April 23, 2025
Fischer Black, Myron Scholes (Black and Scholes 1973), and Robert C. Merton III (Merton 1973), invented a Nobel Prize winning theory showing how to value equity options using a money market account to finance dynamic trading in the underlying stock. In their model, it is possible to exactly replicate the option payoff so the value of the option is the cost of setting up the initial hedge.
Stephen Ross tremendously generalized the B-S/M model in “A Simple Approach to the Valuation of Risky Streams” (Ross 1978) by showing how to approximately value a derivative using any collection of instruments.
He used the Hahn-Banach theorem to show
If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator, L.
Perhaps Ross never won a Nobel prize because his result was so audacious. Maybe people were not ready to believe it was possible to come up with a theory that applied to every market traded instrument, not just a bond, stock, and option.
This note identifies Ross’s valuation operator and suggests an improvement by placing cash flows on equal footing with prices. In some sense, cash flows are more important. The Fundamental Theorem of Asset Pricing puts geometric constraints on arbitrage-free prices given cash flows.
Trading strategies create synthetic market instruments. If a perfect hedge for a derivative instrument exists it is determined by the Fréchet derivative of the option value with respect to price. The Simple Unified Model assumes every hedge has only a finite number of trades, as is the case in the real world. This implies perfect hedges do not, in general, exist.
The SUM specifies a rigorous mathematical model that can be taught at the master’s level. Work remains for future researchers to determine when and how much to hedge in order to better manage the risk of this market reality.
This section reviews the mathematics required for the SUM with a view to how it relates to classical Black-Scholes/Merton machinery. It is especially intended for mathematically sophisticated readers. They tend to have difficulty shedding their hard-won knowledge and accept the fact only elementary math is needed.
The SUM does not involve probability. As Ross showed, the Fundamental Theorem of Asset Pricing is a geometric result. We assume a sample space and filtration, but do not require a probability measure. There is no need for the Ito calculus or partial differential equations. The notion of continuous time hedging is a deleterious mathematical artifact of that model leading to incorrect results.1 Every hedge in the real world consists of a finite number of trades.
Recall the set exponential \boldsymbol{{R}}^I is the set of functions from I to \boldsymbol{{R}}. The dot product of x,y\in\boldsymbol{{R}}^I is x\cdot y = \sum_{i\in I} x(i)y(i) and the norm is \|x\| = \sqrt{x\cdot x}. We assume \|X\| = \sup_{\omega\in\Omega}\|X(\omega)\| is finite.
If an algebra \mathcal{A} of sets on \Omega is finite then the atoms of the algebra form a partition of \Omega and a function X\colon\Omega\to\boldsymbol{{R}} is measurable with respect to \mathcal{A} if and only if it is constant on atoms. In this case X is a function on the atoms and we write X\colon\mathcal{A}\to\boldsymbol{{R}}. A partition provides partial information about outcomes. Full information is knowing which \omega\in\Omega occurred, partial information is knowing only which atom \omega belongs to. The partition of singletons represents complete information. The partition consisting of the sample space represents no information.
The dual of bounded functions on \Omega, B(\Omega)^*, can be identified with finitely additive measures on \Omega(Dunford and Schwartz 1958). We write B(\Omega,\boldsymbol{{R}}^I) for the space of bounded vector-valued functions. Its dual is isomorphic to the space of finitely additive vector-valued measures ba(\Omega,\boldsymbol{{R}}^I) A linear functional L\in B(\Omega)^* defines a measure \lambda(A) = L1_A for A\subseteq\Omega where 1_A(\omega) = 1 if \omega\in A and 1_A(\omega) = 0 if \omega\in\Omega\setminus A. The facts 1_\emptyset = 0 and 1_{A\cup B} = 1_A + 1_B - 1_{A\cap B}, A,B\subseteq\Omega show \lambda is a measure. Its norm is the supremum over all finite partitions \{A_i\} of {\sum_i |\lambda(A_i)|}.
