How many shares can you buy under an average-price cap?

Question

The offer side of the book looks like this:

Side	Price	Size
Ask	$20.00$	$500$
Ask	$20.01$	$500$
Ask	$20.03$	$1000$

You are willing to buy as many shares as possible, but only if your average price is at or below $20.01$ .

What is the largest quantity you can buy, and what average price do you pay?

Accepted Answer

Sweep the two cheapest levels, which are both at or below the cap: - 500 shares at 20.00 = \10{,}000 - 500 shares at 20.01 = \10{,}005 That is 1{,}000 shares for \20{,}005, an average of 20.005, still under the cap. Now add y shares from the 20.03 level and require the blended average to stay at or below 20.01: \frac{20{,}005 + 20.03\,y}{1000 + y} \le 20.01. Multiply out: 20{,}005 + 20.03\,y \le 20{,}010 + 20.01\,y, so 0.02\,y \le 5, giving y \le 250. So you buy 250 of the 1000 shares offered at 20.03, for a total of 1{,}000 + 250 = \mathbf{1{,}250} shares. Total cost = 20{,}005 + 250 	imes …

How many shares can you buy under an average-price cap?

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