March 12, 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 options by dynamically trading a bond and a stock.
This note suggests improvements to Stephen Ross’s paper “A Simple Approach to the Valuation of Risky Streams” (Ross 1978). Ross expanded on the Nobel Prize winning theory developed by Fischer Black, Myron Scholes (Black and Scholes 1973), and Robert C. Merton (Merton 1973), which originally focused on valuing options through dynamic trading of a bond and a stock. Ross greatly expanded this by showing how to value derivatives using any collection of instruments.
We place cash flows on equal footing with prices and show trading strategies create synthetic market instruments. If a perfect hedge 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 does not involve probability measures. 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.
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}.
The dual of bounded functions on \Omega, B(\Omega)^*, can be identified with finitely additive measures on \Omega(Dunford and Schwartz 1958). Recall if P is probability measure then conditional expectation with respect to an algebra is defined by Y = E[X\mid\mathcal{A}] if and only if Y 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}. We do not use conditional expectation in what follows, only restriction of measures to an algebra.
T — totally ordered set of trading times.
I — market instruments.
\Omega — possible outcomes.
(\mathcal{A}_t)_{t\in T} — partitions1 of \Omega indicating the information available at time t\in T.
X_t\colon\mathcal{A}_t\to\boldsymbol{R}^I — bounded prices2 at times t\in T of market instruments.
C_t\colon\mathcal{A}_t\to\boldsymbol{R}^I — bounded cash flows at times t\in T of market instruments.
E.g., coupons, dividends, and futures margin adjustments are cash flows. Futures have price 0.
(\tau_j, \Gamma_j) — trading strategy of strictly increasing stopping times \tau_j and shares \Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{R}^I purchased at \tau_j.
The position resulting from a trading strategy is \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.
V_t = (\Delta_t + \Gamma_t)\cdot X_t — the value, or mark-to-market, of the strategy.
A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t — the amount showing up in the trading account.
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.
Theorem (Fundamental Theorem of Asset Pricing) There is no arbitrage if there exist deflators, positive finitely additive measures3 D_t on \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.
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 (1) 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. With the above notation \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.
Proof: Substitute V_t by its definition on the left 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}. Since the stopping times are strictly increasing, induction can be applied.
Trading strategies create synthetic instruments where price corresponds to value and cash flow corresponds to account.
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 exists4. The value of the derivative is determined by 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 subsequent trading times \tau_j > 05.
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.
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. Zero coupon bond prices are determined by the deflators.
Suppose a bond defaults at a random time \tau and constant fraction R is recovered at default. We augment the sample space to \Omega\times T where the element (\omega, t) indicates default at t. 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. 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 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_0(u).
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 a canonical deflator futures quotes are a martingale.
A limit order is specified by a level. It has price 0 and at most one cash flow equal to the level at the first time the underlying crosses the level.
A partition of \Omega is a collection of pairwise disjoint sets with union \Omega. If \mathcal{A} is a finite algebra of sets on \Omega then the atoms of \mathcal{A} form a partition of \Omega. Partial information is knowing which atom \omega\in\Omega belongs to. A function X\colon\Omega\to\boldsymbol{R} is \mathcal{A}-measurable if and only if it is constant on atoms so X is a function on the atoms of \mathcal{A}.↩︎
Prices are bounded. There is a finite amount of money in the world. Likewise for the number of shares it is possible to trade.↩︎
The dual of bounded functions B(\Omega)^* \cong ba(\Omega) is the space of finitely additive measures on \Omega. L\in B(\Omega)^* corresponds to the measure \lambda(E) = L1_E. If P is a positive measure with mass 1 then Y = E[X|\mathcal{A}] if and only if Y(P|_\mathcal{A}) = (XP)|_\mathcal{A}.↩︎
A perfect hedge never exists.↩︎
Continuous time trading is impossible.↩︎