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 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)?

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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

For asymptotic expansions when T is large you should read the paper by Guéant, Lehalle, and Fernandez-Tapia here or the book of Guéant The financial mathematics of market-liquidity.

Answer 3

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).