The integral \int_\Omega f\,d\lambda is defined for simple functions that are a finite linear sum f = \sum_i a_i 1_{A_i}, A_i\subseteq\Omega, by \int_\Omega f\,d\lambda = \sum_i a_i\lambda(A_i). Dunford and Schwartz show this is well-defined and an isometry so it can be extended to all of B(\Omega).
Define M_f\colon B(\Omega)\to B(\Omega) by M_fg = fg. Its adjoint M_f^*\colon ba(\Omega)\to ba(\Omega) defines multiplication of a measure by a bounded function f\lambda = M_f^*\lambda. The Radon-Nikodym derivative of f\lambda with respect to \lambda is f.
Recall if P is probability measure then conditional expectation of a random variable X\colon\Omega\to\boldsymbol{{R}} with respect to an algebra is defined by Y = E[X|\mathcal{A}] if and only if Y\colon\Omega\to\boldsymbol{{R}} is \mathcal{A}-measurable and \int_A Y\,dP = \int_A X\,dP for A\in\mathcal{A}. This is equivalent to Y(P|_{\mathcal{A}}) = (XP)|_{\mathcal{A}} where the vertical bar indicates restriction of measure. We do not use conditional expectation in what follows, only restriction of measures to an algebra.
Let I be the finite set of market instruments, x\in\boldsymbol{{R}}^I be their prices at the beginning of the period, and X\colon\Omega\to\boldsymbol{{R}}^I be their prices at the end of the period where \Omega is the set of possible outcomes.
The initial cost of purchasing \gamma shares of each instrument is \gamma\cdot x. The proceeds from selling \gamma shares at the end of the period is \gamma\cdot X(\omega) if \omega\in\Omega occurs. Arbitrage exists if there is a \gamma\in\boldsymbol{{R}}^I with \gamma\cdot x < 0 and \gamma\cdot X\ge0 on \Omega: you make money acquiring the shares and never lose money liquidating them. Note this definition of arbitrage does not involve probability2.
Theorem. (One-Period Fundamental Theorem of Asset Pricing) There is no arbitrage if and only if x belongs to the smallest closed cone containing the range of X. If x^* is the nearest point in the cone to x then \gamma = x^* - x is an arbitrage.
Proof. The smallest closed cone containing the range of X is {\{\int_\Omega X\,dD\mid D\ge 0, D\in ba(\Omega)\}}. If x = \int_\Omega X\,dD and \gamma\cdot X\ge0 on \Omega then \gamma\cdot x\ge 0. This establishes the “easy” direction.
If x does not belong to the smallest closed cone containing the range of X there is a unique closest point x^* in the cone. Let \gamma = x^* - x. We will show \gamma\cdot x < 0 and \gamma\cdot y\ge0 for every y in the cone.
First we show \gamma\cdot x < 0. We have {\|x^* - x\| \le \|tx^* - x\|} for t > 0 so {0\le (t^2 - 1)\|x^*\|^2 + 2(t - 1)x^*\cdot x}. The quadratic is non-negative and has a root at t = 1 so its derivative there is {0 = 2\|x^*\|^2 + 2x^*\cdot x = 2x^*\cdot(x^* - x)}. Since {0 < \|x^* - x\|^2 = (x^* - x)\cdot(x^* - x) = -x\cdot(x^* - x)} we have {\gamma\cdot x < 0}.
If y is in the cone then ty + x^* is in the cone for t \ge 0 so {\|x^* - x\|^2 \le \|ty + x^* - x\|^2} and {0\le t^2\|y\|^2 + 2ty\cdot(x^* - x)}. Dividng by t > 0 and letting it go to 0 shows {0\le y\cdot(x^* - x)}.
Something is amiss with the one-period model. If the end of the period is the end of trading then prices must be zero since no future trading is possible. It also makes the assumption all shares are liquidated at the end of the period. The final “prices” are actually cash flows associated with holding the instruments and are paid in proportion to the initial position. The multi-period model clarifies this.
Let T be a finite totally ordered set of trading times, I the finite set of market instruments, \Omega the set of possible outcomes, and (\mathcal{A}_t)_{t\in T} be the partitions3 of \Omega indicating the information available at time t\in T.
