Numeraire change in option pricing

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I'm reading a book on Interest Rate Theory and I have difficulty understanding a certain derivation:

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I don't understand how they changed the numeraire to get the desired expression (2.12) - I feel like there is an extra step omitted there?

Here is the Proposition 2.2 which they refer to - I do understand it, I just don't know how they use it in this case. Especially since the result is somewhat "symmetric" yet the expressions in the conditional expectation aren't.

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