content_image/308735.jpg content_image/308736.jpg content_image/308731.jpg
Then, the objective function of (57) can be more succinctly represented as a function of µ 

<img src='content_image/308735.jpg'>

The optimization problem then becomes: 

<img src='content_image/308736.jpg'>

## 8.4.4 Linearization of Constraint (A) 

Note that constraint (A) in (60) involves the quadratic terms ∑ J j $_{=1}$x$_{j}$ ( A $^{T}$ξ ) $_{j}$. Therefore, in this section, we suggest a way to linearize this constraint. 

Intuitively, x$_{j}$ ( A $^{T}$ξ ) j is simply the market maker’s revenue from the j th order. Let R$_{j}$ denote the market maker’s revenue from the j th order. x$_{j}$ is the number of shares of the option traded. ( A $^{T}$ξ ) j is the market-clearing price of the j th order. To begin the transformation, we first consider the feasible set of x and ξ . We say that the pair 

<img src='content_image/308731.jpg'>

Figure 5 is a three-dimensional graph. The graph shows the market maker’s revenue as a 

