Assume a random walk with no bias. If one puts a limit order at a price below the market where the trading vehicle has a 49% chance of being below after a certain period of time, what are the chances the limit order will be filled during that period of time assuming a normal distribution?

Hint: The solution involves the Reflection Principle. The answer is very simple.

Steve Ellison writes:

The Reflection Principle is that the probability of touching a given price is twice the probability of the final price being below the given price at the end of the period. Thus the probability of the limit order being filled is theoretically 98% before considering the possibility of the order being too far down the order queue or too large to be completely filled.

For what price is there a 49% chance the final price will be lower in a random walk? One tick down?

Phil McDonnell replies:

Steve Ellison has it right. The probability would roughly double to 98% of getting a fill on a limit order because of the Reflection Principle for the given problem.

It is good to note that the probability depends on both volatility of the trading vehicle and the amount of time it has to run. For example SPY has an annualized volatility of .1014. It closed today at 124.48. I show the probability of it being at or above 125 on the third Friday of February as 49%. So if I entered a limit order to sell at 125 and left it in until the third Friday in Feb then I should have about a 98% chance of getting filled.

In comparison the chance of a SPY fill at a limit of 125 by next Friday is only about 78% so time matters as well as price.


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