Probability and Statistics in Quant Finance
Certain types of traders attempt to repeatedly buy and sell the same asset for a profit over a short time period, such as high-frequency “market makers”. For example, if you can repeatedly sell a stock for $8.50 and buy it for $8.49, you will make $0.01 each time. This is known as arbitrage.
If this transaction succeeds with probability 99%, about how many times can this transcation be executed before the probability of at least one failure exceeds 50%?
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1 solution
Since the probability of any transaction succeeding is 0 . 9 9 , the probability of N [independent] transactions all succeeding will be 0 . 9 9 N . Therefore, the probability of at least one failure is 1 − 0 . 9 9 N .
If this exceeds 0 . 5 0 , then we can solve for N as 1 − 0 . 9 9 N > 0 . 5 0 . 9 9 N < 0 . 5 N ⋅ ln ( 0 . 9 9 ) < ln ( 0 . 5 ) N > ln ( 0 . 9 9 ) ln ( 0 . 5 ) ≈ 6 8 . 9 7
It follows that the smallest value of N is 6 9 . Looking at the given values, the nearest choice is 7 0 .