# 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.

• 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.

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

• 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?

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 $$\kappa$$.
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 $$\kappa$$.