Break-even fill quality for a resting bid

MediumMarket Making & Betting Games~3m

You leave a resting buy limit at $99 while the fair mid is $100. A resting quote is a free option you've written: it only executes when someone chooses to trade against it. Two kinds of fills occur, and you must value each yourself from the prices. A good (uninformed) fill leaves the fair value unchanged at $100; a picked-off (informed) fill happens only because the value has really dropped to $97. In each case your P&L is the true value minus the $99 you paid. Let $p$ be the probability a given fill is the good kind. Below what value of $p$ is leaving the bid a losing proposition (find the break-even $p$ , to four places)?

Show hints (2)+

Write $E[\text{P\&L}\mid\text{fill}] = p(+1) + (1-p)(-2)$ and set it to zero.
The $1/$2 payoff asymmetry forces the break-even above one-half.

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0.6667 (± 0.001)

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Break-even fill quality for a resting bid

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