Is the variance calculation correct in the book?

Question

I'm reading the book "Financial Markets Under the Microscope" for my market microstructure studies. In the book, the variance of the market maker's gain is calculated as follows:

Assume that with probability $φ$, an arriving trade is informed, and correctly predicts a price jump of size $±J$. Otherwise (with probability $1 − φ$), the arriving trade is uninformed and does not predict a future price change. The variance $σ^2$ of the market-maker’s gain per trade is then given by

$σ^2 = (1 − φ) × 0 + φ × J^2$

But I calculate the variance as $σ^2 = φ × J^2 - φ^2 × J^2$ which is different from the book.

Do I miss something?

Answer 1

As I see it, a reasonable possibility is that it is a directional bet; money will only be made if the direction of the jump is correct, else the bet is lost.

So if the random outcome is $X \in \{-J, 0, J\}$ with

$\mathbb{P}(+J) = \mathbb{P}(-J) = \varphi /2 \;$ such that $\mathbb{P}(\pm J) = \varphi$,

then it follows that $\mathbb{E}[X] = 0$, and $\mathbb{V}[X] = \varphi \times J^2$. Q.E.D.