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*S*and_{1}, ..., S_{T}*B*, respectively, is arbitrage-free under the_{0}, ..., B_{T}__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**X*:=_{0}*S*:=_{0}/B_{0}, ..., X_{T }*S*is a martingale under_{T}/B_{T}*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 [1]. The expectation of future prices is taken against the filtration.

[1] 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!

## No comments:

Post a Comment