For each t\in T let X_t\colon\mathcal{A}_t\to\boldsymbol{{R}}^I be the prices4 of market instruments and C_t\colon\mathcal{A}_t\to\boldsymbol{{R}}^I the cash flows of market instruments. E.g., coupons, dividends, and futures margin adjustments are cash flows. Futures have price 0. We assume at each time t\in T that prices and cash flows are bounded.
A trading strategy (\tau_j, \Gamma_j) is a finite number of strictly increasing stopping times \tau_j and shares to purchase in each instrument \Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{{R}}^I at \tau_j.
Trades accumulate to a position \Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s where {\Gamma_s = \Gamma_j 1_{\{\tau_j = s\}}}. Note the strict inequality. It takes time for a trade to settle and become a part of the position.
The value, or mark-to-market, of a trading strategy is {V_t = (\Delta_t + \Gamma_t)\cdot X_t} and represents how much liquidating the current position and trades yet to settle at current market prices would yield. Trading strategies incur cash flow amounts in the trading account of {A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t}: you receive cash flows proportional to existing positions and pay the current market price for trades just executed.
The one-period model corresponds to a trading strategy (t_0, \Gamma_0), (t_1, -\Gamma_0). At t_0 the trading account is charged A_{t_0} = -\Gamma_0\cdot X_{t_0} and has value V_{t_0} = \Gamma_0\cdot X_{t_0}. This makes the unrealistic assumption it is possible to instantly buy and sell with no transaction costs. If there are no intermediate cash flows and then A_t = 0 for t_0 < t < t_1. Unwinding at t_1 results in {A_{t_1} = \Gamma_0 C_{t_1} + \Gamma_0\cdot X_{t_1}}. You get the cash flow at t_1 proportional to the position put on at t_0 and close the position \Gamma_0 at the current price.
Arbitrage exists if there is a trading strategy with A_{\tau_0} > 0, A_t \ge0, t > \tau_0, and \sum_j \Gamma_j = 0. You make money on the first trade and never lose money until the position is closed.
The Fundamental Theorem of Asset Pricing states there is no arbitrage if and only if there exist positive finitely additive measures D_t\in ba(\mathcal{A}_t), t\in T, with \sum_{t\in T} \|D_t\| < \infty and \tag{1} X_t D_t = \sum_{u > t} (C_s D_s)|_{\mathcal{A}_t}. We call such (D_t)_{t\in T} a deflator. Deflators are the (not generally unique) valuation operator Ross identified. We will see later that if repurchase agreements are available there is a canonical deflator that corresponds to the stochastic discount. A money market account corresponds to rolling over at the repo rate. Its reciprocal is the stochastic discount.
Equation (1) is the Graham-Todd “Securities Analysis” ansatz that the value of an investment is the present value of future cash flows.
Lemma. Equation (1) is equivalent to \tag{1a} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_u D_u)|_{\mathcal{A}_t}, t\le u.
The proof is trivial. If cash flows are zero this implies discounted prices are a martingale.
Claim. If M_t = M_u|_{\mathcal{A}_t}, t\le u, is a \boldsymbol{{R}}^I-valued martingale measure and D_t\in ba(A_t) are positive measures then {X_t D_t = M_t - \sum_{s\le t} C_s D_s} is arbitrage-free.
Proof. Substitute X_u D_u by this formula in (1a) and cancel terms in the sums.
Example. (Black-Scholes/Merton) M_t = (r, se^{\sigma B_t - \sigma^2t/2})P, C_t = (0,0), D_t = e^{-\rho t}P where (B_t) is standard Brownian motion and P is Wiener measure.
Lemma. Assuming (1) and the definitions of value and amount \tag{2} V_t D_t = \sum_{u > t} (A_u D_u)|_{\mathcal{A}_t}.
Proof. Substitute V_t by its definition on the left-hand side of (2) and use X_t D_t from (1). Note {\Delta_t + \Gamma_t = \Delta_{t+\epsilon}} for {\epsilon > 0} sufficiently small. In this case {V_t D_t = (V_u D_u + A_u D_u)|_{\mathcal{A}_t}} for {u = t+\epsilon}. The result follows by induction.
