# discounted asset price is MG but "discount" means…@@

The Fundamental Theorem

A financial market with time horizon T and price processes of the risky asset and riskless bond (I would say a money-market-account) given by S1, …, ST and B0, …, BT, respectively, is arbitrage-free under the real world probability P if and only if there exists an equivalent probability measure Q (i.e. risk neutral measure) such that
The discounted price process, X0 := S0/B0, …, XT := ST/BT is a martingale under Q.

#1 Key concept – divide the current stock price by the current MMA value. This is the essence of “discounting“, different from the usual “discount future cashflow to present value
#2  key concept – the alternative interpretation is “using MMA as currency, then any asset price S(t) is a martingale”
I like the discrete time-series notation, from time_0, time_1, time_2… to time_T.
I like the simplified (not simplistic:) 2-asset world.
This theorem is generalized with stochastic interest rate on the riskless bond:)
There’s an implicit filtration. The S(T) or B(T) are prices in the future i.e. yet to be revealed . The expectation of future prices is taken against the filtration.
 though in the case of T-forward measure, B(T) = 1.0 known in advance.
–[[Hull]] P 636 has a concise one-pager (I would skip the math part) that explains the numeraire can be just “a tradable”, not only the MMA. A few key points:

) both S and B must be 2 tradables, not something like “fwd rate” or “volatility”
) the measure is the measure related to the numeraire asset
) what market forces ensure this ratio is a MG? Arbitragers!