If you know the future prices of a stock, what's the best time to buy and sell?

Question

Interview Question by a financial software company for a Programmer position

Q1) Say you have an array for which the ith element is the price of a given stock on day i.

If you were only permitted to buy one share of the stock and sell one share of the stock, design an algorithm to find the best times to buy and sell.

My Solution : My solution was to make an array of the differences in stock prices between day i and day i+1 for arraysize-1 days and then use Kadane Algorithm to return the sum of the largest continuous sub array.I would then buy at the start of the largest continuous array and sell at the end of the largest continous array.

I am wondering if my solution is correct and are there any better solutions out there???

Upon answering i was asked a follow up question, which i answered exactly the same

Q2) Given that you know the future closing price of Company x for the next 10 days , Design a algorithm to to determine if you should BUY,SELL or HOLD for every single day ( You are allowed to only make 1 decision every day ) with the aim of of maximizing profit

Eg: Day 1 closing price :2.24
Day 2 closing price :2.11
...
Day 10 closing price : 3.00

My Solution: Same as above

I would like to know what if theres any better algorithm out there to maximise profit given that i can make a decision every single day

Answer 1

Q1 If you were only permitted to buy one share of the stock and sell one share of the stock, design an algorithm to find the best times to buy and sell.

In a single pass through the array, determine the index i with the lowest price and the index j with the highest price. You buy at i and sell at j (selling before you buy, by borrowing stock, is in general allowed in finance, so it is okay if j < i). If all prices are the same you don't do anything.

Q2 Given that you know the future closing price of Company x for the next 10 days , Design a algorithm to to determine if you should BUY,SELL or HOLD for every single day ( You are allowed to only make 1 decision every day ) with the aim of of maximizing profit

There are only 10 days, and hence there are only 3^10 = 59049 different possibilities. Hence it is perfectly possible to use brute force. I.e., try every possibility and simply select the one which gives the greatest profit. (Even if a more efficient algorithm were found, this would remain a useful way to test the more efficient algorithm.)

Some of the solutions produced by the brute force approach may be invalid, e.g. it might not be possible to own (or owe) more than one share at once. Moreover, do you need to end up owning 0 stocks at the end of the 10 days, or are any positions automatically liquidated at the end of the 10 days? Also, I would want to clarify the assumption that I made in Q1, namely that it is possible to sell before buying to take advantage of falls in stock prices. Finally, there may be trading fees to be taken into consideration, including payments to be made if you borrow a stock in order to sell it before you buy it.

Once these assumptions are clarified it could well be possible to take design a more efficient algorithm. E.g., in the simplest case if you can only own one share and you have to buy before you sell, then you would have a "buy" at the first minimum in the series, a "sell" at the last maximum, and buys and sells at any minima and maxima inbetween.

The more I think about it, the more I think these interview questions are as much about seeing how and whether a candidate clarifies a problem as they are about the solution to the problem.