Avellaneda -Stoikov market making model

Question

I am reading paper High-frequency trading in a limit order book by Marco Avellaneda and Sasha Stoikov. At the end of the paper they obtain a closed-form solution to the optimal market-maker quotes under diffusion without drift. They found that the optimal behaviour of the market-maker would be to set a bid/ask spread of size:

$$ spread = \gamma\sigma^2(T-t) + \frac{2}{\gamma}ln(1+\frac{\gamma}{k}), $$ where $\gamma$ is a discount factor, $\sigma^2$ is the variance of the process, $k$ is the parameter corresponing to the intensity of arrival of market orders, $T$ is terminal time and $t$ is curent time, around a reservation price given by:

$$ price = s - q\gamma\sigma^2(T-t), $$

where $q$ is the state of the inventory and $s$ is the current price.

However, I do not see any specification of bounds for this reservation price and therefore I think there is no guarantee that ask prices computed by the market-maker will be higher or bid prices will be lower than the current price of the process.

How is therefore this necessity of market makers' ask prices being higher and bid prices being lower than the actual price enforced in their model (e.g. in their simulations)?

Edit: To be more concrete, I just specify, that in my opinion, it needs to hold that:

$$ price + spread/2 - s > 0 $$

Lets denote $price$ by $p_{mm}$ and $spread/2$ by $s_{mm}$. Then

$$ p_{mm} + s_{mm} - s > 0, \\ s - q\gamma\sigma^2(T-t) + \frac{\gamma\sigma^2(T-t)}{2} + \frac{1}{\gamma}ln(1+\frac{\gamma}{k}) - s >0 \\ (...) \\ \frac{1}{2} + \frac{ln(1+\frac{\gamma}{k})}{\gamma^2\sigma^2(T-t)} > q $$

However, this situation does not need to happen, so there is no guarantee he will set prices compatible with current market prices.

Answer 1

The market-maker makes a bid-ask spread $\delta$ around the reservation price $r$. So at any time, the market-maker quotes the bid price $$ p_b = r - \delta/2, $$ and the ask price $$ p_a = r + \delta/2. $$ Bid price is hence always below the reservation price and ask price is always above the reservation price. The reservation price $$ r = s - q\gamma\sigma^2(T-t) $$ is the market price minus a term that depends on the inventory $q$ that the market-maker is holding. If $q$ is positive the reservation price moves lower (below the market price) and vice-versa for negative $q$, reflecting the risk of inventory.

If the inventory grows the reserve price will eventually move to a point where the market-maker quotes starts to attract orders to liquidate inventory, which will again result in a change in the reserve price. Orders arrive with probability $$ \lambda_a(\delta^a)dt = Ae^{-k\delta^a}dt, $$ for the ask price and similarly for the bid price. Here $\delta^a$ is the distance of the ask quote from the market price. So if the bid-price gets high enough it will be executed with probability 1, and equivalently if the ask-price gets low enough. The reserve price will hence settle into a state of equilibrium reflecting the risk of inventory.

Note that there is no requirement that $p_b<s$, or $p_a>s$. The market-maker can post competitive bid and ask prices that improves on the current market price in order to manage the inventory.

Answer 2

The reservation price is highly influenced by the election of the parameter T isn't it? So, if T is high enough, each step in which q is not zero, the reservation price could be too high (or too low), and so the election of bid and ask quotes (both above or below the mid-price).

Answer 3

For asymptotic expansions when T is large you should read the paper by Guéant, Lehalle, and Fernandez-Tapia (https://arxiv.org/abs/1105.3115) or the book of Guéant (https://www.goodreads.com/book/show/29933253-the-financial-mathematics-of-market-liquidity)