# Determine the right order size with market making strategy

In a market market strategy https://web.stanford.edu/class/msande448/2017/Final/Reports/gr4.pdf, how can we determine the right order size? Assuming I use a market making strategy and on a specific stock at a time t, I place a limit buy order at price p_1 with volume v_1 and limit sell order at price p_2 with volume v_2. In taking into account that v_1 = v_2, how can we determine the right order size?

The author of that paper told me : "That's a question I don't fully address. But the goal is typically to manage inventory, so you will never fall outside +/- x around a neutral position. Therefore if you are long x, then you will typically place an order to sell x and return to a neutral position (sometimes market orders are placed to do this more aggressively). "

I did not fully understand what he meant.

Be aware that I possess the full market depth.

"I need to get an algo or a formula to determine to right quantity to trade each time I place the pair (limit_buy_order, limit_sell_order)."

Actually, you need a formula for determination of the optimal prices, not quantities.

For example, if the market goes down and you have long positions in inventory, you should reduce ask price to attract more buy orders and close long positions. And the optimal price reduction step depends on how many orders you want to attract.

• I could never understand the point of this paper. E.g. in their example they have a stock priced ~ USD101 and they are placing bids at USD100.5 and asks at USD101.5 aiming at USD1 spread. What if the market spread is 10c? In that case they'll just never get filled, so what's the point? – LazyCat Aug 14 '18 at 14:19
• @LazyCat quant.stackexchange.com/a/40898 "You might just get lucky and a deep-pocketed liquidity seeker might trade against all better prices and yours. If this happens you get a better price than the situation you would just quote a narrow spread (and hope that the market hasn't moved against you permanently)." – Alexey Golyshev Aug 14 '18 at 14:52
• So "you might just get lucky" is the only justification of using these fancy formulas in practice? – LazyCat Aug 14 '18 at 15:20
• @LazyCat Have you read the original answer? I quoted only a part. You are not alone in the limit order book. You are competing with other market makers. You can deviate from the optimal prices that are predicted by these formulas. You can reduce the spread, reduce your margin by taking more risk, but increase the volume traded. Or you can increase the spread and trade less often. – Alexey Golyshev Aug 14 '18 at 15:53
• I've read it in the context of my question: what's the use of this model, if it suggests to quote a one dollar spread, while in reality the stock trades at a 1c or 10c spread? I don't see an answer there except for "get lucky" part you quoted. Do you? – LazyCat Aug 14 '18 at 18:10

He meant the following:

Market makers (MMs) seek to make money by simultaneously selling high and buying low. The risk they run is that those trades are not simultaneous, instead there is an intermittent period between buy and sell when the market may move adversely for the MM, impacting overall profits.

If the mid market price of some marketable instrument is 0, and the MM has bid -1 and offered +1 and been hit so that he is now long at -1 he has greater inclination to sell and reduce risk rather than buy more and increase risk exposure so that his marketable prices might now be -1.5 +0.5, even if the mid-price is still believed to be 0. This is called skew.

The author is suggesting that the size of (skewed) positions is such that they offset the current risk position of the portfolio after aggregating trades.

• I still don't understand the right quantity to invest. If the midprice is simply (a+b)/2 and the signal from my model tell my that the midprice is sceptical to increase, what should be the right size? Can you give me an example? – user1050421 Aug 7 '18 at 23:45

Given your data, you have absolutely no way of determining the right order size. Just assume, that you place 100 shares on each side (or 1 share if it makes your calculations simpler). If you want to compare different market-making strategies, just assume, the same risk/position for each strategy.

• How to deal with the stock price? It is much more expensive to deal with a stock X at 1000 USD per share than a stock Y at 5 USD per share. So taking 100 share a 1000 cost 100K USD while the other cost 500 USD for 100 shares. – user1050421 Aug 8 '18 at 1:44
• If that's your concern, fix a dollar amount per stock, e.g. 1000USD per stock per side. However, in practice, if a stock is >> 100USD/share, it's likely to be a wide spread and require completely different treatment from a 5USD/share stock anyways. – LazyCat Aug 8 '18 at 1:53

As the author suggested you need an upper limit $X$ on the dollar value of the inventory you are willing to hold. This depends on the amount of capital that you are willing to employ in the market-making business. (If the limit is reached market-makers take strong action, even market orders, to reduce the inventory. This could cause significant loss).

The order size you display at any time can be a fraction $p$ of that limit. The arrival of $\frac{1}{p}$ orders on the same side (consecutive buys or consecutive sells) would bring your inventory to the limit. You want this to be fairly unlikely in a random situation and you can set $p$ small enough accordingly.

Let's say you have 1 million dollars and you think $p=0.1$ is safe enough, then you can post limit orders for 100,000 dollars worth of stock.