Jun 9, 2026
This note provides a simple and rigorous mathematical model for valuing, hedging, and managing the risk of all derivative instruments. It is based on (Ross 1978) “A Simple Approach to the Valuation of Risky Streams” where he showed
If there are no arbitrage opportunities in a market, then there must exist a (not generally unique) positive linear operator that can be used to value all marketed assets.
One such operator is the stochastic discount, the reciprocal of a money market account. We call any such positive linear operator a deflator.
Unlike (Black and Scholes 1973) and (Merton 1973) Ross made no assumptions on the underlying stochastic processes for prices. He does not require Brownian motion, the Itô calculus, or partial differential equations. His insight was to realize option valuation has nothing to do with probability and used the Hahn-Banach theorem to show prices are constrained by the geometry determined by model cash flows.
There is a clear trajectory in Mathematical Finance from simple closed-form models to those allowing more realistic features to be incorporated. Classical models assume there is no bid-ask spread (perfect liquidity) and prices are a real numbers (infinitely divisible). See Unified Finance for a rigorous model that can incorporate these features.
The Achilles Heel of the Black-Scholes/Merton model is that it assumes continuous-time trading. This leads to infeasible trading strategies and obviously incorrect valuations. The doubling strategy pointed out by (Harrison and Kreps 1979) is impossible to carry out. Merton derived a closed-form solution for barrier options. The value of a barrier option that knocks in the second time the barrier is hit must have a value less than or equal to a one-hit barrier option. According to the B-S/M theory, the value of a barrier option that knocks in the n-th time the barrier is hit is the same for all n>1. If Brownian motion hits level a at time \tau then it hits a infinitely often in the interval [\tau, \tau + \epsilon] for any \epsilon > 0. See (Karatzas and Shreve 1991) Section 3.2. This is why Brownian motion is an unrealistic model of instrument prices.
Every hedging strategy involves only a finite number of trades. This remedies the untenable results above. It does not solve the fundamental problem every trader faces the first day on the job: when and how many market instruments to buy in order to manage risk. This note provides a simple unified model that provides a rigorous mathematical foundation for addressing this problem. Fundamental research remains to be done. See [#future-research]
Let T be the finite set of possible trading times and \Omega be a sample space representing everything that can happen. Information at time t is modeled by a partitions (\mathcal{A}_t)_{t\in T} of \Omega. We assume every atom of \mathcal{A}_t is a union of atoms of \mathcal{A}_u for t < u. This is true if A_u is a refinement of A_t. Finally, let I be the finite set of market instruments under consideration.
Instruments have prices and cash flows indexed by T. The price at time t\in T is a bounded \boldsymbol{{R}}^I-valued function X_t\in B(\Omega,\boldsymbol{{R}}^I). The cash flows that result from owning an instrument are also bounded \boldsymbol{{R}}^I-valued functions C_t\in B(\Omega,\boldsymbol{{R}}^I). Examples of cash flows are stock dividends, bond coupons, and margin adjustments from futures. The price of a futures contract is always zero.
Ross thought of prices as right-continuous functions and identified the jumps as the cash flow/dividend. Calling the close-to-open price gap a dividend is a categorical mistake. It is market movement, not a payout. In fact, stocks trade at integral multiples of their minimum trading increment so every price change would be considered a dividend payment under Ross’s definition. It also fails if the trading times are discrete.
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 when u - t sufficiently small.
The value, or mark-to-market, of a position is how much would be made if the position was liquidated at current prices, {V_t = (\Delta_t + \Gamma_t)\cdot X_t}. Trade \Gamma_t is not part of the position at time t but will be at t + \epsilon and needs to 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}: cash flows proportional to position are credited and trades just executed are debited at the current market price.
We define arbitrage (for this model) as the existence of a trading strategy with A_{\tau_0} > 0 and A_t\ge0, for t > \tau_0: the first trade makes money and subsequent trades never lose money. We also require the strategy to eventually closes out, \sum_j \Gamma_j = 0, otherwise borrowing a dollar every day would be an arbitrage.1 Note our definition of arbitrage does not involve a probability measure.
Theorem. (Fundamental Theorem of Asset Pricing) The Simple Universal 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.
The proof relies on the following two lemmas.
Lemma. Using the above definitions $$ V_t D_t = (V_u D_u + {t < s u} A_s D_s)|{_t}, tu. there is no need for that. Every arbitrage-free model used in practice has the above parameterization.
This is the skeleton key to valuing, hedging, and managing the risk of any derivative instrument.
Trading strategies create synthetic instruments where price, X, corresponds to value, V, and cash flow, C, corresponds to amount, A.
