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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.

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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.

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    $\begingroup$ Thanks. However, I am still perplexed as the market-maker needs to quote prices w.r.t. the market prices. So he obtains his reservation prices, that are derived from the market price, but not equal to them, and then he adjusts them by half of the computed spread. However, in order for his price to be market-eligible price, it needs to hold that $p_a$ > $s$ and $p_b$ < s. However, there is no guarantee that this will hold after computing $p_a$ and $p_b$, right? $\endgroup$ – ragoragino Oct 12 '17 at 8:20
  • $\begingroup$ It is not necessary the $p_a>s$ and $p_b<s$. The market-maker can quote any prices he wants. If $p_b$ is high enough he will get executed with probability 1, and equivalently if $p_a$ is low enough. $\endgroup$ – RRG Oct 12 '17 at 8:32
  • $\begingroup$ It seems unrealistic for the market-maker to quote $s<p_b<p_a$ or $p_b<p_a<s$, doesn't it? Maybe it is a limitation of this model. $\endgroup$ – Daneel Olivaw Oct 12 '17 at 8:40
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    $\begingroup$ @Daneel Olivaw, it is not that unrealistic. Assume a stock is trading bid $10$ ask $10.1$. So the midprice $s=10.05$. A market maker who holds short inventory might quote $p_b=10.06>s$, $p_a=10.15$ in order to attract orders to its bid (to reduce short inventory) and reduce the probability to be hit on its ask (which would increase the short inventory). $\endgroup$ – RRG Oct 12 '17 at 8:51
  • $\begingroup$ @RRG Right, this makes sense that the market-maker can place quotes improving on the current midprice. So I guess the fact that the plot in the original paper does not show crossing between the quotes of the market-maker and the midprice is just a matter of coincidence. $\endgroup$ – ragoragino Oct 12 '17 at 10:01

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