I am reading a paper High-frequency trading in a limit order book by Avellaneda and Stoikov.
I verified the formula (6) should be correct. However it doesn't make sense to me when I use it for different stock price.
Let's say stock ABC is traded at $100 per share with sigma = 2. Given r=0.1, Holding 100 share at beginning of time gives the reservation price 100 + (1-2*100) * (0.1*2^2*1)/2 = 60.2.
Now the board decides to merge the stock 100:1. So the new price is $10000 with sigma =200 , holding 1 share at beginning of time gives the reservation price 10000 + (1-2*1) * (0.1*200^2*1)/2 = 8000.
So with nothing changed except the price, holding the same dollar amount of stock actually gives reservation ask price at 60% v.s. 80% of the current price.
Did I misunderstand anything? Thanks.