Feb 15, 2026
Let’s start with a brief description of the Simple Unified Model. It will probably not make much sense on first reading. If you read to the end and come back again it will make complete sense and seem perfectly obvious, if I did my job right.
Let T be the set of possible trading times and I be the set of tradeable market instruments. For t\in T let X_t be the vector of prices indexed by I, C_t be the vector of cash flows indexed by I. You can buy or sell instruments at prices X_t and receive cash flows C_t proportional to your position in each instrument at time t.
A trading strategy is a finite number of times \tau_0 < \cdots < \tau_n depending on information available and trading sizes \Gamma_j at time \tau_j. These are vectors indexed by I indicating how much you bought or sold in each available instrument. Trades accumulate into positions \Delta_t = \sum_{\tau_j < t} \Gamma_j.
The value (mark-to-market) of your position at time t is V_t = (\Delta_t + \Gamma_t)\cdot X_t. The amount showing up in your brokerage account at time t is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t.
Arbitrage is a trading strategy with \sum_j \Gamma_j = 0 where A_{\tau_0} > 0 and A_t \ge 0 for t > \tau_0. You make money on the first trade and never loose money until the position is closed.
Every arbitrage-free model of prices and cash flows has the form X_t D_t = M_t - \sum_{s\le t} C_s D_s where D_t are positive measures depending only on information available at time t and (M_t)_{t\in T} is a vector-valued martingale measure indexed by I.
It is trivial to show X_t D_t = (X_u D_u + \sum_{s\in (t, u]} C_s D_s)|_{A_t} where |_{A_t} means given information at time t. It is less trivial to show V_t D_t = (V_u D_u + \sum_{s\in (t, u]} A_s D_s)|_{A_t} but this is the skeleton key to pricing derivative instruments. Note how prices X_t correspond to values V_t and cash flows C_s correspond to amounts A_s.
Every trading strategy creates a synthetic market instrument.
Derivative instruments are specified by a contract. The buyer wants amounts \hat{A}_j at times \hat{\tau}_j. The seller quotes a price and is obligated to use that to trade available market instruments to satisfy the contract.
The seller needs to find a trading strategy (\tau_j, \Gamma_j) with A_{\hat{\tau}_j} = \hat{A}_j and A_t = 0 otherwise.
The SUM provides an answer for the first trade. Since V_0 = \Gamma_0\cdot X_0 the derivative with respect to initial prices X_0 is \Gamma_0. It only provide a rigorous mathematical model for reasoning about the yet unsolved problem of what subsequent trades you can plausible suggest to traders and risk managers to convince them you are adding value to their business.
(Ross 1978) demonstrated how to replace the untenable probabilistic theory of (Black and Scholes 1973) and (Merton 1973) with a much simpler and more general 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 also generalized from a bond, stock, and option to any collection of instruments. There is no need to require a so-called “real world” probability measure, only to get immediately thrown out for a “risk neutral” probability measure that is not the probability of anything. All he needed was the Hahn-Banach theorem: if a point is not in a convex set then there exists a hyperplane with the point on one side and the convex set on the other.
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, as is the case in our actual real world, 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 positive adapted measures. These are what Ross called the valuation operator. We call them deflators. If the model has repurchase agreements then the deflator is the stochastic discount.
This section collects the mathematical facts required to prove every arbitrage free model is parameterized by a vector-valued martingale measure and a deflator.
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 vector space dual of B(S) can be identified with ba(S), the set of finitely additive measures on S.
For any bounded linear functional 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 \lambda(\emptyset) = L(1_\emptyset) = L(0) = 0 we have \lambda is a finitely additive measure.
For any countably additive measure \lambda\in ba(S) define L\colon B(S)\to\boldsymbol{R} by L(1_A) = \lambda(A). This can be linearly extended to linearly to finite linear combinations of characteristic function. These are uniformly dense in B(S).
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. We write M_f^*(\lambda) as f\lambda.
An algebra of sets on the set S is a collection of subsets closed under union and complement. If the algebra is finite then its atoms form a partition of S. A partition is a disjoint collection of subsets of S with union S and represents partial information. Elements of a partition are called atoms. Full knowledge is represented by the collection of singletons \{\{s\}\mid s\in S\}. No knowledge is represented by the singleton \{S\}. Partial knowledge of s\in S is represented by the atom to which s belongs A_s = \cap\{A\mid s\in A\}.
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 T be a subset of the real numbers consisting of possible trading times and I be the set of tradeable instruments. Let \Omega be the set of what can happen in a market. For each t\in T we specify {\mathcal{A}}_t to be a partition representing information available at time t.