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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.

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**edit after reading the paper. Since reservation bid price is defined as the price that the agent would be indifferent between current inventory and current inventory +1, the effect of adding a $\$10000$ per share stock is different than that of adding a $\$100$ per share stock. Hence the difference.

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  • $\begingroup$ thanks for your reply. For the \$100 per share stock, I calculated using 100 shares so the dollar amount is the same \$10000, as hold a share of \$10000 per stock. $\endgroup$ – frank Dec 11 '19 at 2:55
  • $\begingroup$ yes, but if you look at the equation it changes very differently because you have number of shares and unit price at different "multiplier" level $\endgroup$ – numerairX Dec 11 '19 at 14:33
  • $\begingroup$ I get it. The pricing not only includes consideration of current inventory but also the possibility of future transaction, so the stock price of a share does matters. Thanks! $\endgroup$ – frank Dec 12 '19 at 6:53

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