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Question

  • What is the difference between "reservation price" and "optimal bid and ask quotes"?
  • Are they the same thing?

(1) Reservaton price

  • In the paper High-frequency trading in a limit order book, the ask reservation price (ra) and bid reservation price (rb) is introduced.
  • If we put limit orders on these two reservation prices, we can manage the inventory effectively, while performing market making.

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  • I thought this is the academic contribution of this academic paper, but the paper continued and mentioned "optimal bid and ask quotes". But I do not know why it is there.

(2) Optimal bid and ask quotes

  • In the sub-chapter "2.4 Limit orders", the bid price pa and ask price pa are introduced. The difference between market price and these two prices, δa and δb, are included in the mathematical equation to solve.

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  • If we solve the above equation using Hamilton-Jacobi-Bellman(HJB) equation, following answers can be deduced.

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  • The paper seems to not clearly clarify the difference between why we have optimal bid and ask quotes, while the advantages of reservation prices are clearly stated.
  • Can anyone tell me why the author introduced the "optimal bid and ask quotes", please?
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1 Answer 1

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The paper says the reservation bid/ask prices are the quotes in "the setting where no trading is allowed." I interpret this to mean "no competition." By taking the average of the bid and ask prices, you get an ideal mid-price for the market maker based on their inventory and their risk tolerance. Quoting symmetrically around this mid-price should take care of the inventory risk management. Note that this ideal mid-price does not account for other orders in the book (competition), which leaves the question "how far from the reservation price should I quote?"

Addressing this question regarding the bid-ask spread is where the order book's liquidity is taken into account with a new variable $\kappa$. The optimal bid-ask spread is defined as: $$\delta^a+\delta^b=\gamma\sigma^2(T-t)+\frac{2}{\gamma}\ln(1+\frac{\gamma}{\kappa})$$ The variable $\kappa$ is representative of the order book's liquidity. Intuitively, greater liquidity means more competition in the market, meaning the market maker must quote narrowly. Conversely, lower liquidity allows for wider quotes. This is seen in the inverse relation between the spread and $\kappa$.

Answering your questions directly
Q: What is the difference between "reservation price" and "optimal bid and ask quotes"?
A: The reservation price is a "personalized mid-price" according to one's inventory, allowing symmetric quotes around this mid-price to manage inventory risk.

Q: Are they the same thing?
A: They are not the same because the optimal bid and ask quotes are an extension of the reservation quotes that takes into account order book liquidity with $\kappa$.

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