defaults (inventory 6, half-way through the day, $\gamma = 0.1$ ) the reservation price is 98.80, the spread 1.49, the bid 98.05 and the ask 99.55. At the start of the day the spread is 1.51 for $\gamma = 0.05$ , 2.05 for $\gamma = 0.2$ and 4.27 for $\gamma = 0.8$ . At the close the order reverses: 1.31, 1.25 and 1.07.

Reservation price and optimal quotesOpen in Avellaneda–Stoikov

Inventory q6

-1010

Time of day t (T = 1)0.50

Risk aversion γ0.10

-20

Total spread against the time left (T − t) for three values of γ; the vertical line marks the current time.

γ = 0.05γ = 0.2γ = 0.8

Reservation price: 98.80
Total spread: 1.49
Bid: 98.05
Ask: 99.55

Symmetric and inventory strategies

Avellaneda and Stoikov compare the inventory strategy above with a symmetric one that always centres the same spread on the mid, ignoring inventory. In their simulations the symmetric maker earns slightly more on average, since it never gives up a fill to manage a position. But its inventory wanders freely, so its final profit varies much more from one day to the next. The inventory strategy gives up a little profit for a large reduction in variance, which is what a risk-averse maker with this utility function wants.

The model assumes a constant $\sigma$ , a smooth exponential fill rate and no informed customers. The last is a real simplification: with them the fill-rate term should be wider, as in "Adverse selection and informed trading".

Summary

The maker has exponential utility, a Brownian mid with volatility $\sigma$ and fills at rate $A e^{-k\delta}$ at distance $\delta$ from the mid.
The reservation price $r = S - q\gamma\sigma^2(T-t)$ is the average of the prices at which the maker is indifferent to buying or selling one more share.
To first order the total spread is $\gamma\sigma^2(T-t) + \tfrac{2}{\gamma}\ln(1+\gamma/k)$ , an inventory-risk term plus a fill-rate term.
The quotes are $r \mp \tfrac12$ of the spread: a long maker moves both below the mid, and both effects fade towards the close.
Higher risk aversion widens the spread early in the day and narrows it at the close, and the inventory strategy trades some mean profit for much lower variance.

2 questions on this lesson at the end of the series

Optimal quoting: the Avellaneda–Stoikov model

Lesson 10 of 111 min read

Set-up

U = -e^{-\gamma\,(x + q S_T)},\qquad \gamma > 0.

The reservation price

Suppose the maker does not trade at all until $T$ . With $\tau = T - t$ time left, $S_T = S + \sigma\sqrt{\tau}\,Z$ with $Z$ standard normal, and since $E[e^{cZ}] = e^{c^2/2}$ ,

E\bigl[-e^{-\gamma(x + qS_T)}\bigr] = -e^{-\gamma(x + qS)}\;e^{\gamma^2 q^2\sigma^2\tau/2}.

-\gamma\,(x - r^b + (q+1)S) + \tfrac12\gamma^2 (q+1)^2\sigma^2\tau = -\gamma\,(x + qS) + \tfrac12\gamma^2 q^2\sigma^2\tau.

Since $(q+1)^2 - q^2 = 2q + 1$ , this solves to the indifference bid below. The same steps for selling one share, with cash $x + r^a$ and inventory $q - 1$ , give the indifference ask:

r^b = S - \frac{(1 + 2q)\,\gamma\sigma^2\tau}{2},\qquad r^a = S + \frac{(1 - 2q)\,\gamma\sigma^2\tau}{2}.

Their average is the reservation price:

r = S - q\,\gamma\,\sigma^2\,(T - t).

The optimal spread

\delta^a + \delta^b = \gamma\,\sigma^2\,(T - t) + \frac{2}{\gamma}\ln\!\Bigl(1 + \frac{\gamma}{k}\Bigr).

This is the paper's approximation, not an exact solution, but it is accurate in the range where the model is used. Each term has a meaning.

The first term, $\gamma\sigma^2(T-t)$ , is the price of inventory risk: it grows with risk aversion, with variance and with the time left to hold a position, and it vanishes at the horizon.
The second term, $\tfrac{2}{\gamma}\ln(1 + \gamma/k)$ , comes from the fill rate. As $\gamma \to 0$ it tends to $2/k$ , which is the spread of a risk-neutral maker who just maximises the income $2\delta A e^{-k\delta}$ : the half-spread $1/k$ of "Adverse selection and informed trading" when every customer is noise.

Quotes

The maker centres the spread on the reservation price, not on the mid:

b = r - \tfrac12(\delta^a + \delta^b),\qquad a = r + \tfrac12(\delta^a + \delta^b).

Worked example

Quotes at the middle of the day

Take $\gamma = 0.1$ , $\sigma = 2$ , $k = 1.5$ , mid $S = 100$ , $T - t = 0.5$ and an inventory of $q = 6$ shares. The inventory shift is:
$\displaystyle q\gamma\sigma^2(T-t) = 6 \cdot 0.1 \cdot 4 \cdot 0.5$ 1.2
The reservation price is the mid less the shift:
$\displaystyle r = 100 - 1.2$ 98.8
The inventory-risk term of the spread is:
$\displaystyle \gamma\sigma^2(T-t) = 0.1 \cdot 4 \cdot 0.5$ 0.2
The fill-rate term of the spread is:
$\displaystyle \tfrac{2}{0.1}\ln(1 + 0.1/1.5) = 20 \cdot 0.064539$ 1.2908
The total spread is their sum:
$\displaystyle 0.2 + 1.2908$ 1.4908
The bid is half a spread below the reservation price:
$\displaystyle b = 98.8 - 0.7454$ 98.0546
The ask is half a spread above it, well below the 100.75 a flat maker would post:
$\displaystyle a = 98.8 + 0.7454$ 99.5454
At the close, $T - t = 0$ , the spread is only the fill-rate term. With $\gamma = 0.5$ it is:
$\displaystyle \tfrac{2}{0.5}\ln(1 + 0.5/1.5)$ 1.1507

The effect of risk aversion

Reservation price and optimal quotesOpen in Avellaneda–Stoikov

Inventory q6

-1010

Time of day t (T = 1)0.50

Risk aversion γ0.10

-20

Total spread against the time left (T − t) for three values of γ; the vertical line marks the current time.

γ = 0.05γ = 0.2γ = 0.8

Reservation price: 98.80
Total spread: 1.49
Bid: 98.05
Ask: 99.55

Symmetric and inventory strategies

Summary

The maker has exponential utility, a Brownian mid with volatility $\sigma$ and fills at rate $A e^{-k\delta}$ at distance $\delta$ from the mid.
The reservation price $r = S - q\gamma\sigma^2(T-t)$ is the average of the prices at which the maker is indifferent to buying or selling one more share.
To first order the total spread is $\gamma\sigma^2(T-t) + \tfrac{2}{\gamma}\ln(1+\gamma/k)$ , an inventory-risk term plus a fill-rate term.
The quotes are $r \mp \tfrac12$ of the spread: a long maker moves both below the mid, and both effects fade towards the close.
Higher risk aversion widens the spread early in the day and narrows it at the close, and the inventory strategy trades some mean profit for much lower variance.

2 questions on this lesson at the end of the series