July 23, 2025
Let \Omega be a sample space representing all possible instrument prices. Information at time t is modelled by a partition \mathcal{A}_t of \Omega. We assume \mathcal{A}_u is a refinement of \mathcal{A}_t whenever u > t. The possible trading times T are a subset of the real numbers. To avoid doubling strategies (Harrison and Kreps 1979) we assume \epsilon = \inf\{|t - u|\mid t,u\in T, t\not=u\} > 0. Finally, let I be the set of all market instruments.
We use B(\Omega) for the space of bounded functions on \Omega. Recall its vector space dual can be identified with the space of finitely addititve measures on \Omega denoted ba(\Omega).
Every instrument has prices and cash flows indexed by T. We assume instruments can be bought or sold in any amount at time t with price X_t\in B(\Omega,\boldsymbol{{R}}^I), t\in T. Cash flows associated with owning an instrument are C_t\in B(\Omega,\boldsymbol{{R}}^I) at time t\in T. Examples of cash flows are stock dividends, bond coupons, and margin adjustements from futures. The price of a futures contract is always zero. Usually C_t = 0 for all but a finite number of times t\in T.
A trading strategy is a finite number of strictly increasing stopping times \tau_j\colon\Omega\to T, 0\le j\le n, and number of shares to buy at \tau_j denoted \Gamma_j\colon\Omega\to\boldsymbol{{R}}^I where \Gamma_j is constant on \{\tau_j = t\} for all t\in T. Shares accumulate to position \Delta_t = \sum_{\tau_j < t} \Gamma_j. Note the strict inequality. Trades take some time to settle before becoming part of the position. We also write this as \Delta_t = \sum_{s < t} \Gamma_s where \Gamma_s = \Gamma_j when s = \tau_j and is zero otherwise. Note \Delta_t + \Gamma_t = \Delta_u for u > t and u - t sufficiently small.
The value, or mark-to-market, of a position is how much you would make if you liquidated at current prices, {V_t = (\Delta_t + \Gamma_t)\cdot X_t}. You don’t own \Gamma_t at time t but you will at t + \epsilon and should be accounted for. The amount associated with a trading strategy at time t is {A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t}: you receive cash flows proportional to your position and have to pay for trades just executed.
We define arbitrage (for this model) as the existence of a trading strategy that closes out (\sum_j \Gamma_j = 0) with A_{\tau_0} > 0 and A_t\ge0, for t > \tau_0: you make money on the first trade and never lose money therafter. If you don’t require the strategy to close out then borrowing a dollar every day would be an arbitrage. Note our definition of arbitrage does not involve a probablility measure.
Theorem. (Fundamental Theorem of Asset Pricing.) The Simple Unified Model is arbitrage free if and only if there exist a deflator , positive measures D_t in ba(\mathcal{A}_t), t\in T, with \tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{\mathcal{A}_t}, t\le u where | indicates restriction of measure.
Lemma. Using the above definitions \tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\mathcal{A}_t}, t\le u.
Trading strategies create synthetic instruments where price corresponds to value and cash flow corresponds to amount.
Proof: We have {X_t D_t = (X_u D_u + C_u D_u)|_{\mathcal{A}_t}} and {\Delta_t + \Gamma_t = \Delta_u} for some u\in T with {u - t \ge \epsilon} sufficiently small.
We have \begin{aligned} V_t D_t &= (\Delta_t + \Gamma_t)\cdot X_t D_t \\ &= (\Delta_u\cdot X_u D_u + \Delta_u C_u D_u)|_{\mathcal{A}_t} \\ &= (\Delta_u\cdot X_u + (\Gamma_u\cdot X_u + A_u)D_u|_{\mathcal{A}_t} \\ &= (\Delta_u + \Gamma_u)\cdot X_u + A_u)D_u|_{\mathcal{A}_t} \\ &= (V_u + A_u)D_u|_{\mathcal{A}_t} \\ \end{aligned} The lemma follows by finite induction since \epsilon > 0.
We say (M_t)_{t\in T} is a martingale measure if M_t\in ba(\mathcal{A}_t) and M_t = M_u|_{\mathcal{A}_t} for u > t.
Lemma. If (M_t)_{t\in T} is an \boldsymbol{{R}}^I-valued martingale measure and D_t\in ba(\mathcal{A}_t) are positive then X_t D_t = M_t - \sum_{s\le t} C_s D_s is an arbitrage-free model of prices and cash flows.
Proof: Since {V_{\tau_0} = (\sum_{u > \tau_0} A_u D_u)|_{\mathcal{A}_t}} and if A_t \ge 0 for t > \tau_0 then V_{\tau_0} \ge 0. Since \Delta_{\tau_0} = 0 we have {V_{\tau_0} = \Gamma_{\tau_0}\cdot X_{\tau_0}} and {A_{\tau_0} = -\Gamma_{\tau_0}\cdot X_{\tau_0}} so {A_{\tau_0} = -V_{\tau_0} \le 0}. This shows the model is arbitrage-free and proves the “easy” direction of the FTAP. The contrapositive involves the Hahn-Banach theorem, but given the plethora of arbitrage-free models why bother?
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 exists. 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 determine the trading times \tau_j.
The Black-Scholes(Black and Scholes 1973) and Merton(Merton 1973) model has \Omega = C[0,\infty), T = [0,\infty) with instruments a bond and a stock having no associated cash flows. Our martingale measure is M_t = (r, se^{\sigma B_t - \sigma^2t/2})P where (B_t) is Brownian motion and P is Weiner measure. The deflator is D_t = e^{-\rho t}P.
If repurchase agreements are availble they determine a canonical deflator. 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.
The set exponential B^A is the set of all functions from A to B. For any set I, \boldsymbol{{R}}^I is a vector space with scalar multiplication and vector addition defined pointwise.
If \mathcal{A} is a sigma algebra on \Omega then it is an algebra. If it is finite then the atoms of \mathcal{A} form a partition of \Omega. In this case a function X\colon\Omega\to\boldsymbol{{R}} is \mathcal{A}-measurable if and only if X is constant on atoms of \mathcal{A}. In this case it is a function on the atoms of \mathcal{A} and write B(\mathcal{A}).
Our definition of arbitrage is not sufficient. Recall A_0 = -\Gamma_0\cdot X_0. Even if this is strictly positive traders and risk managers will compare it to |\Gamma_0|\cdot|X_0| as a measure of how much capital will be tied up in the trading strategy.