How Stock Price Scale Affects Avellaneda–Stoikov Reservation Prices

Summary

This exchange considers how the Avellaneda–Stoikov reservation price responds to a stock's price scale and an inventory measured in shares. The question compares a stock before and after a large share consolidation, where the share price and volatility per share both change while the dollar value of the position stays similar. Plugging the values into the reservation-price formula appears to produce different reservation prices as percentages of the market price.

The reply explains that reservation price is defined through indifference between the current inventory and an inventory increased by one share. Adding one share of a high-priced stock changes wealth by more dollars than adding one share of a lower-priced stock, so the marginal inventory unit is not economically equivalent across the two cases. The brief answer does not further examine how the model's inventory and volatility units should be rescaled under a corporate action.

Key ideas

• The reservation price reflects the value of adding one share to the current inventory.
• One share represents different dollar exposure when the stock price changes substantially.
• A share consolidation changes the economic size represented by a single inventory unit.
• Comparisons across price scales require interpreting inventory and volatility in consistent units.

Tags

# Why it's related to stock price


# Why it's related to stock price












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.

## Answer by numerairX (score 1, accepted)

https://quant.stackexchange.com/a/50188

**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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