Trading strategies create synthetic instruments where price corresponds to value and cash flow corresponds to amount.
A (cash settled) derivative contract is specified by stopping times {\hat{\tau}_j} and cash flows \hat{A}_j. If there exists a trading strategy (\tau_j,\Gamma_j) with {\sum_j \Gamma_j = 0}, {A_{\hat{\tau}_j} = \hat{A}_j} and {A_t = 0} (self-financing) otherwise, then a perfect hedge exists5. The value of the derivative instrument is determined by \tag{3} V_t D_t = (\sum_{\hat{\tau}_j > t} \hat{A}_j D_{\hat{\tau}_j})|_{\mathcal{A}_t}. Note the right hand side is determined by the contract specifications and deflator. Assuming \tau_0 = 0, V_0 = \Gamma_0\cdot X_0 so the initial hedge \Gamma_0 is the Fréchet derivative D_{X_0}V_0 with respect to X_0. Since V_t = (\Gamma_t + \Delta_t)\cdot X_t we have \Gamma_t = D_{X_t}V_t - \Delta_t. Note \Delta_t is settled prior to time t. This does not specify trading times \tau_j > 0.
The Black-Sholes/Merton model make the unrealistic assumption that continuous time trading is possible. The SUM models what traders actually do, but does not provide an answer to the question they all have, “When and how much should I trade?” This model only provides a rigorous framework for additional work on this fundamental question.
A zero coupon bond maturing at time u has a single unit cash flow at u, C_u^{D(u)} = 1 and C_t^{D(u)} = 0 for t\not=u. In an arbitrage free model its price at time t\le u, X_t^{D(u)} = D_t(u), satisfies D_t(u) D_t = D_u|_{\mathcal{A}_t} so D_t(u) is the Radon-Nikodym derivative of D_u|_{\mathcal{A}_t} with respect to D_t. If we assume \mathcal{A}_0 is \{\Omega\} and D_0(\Omega) = 1, then D_0(u) = D(u) = D_u(\Omega).
Zero coupon bond prices are determined by the deflators. These determine the value of any fixed income instrument. If the instrument pays (c_j) at time (u_j) then its value at time t is V_t = \sum_{u_j>t} c_j D_t(u_j)
A forward contract on an underlying with price S_t at t\in T is specified by a forward f\in\boldsymbol{{R}} and an expiration u\in T. It has a single cash flow S_u - f at u. Its value at time t < u satisfies V_t D_t = ((S_u - f)D_u)|_{\mathcal{A}_t}. If the underlying has no cash flows then S_t D_t is a martingale measure so V_t = S_t - fD_t(u). If V_0 = 0 then f is the at the money forward and the cost of carry is S_0 = fD(u).
A European option expiring at time u\in T has a single cash flow \nu(S_u) at u where \nu is the payoff function. The value of the option at time t\in T satisfies {V_t D_t = (\nu(S_u) D_u)|_{\mathcal{A}_t}}, t\le u. Since D_u(\Omega)/D(u) = 1 we can have V_0 = E[\nu(S_u)]D(u) under this “probability” measure.
Any positive random variable F with finite mean and log variance can be written F = fe^{sX - \kappa(s)} where X has mean 0 and variance 1 and \kappa(s) = \log E[e^{sX}] is the cumulant. Note E[F] = f.
If an option pays shares of the underlying instead of cash we need to calculate E[F\nu(F)] = fE[e^{sX - \kappa(s)}\nu(F)] = fE^s[\nu(F)] where E^s is the share measure with dP^*/dP = e^{sX - \kappa(s)}. Since E[e^{sX - \kappa(s)}] = 1 we have the share measure P^s is a “probability” measure.
