We have so far focused on risk-neutral valuation using the intuitive ″money-market″ numeraire. In this framework, the main pricing operator is
which we have interpreted as the expected value of terminal option payoff, when stochastically discounted back to valuation date. Written another way,
where *M*(*t, T, ω*) =is the value of a money-market account with initial unit deposit at time *t*, and continually rolled over until *T* along the series of instantaneous short rates *r*(*u, ω*). Since *M*(*t, t, ω*) = 1, the above two formulae are equivalent. However, the second formula highlights the required martingale property: It states that having chosen the money-market account as the currency (numeraire), then all relative prices in the future should have 0 expected P&L. This requirement pins down the risk-neutral probabilities used when taking expected values.

Derivatives prices being *relative* prices, it turns out that a more general result holds, stating that no matter what the chosen currency/numeraire, then relative prices of all assets (underlyings and derivatives) with respect to that numeraire should have 0 expected P&L, that is, relative ...

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