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?