The (forward) value of a put option with strike k and underlying F at expiration is \begin{aligned} p &= E[\max\{k - F, 0\}] \\ &= E[(k - F)1(F \le k)] \\ &= kP(F\le k) - E[F1(F \le k)] \\ &= kP(F\le k) - fP^s(F \le k) \\ \end{aligned} Note F\le k if and only if X \le (\log k/f + \kappa(s))/s. We call x(f,k,s) = (\log k/f + \kappa(s))/s the moneyness of the option. Let \Psi^s be the cumulative distribution function of X under the share measure. Note \Psi^0 = \Psi is the cumulative distribution function of X. The put value is p = k\Psi(x) - f\Psi^s(x) where x = x(f,k,s). If X is standard normal then \kappa(s) = s^2/2 and this reduces to the Black(Black 1976) formula.
Delta is the derivative of option value with respect to f \begin{aligned} \partial_f p &= \partial_f E[\max\{k - F, 0\}] \\ &= E[-1(F \le k)\partial_f F] \\ &= E[-1(F \le k)e^{sX - \kappa(s)}] \\ &= -P^s(F \le k) \\ &= -\Psi^s(x) \\ \end{aligned}
Gamma is the second derivative of option value with respect to f \begin{aligned} \partial_f^2 p &= -\partial_f \Psi^s(x) \\ &= -\partial_x \Psi^s(x)\partial_f x \\ &= -\partial_x \Psi^s(x)/\partial_x f \\ &= \partial_x \Psi^s(x)/f s\\ \end{aligned}
Vega is the derivative of option value with respect to s. The vega of a put option is \begin{aligned} \partial_s p &= \partial_s E[\max\{k - F, 0\}] \\ &= E[-1(F \le k)\partial_s F] \\ &= -E[1(X\le x)F(X - \kappa'(s))] \\ &= -f\partial_s \Psi^s(x) \\ \end{aligned} The last equality follows from \partial_s E^s[1(X\le x)] = E[1(X \le x)e^{sX - \kappa(s)}(X - \kappa'(s))].
This shows put option value and greeks can be computed from the share cumulative distribution \Psi^s(x) and its derivatives with respect to x and s. This is the case if \Psi is a generalized hypergeometic function. For example, \Phi^s(x) = \frac{1}{2} + \frac{x-s}{\sqrt{2\pi}} \, _1F_1\left(\frac{1}{2}, \frac{3}{2}, -\frac{(x-s)^2}{2}\right) where \Phi is the standard normal cumulative distribution.
If repurchase agreements are available then a canonical deflator exists. A repurchase agreement over the interval [t_j, t_{j+1}] is specified by a rate f_j known at time t_j. The price at t_j is 1 and it has a cash flow of {\exp(f_j\Delta t_j)} at time t_{j+1} where \Delta t_j = t_{j+1} - t_j. By equation (1) we have {D_j = \exp(f_j\Delta t_j)D_{j+1}|_{\mathcal{A}_j}}. If D_{j+1} is known at time t_j then {D_{j+1} = \exp(-f_j\Delta t_j)D_j} and {D_n = \exp(-\sum_{j < n} f_j\Delta t_j) D_{t_0}} is the canonical deflator at time t_n.
The continuous time analog is D_t = \exp(-\int_0^t f(s)\,ds)D_0 where f is the continuously compounded instantaneous forward rate. This is commonly referred to as the stochastic discount.
Suppose a bond defaults at a random time \tau and fraction \rho is recovered at default. We augment the sample space to \Omega\times T\times [0,1] where the element (\omega, t, r) indicates default at t with recovery r. Information about default at time t is represented by the partition of T consisting of singletons, \{s\}, {s < t}, and the set \{u\ge t\}. If default occurs before t then we know \tau exactly, otherwise we only know \tau \ge t. It is customary to assume \rho is a constant. The value of a risky zero coupon at time t is determined by D^{\tau,R}_t(u)D_t = (RD_\tau 1(t \le \tau \le u) + D_u 1(\tau > u))|_{\mathcal{A}_t}, t > \tau.
A futures contract on an underlying with price S_t, t\in T, is specified by quote times t_0 < \cdots < t_n = u where u is the futures expiration. The futures quote at expiration is equal to the price of the underlying \Phi_u = S_u. The price of a futures is always zero and has cash flows \Phi_{t_{j+1}} - \Phi_{t_j} at t_{j+1}\le u where the quote \Phi_t is determined by the market prior to expiration.
