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Expected Value Against an Optimiser

Here is a calculation almost everyone gets wrong the first time, and it is the arithmetic underneath every make-a-market exercise.

The wrong calculation

You quote 98 at 102 on something you believe is worth 100. What is your expected profit per trade?

The tempting answer: you buy at 98 and sell at 102, each two ticks from fair value, so you make 2 either way. Expected profit, 2.

That calculation assumes the trade arrives independently of your price. It does not. The counterparty sees 98 at 102 before deciding, and chooses the side that suits them.

The right one

Condition on the trade happening.

Expected value of a fill
E[profit]=E[value∣they lifted]−your askE[\text{profit}] = E\big[\text{value} \mid \text{they lifted}\big] - \text{your ask}

The expectation is over the value given that this particular side traded, not over the value in general. Dropping the conditioning is the error.

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