Maximizing profits for given stock quotes and volumes - algorithm

Question

Given an array of stock quotes Q[0], ..., Q[n-1] in chronological order and corresponding volumes V[0], ..., V[n-1] as numbers of shares, how can you find the optimal times and volumes to buy or sell when the volumes of your trades are each limited by V[0], ..., V[n-1]?

Answer 1

I assume that you want to start and end with 0 shares in each stock and that you have unlimited capital.
The problem can be boiled down to buying at the lowest prices available and selling at the highest, with the side condition that buying a share has to be done prior to selling it.
I would process the data in time order and add purchases as long as there is available volumes with a higher price in the future (for each purchase you need to tick off the same amount of shares as sold using the highest future price available).
Continue to move forward in time add add buys as long as there is a profitable time to sell in the future. If there is surplus volume available volume but no available profitable selling spot in the future, you need to look back to see if the current price is lower than any purchases already made. In that case, exchange the most expensive shares from the past for the cheaper ones, but only if there is a future selling point available. Also check if there is any profitable selling point available for the scrapped purchase order.
Example:
Day Price Volume
1 100 1000
2 80 1000
3 110 1000
4 70 1000
5 120 2000
Day 1:
Purchase 1000 at 100 per share. Sell 1000 day 4 at 120.
Day 2:
Purchase 1000 at 80 per share. Sell 1000 day 4 at 120.
Day 3:
No available profitable selling opportunity because all future shares at prices above 70 are already booked!
Look back and see if you have purchased at prices above 110.
You havent, so there is no purchase.
Day 4:
No available profitable opportunity because all future volumes at prices above 70 are already booked!
Look back and see if you have purchased at prices above 70.
Replace purchase of 1000 shares day 1 with purchase of 1000 shares at 70 day 4.
Re-examine the shares of day one and check if there is any other profitable sale available (you only need to consider the timeline up to day 4).
There is, so purchase 1000 at 100 per share day 1 and sell them at 110 per share day 3.
The final order book is:
Day Price Volume Order type shares owned
1 100 1000 Buy 1000
2 80 1000 Buy 2000
3 110 1000 Sell 1000
4 70 1000 Buy 2000
5 120 2000 Sell 0
Total profit: 10000