Calculating Failure Risk Across Repeated Trades

Summary

The document gives a probability calculation for repeated transactions that each succeed with the same stated probability. To find the chance of at least one failure over a chosen number of attempts, it uses the complement of the event that every attempt succeeds. Under the assumption that outcomes are independent, the probability of no failures is the per-attempt success probability raised to the number of attempts.

Setting the probability of at least one failure to one half and solving for the number of attempts gives the threshold reported in the answer. The example is framed around repeated profitable trades, but the method is a general probability result rather than a trading strategy or a detailed market-risk model. Its usefulness depends on the constant success probability and independence assumptions; correlated outcomes or changing execution conditions would require a different model.

Key ideas

• The chance of at least one failure is one minus the chance that every attempt succeeds.
• With independent attempts and constant success probability, the probability of all successes is exponential in the attempt count.
• Solving for a target cumulative failure probability gives the maximum repetition threshold for the stated assumptions.
• The calculation does not account for dependence between trades or changing probabilities over time.

Tags

# Probability and statistics in Quantitative Finance


# Probability and statistics in Quantitative 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 transaction be executed before the probability of at least one failure exceeds 50%?

## Answer by ZRH (score 3, accepted)

https://quant.stackexchange.com/a/44412

Imho that's more of a probability question than finance really.

If you take $N$ attempts, then the probability of at least one (or more) failures is the complementary probability of never failing on those $N$ attempts:

$p=1-p_{pass}^N$

Solving for $p=50\%$, you need to evaluate $p_{pass}^{N}=0.5$, where $p_{pass}=0.99$, getting that $N\geq 69$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.