In an arbitrage-free model 0D_{t_j} = ((\Phi_{t_{j+1}} - \Phi_{t_j})D_{t_{j+1}})|_{\mathcal{A}_{t_j}}. Under the canonical deflator futures quotes are a martingale.
A limit order is specified by a level. When the underlying reaches the level the holder receives one share of stock by paying the level. This corresponds to a cash flow C_{\tau_L} = (-L, 1) where \tau_L is the first time level L is reached and the components correspond to the currency used for the underlying quote and the underlying.
In this section we prove the FTAP for multi-period models. We start with a proof for the one-period model that can be generalized.
For the one-period model consider the amount operator {A\colon\boldsymbol{{R}}^I\to\boldsymbol{{R}}\oplus^\infty B(\Omega)} defined by {A\gamma = -\gamma\cdot x \oplus^\infty \gamma\cdot X}. The norm on the range is \|p\oplus^\infty P\|_\infty = \max\{|p|,\|P\|\} for p\in\boldsymbol{{R}}, P\in B(\Omega). The first component is the amount made purchasing \gamma shares and the second is the amount made when closing the postion at current market prices. Arbitrage exists if there is a \gamma\in\boldsymbol{{R}}^I with A\gamma in the cone {\mathcal{P}= \{p\oplus P\mid p > 0, P\ge0\}\subseteq \boldsymbol{{R}}\oplus^\infty B(\Omega)}
The adjoint A^*\colon\boldsymbol{{R}}\oplus^1 ba(\Omega)\to\boldsymbol{{R}}^I is A^*(\pi\oplus^1 \Pi) = -x\pi + \langle X,\Pi\rangle where we identify (\boldsymbol{{R}}^I)^* with \boldsymbol{{R}}^I by x\in\boldsymbol{{R}}^I corresponds to x^*\in(\boldsymbol{{R}}^I)^* using \langle y, x^*\rangle = y\cdot x, y\in\boldsymbol{{R}}^I. The norm on the domain is \|\pi\oplus^1 P\|_1 = |\pi| + \|\Pi\|. The calculation \begin{aligned} \langle \gamma, A^*(\pi\oplus^1 \Pi) \rangle &= \langle A\gamma, \pi\oplus^1 \Pi \rangle \\ &= \langle -\gamma\cdot x\oplus^\infty \gamma\cdot X, \pi\oplus^1 \Pi \rangle \\ &= \langle -\gamma\cdot x, \pi\rangle + \langle \gamma\cdot X, \Pi \rangle \\ &= \langle \gamma, -x\pi + \langle X,\Pi\rangle \rangle \\ \end{aligned} shows A^*(\pi\oplus^1 \Pi) = -x\pi + \langle X,\Pi\rangle\in\boldsymbol{{R}}^I.
The range of A is closed, since it is finite dimensional, and \mathcal{P} has an interior point so by the Hahn-Banach theorem there exists a hyperplane H\supseteq\operatorname{ran}A that does not meet \mathcal{P}. Every hyperplane is the preanhiliator of an element of the dual, say d\oplus^1 D\in\boldsymbol{{R}}\oplus ba(\Omega), so {H = {}^\perp\{d\oplus^1 D\} = \{p\oplus^\infty P\mid \langle p\oplus^\infty P, d\oplus^1 D\rangle = 0\}}. Since \operatorname{ran}A = {}^\perp(\operatorname{ker}A^*)\subseteq H we have d\oplus^1 D\in\operatorname{ker}A^* so 0 = -xd + \langle X,D\rangle. We now show d and D are positive.
If there exist p_+\oplus^\infty P_+\in\mathcal{P} and p_-\oplus^\infty P_-\in\mathcal{P} with {\langle p_+\oplus^\infty P_+,d\oplus^1 D\rangle > 0} and {\langle p_-\oplus^\infty P_-,d\oplus^1 D\rangle < 0} then there is a convex combination in H, which contradicts the fact H does not meet the cone \mathcal{P}. This shows we can assume {\langle p\oplus^\infty P, d\oplus^1 D\rangle > 0} for all {p\oplus^\infty P\in\mathcal{P}}. It follows that d > 0 and D > 0.
