Jan 29, 2026
(Ross 1978) demonstrated how to replace the probabilistic theory of (Black and Scholes 1973) and (Merton 1973) with a much simpler geometric theory.
If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator.
Ross showed there is no need for Ito processes or partial differential equations and generalized from a bond, stock, and option to any collection of instruments.
Ross made the deleterious assumptions that continuous time trading is possible and conflated cash flows with jumps in stock price. Assuming only a finite number of trades are possible and introducing an explicit variable for cash flows leads to an even simpler theory.
Every arbitrage free model can be parameterized by a vector-valued martingale measure and a positive adapted measure.
The vector space of bounded functions on the set S is B(S) = \{f\colon S\to\boldsymbol{R}\mid \|f\| = \sup_{s\in S} |f(s)| < \infty\}. The dual of B(S) can be identified with ba(S), the set of finitely additive measures on S. For bounded L\colon B(S)\to\boldsymbol{R} define \lambda(A) = L(1_A) for A\subseteq S where 1_A(s) = 1 if s\in A and 1_A(s) = 0 otherwise. Since \lambda(A\cup B) = L(1_{A\cup B}) = L(1_A + 1_B - 1_{A\cap b} = \lambda(A) + \lambda(B) - \lambda(A\cap B) and \lamba(\emptyset) = L(1_\emptyset) = L(0) = 0 we have \lambda is a finitely additive measure.
For f\in B(S) define M_f\colon B(S)\to B(S) by M_fg = fg to be the linear operator of multiplication by f. Its adjoint M_f^*\colon ba(S)\to ba(S) defines how to multiply a finitely additive measure by a bounded function.
An algebra of sets on a set S is a collection of subsets closed under union and complement. If the algebra is finite then its atoms form a partition. Conditional expectation can be defined by restriction of measure. Given a set \Omega, an algebra {\mathcal{A}} of sets on \Omega, and a probability measure P on {\mathcal{A}}. If Y = E[X\mid{\mathcal{A}}] is the conditional expectation of X, then Y(P|_{\mathcal{A}}) = (XP)|_{\mathcal{A}} where the vertical bar indicates restriction.
Let I be the set of tradeable instruments and T\subset [0,\infty) be possible trading times.