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Depth-weighted mid

Market makingHard

The bid side has $30$ units at $100$ and $10$ units at $99$ . The ask side has $10$ units at $101$ and $20$ units at $103$ . Let $B$ be the volume-weighted average price of all bid units and $A$ that of all ask units, with total depths $D B$ and $D A$ . The depth-weighted mid is $(B \times D A + A \times D B) \div (D B + D A)$ . What is its value? Give an exact fraction.

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Hint

Find B, A and the depths first.

Solution

The bid side has depth $D B = 40$ and average price $B = \frac{399}{4}$ ; the ask side has depth $D A = 30$ and average price $A = \frac{307}{3}$ .
The numerator is $\frac{399}{4} \times 30 + \frac{307}{3} \times 40 = \frac{42515}{6}$ .
Dividing by $70$ gives the depth-weighted mid price $\frac{8503}{84}$ .

Answer: $\frac{8503}{84}$

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