This shows there is no arbitrage if and only if {xd = \langle X,D\rangle} for some positive {d\in\boldsymbol{{R}}} and {D\in ba(\Omega)}.
For the multi-period model we assume discrete trading times (t_j)_{j\in\boldsymbol{{N}}}. Define the amount operator A\colon\oplus_{j\in\boldsymbol{{N}}} B(\mathcal{A}_j, \boldsymbol{{R}}^I)\to\oplus_{j\in\boldsymbol{{N}}} B(\mathcal{A}_j) by A(\oplus_j \Gamma_j) = \oplus_j A_j where A_j = \Delta_j\cdot C_j - \Gamma_j\cdot X_j and {\Delta_j = \sum_{i < j} \Gamma_j}. Note {\Delta_0 = 0} and \Gamma_j = \Delta_{j+1} - \Delta_j. The direct sum uses the sup norm \|\oplus_j \Gamma_j\|_\infty = \sup_j \|\Gamma_j\|.
We now compute the adjoint A^*\colon\oplus_{j\in\boldsymbol{{N}}} ba(\mathcal{A}_j)\to\oplus_{j\in\boldsymbol{{N}}} ba(\mathcal{A}_j, \boldsymbol{{R}}^I). For \oplus_j D_j\in \oplus_j ba(\mathcal{A}_j) with sum norm \|\oplus_j D_j\|_1 = \sum_j \|D_j\| \begin{aligned} \langle \oplus_j \Gamma_j, A^*(\oplus_j D_j)\rangle &= \langle A(\oplus_j \Gamma_j), \oplus_j D_j\rangle \\ &= \sum_j \langle \Delta_j\cdot C_j - \Gamma_j\cdot X_j, D_j\rangle \\ &= \sum_j \langle (\sum_{i < j}\Gamma_i)\cdot C_j - \Gamma_j\cdot X_j, D_j\rangle \\ &= \sum_j \sum_{i < j} \langle \Gamma_i\cdot C_j, D_j\rangle - \sum_j \langle \Gamma_j\cdot X_j, D_j\rangle \\ &= \sum_i \sum_{j > i} \langle \Gamma_i\cdot C_j, D_j\rangle - \sum_j \langle \Gamma_j\cdot X_j, D_j\rangle \\ &= \sum_j \sum_{i > j} \langle \Gamma_j\cdot C_i, D_i\rangle - \sum_j \langle \Gamma_j\cdot X_j, D_j\rangle \\ &= \sum_j \langle \Gamma_j\cdot \sum_{i > j} C_i, D_i\rangle - \sum_j \langle \Gamma_j\cdot X_j, D_j\rangle \\ &= \sum_j \langle \Gamma_j, \sum_{i > j} C_i D_i\rangle - \sum_j \langle \Gamma_j, X_j D_j\rangle \\ \end{aligned}
Lemma. If X\in B(\mathcal{A}) then \langle X, \Pi\rangle = \langle X, \Pi|_\mathcal{A}\rangle for \Pi\in ba(\Omega).
Proof. This follows from \langle 1_A,\Pi\rangle = \langle 1_A,\Pi|_\mathcal{A}\rangle for A\in\mathcal{A} and every X\in B(\mathcal{A}) is the norm limit of simple functions.
This shows the dual of the amount operator is {A^*(\oplus_j D_j) = \oplus_j \sum_{i > j} (C_i D_i)|_{\mathcal{A}_j} - X_j D_j}.
The subspace of closed-out trading strategies is \mathcal{G}_0 = \oplus^\infty_{j\in\boldsymbol{{N}}} B(\mathcal{A}_j,\boldsymbol{{R}}^I) having a finite number of non-zero terms that sum to 0. If \mathcal{P}= \{\oplus^1_{j\in\boldsymbol{{N}}} P_j\mid P_j\in ba(\mathcal{A}_j), P_0 > 0, P_j\ge 0, j>0\} is the cone of positive measures then arbitrage exists if A(\mathcal{G}_0) meets \mathcal{P}.