Proof: If at most one cash flow occurs in the interval (t,u] then {X_t D_t = (X_u D_u + C_u D_u)|_{\mathcal{A}_t}} and {\Delta_t + \Gamma_t = \Delta_u} is \mathcal{A}_t measurable. Using \Delta_u\cdot C_u = \Gamma_u\cdot X_u + A_u we have \begin{aligned} V_t D_t &= (\Delta_t + \Gamma_t)\cdot X_t D_t \\ &= \Delta_u\cdot (X_u D_u + C_u D_u)|_{\mathcal{A}_t} \\ &= (\Delta_u\cdot X_u D_u + \Delta_u\cdot 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 is established by finite induction.
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 \tag{3} X_t D_t = M_t - \sum_{s\le t} C_s D_s is an arbitrage-free model of prices and cash flows.
For example, the Black-Scholes/Merton model for the bond and stock with no dividends is parameterized by {M_t = (1, e^{\sigma B_t - \sigma^2t/2})P}, {D_t = e^{-\rho t}P} where P is Weiner measure on \Omega = C([0,\infty)), the space of continuous functions from [0,\infty) to the real numbers, and B_t is standard Brownian motion.
Proof: This follows from substituting X_u D_u = M_u - \sum_{s\le u} C_s D_s in equation (1) to obtain the arbitrage-free condition: \begin{aligned} (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{\mathcal{A}_t} &= (M_u - \sum_{s\le u} C_s D_s + \sum_{t < s \le u} C_s D_s)|_{\mathcal{A}_t} \\ &= (M_u - \sum_{s\le t} C_s D_s)|_{\mathcal{A}_t} \\ &= M_t - \sum_{s\le t} C_s D_s \\ &= X_t D_t \\ \end{aligned}
Proof: (FTAP): Since {V_{\tau_0}D_{\tau_0} = (\sum_{u > \tau_0} A_u D_u)|_{\mathcal{A}_{\tau_0}}} 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.
Ross’s proof of the contrapositive involves the Hahn-Banach theorem, but there is no need for that. Every arbitrage-free model used in practice is parameterized by equation (3).
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{4} 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 known prior to time t.
\Delta is delta and \Gamma is gamma.
What this theory does not tell us is when to trade. We could let \tau_j = j\Delta t for some time increment \Delta t and this results in the usual Black-Scholes/Merton delta hedge as the time increment goes to zero.
A smarter idea might be to set a price increment \Delta X and determine the next trading time \tau_{j+1} from the last trading time \tau_j by {\tau_{j+1} = \inf \{t > \tau_j\mid \|X_t - X_{\tau_j}\|_\infty > \Delta X\}}.
What this theory does allow us to do is rigorously analyze any trading strategy. For instance, the above \Delta X strategy could be implemented using limit orders. At time \tau_j place limit orders at {X_{\tau_j} \pm \Delta X} of size \Gamma_{j+1}. We do not know \Gamma_{j+1} at time \tau_j since it depends on \tau_{j+1} and X_{\tau_{j+1}}, however limit orders cost nothing to place and cancel so we can keep adjusting their size as t\to\tau_{j+1}.
The B-S/M model obscures the essential problem faced by every trader: when and how how much of each instrument are required to hedge. It is simply not possible to hedge continuously. The SUM forces you to attune to this fundamental question.
Semi-static hedging involves finding an initial static hedge that approximately replicates the option, then finding a dynamic hedge to improve on the difference. See (Carr, Ellis, and Gupta 1998) for early work and (Sauldubois and Touzi 2024) for recent improvements of this technique. The SUM can be applied to find the non-static hedge.
If repurchase agreements are available 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)\approx 1 + 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 and the reciprocal R_t = 1/D_t is called the money market account. It is convenient to assume there is an instrument with price R_t and no cash flows exists in order to finance trading strategies.
It is rare for market participants to use repos to finance their trading. Traders working for large banks have access to a funding desk that charges them the repo rate plus a few basis points. Day traders using a credit card to fund trading get charged at the APR they qualifiy for.
The SUM model does not assume a risk-free rate exists. If a funding instrument is being used then it, or all the instruments used to mimic it, should be added to the set of instruments I.
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.
Note D_0(u)D_0 = D_u|_{\mathcal{A}_0} so the discount to time u is D_0(u) = D_u(\Omega)/D_0(\Omega) = D_u(\Omega) assuming, without loss of generality, that D_0(\Omega) = 1. This technically makes D_0 a “probability measure,” but it is not the probability of anything.
Zero coupon bond prices are determined by the deflators. The (Brace, Gatarek, and Musiela 1997) model corresponds to using the canonical deflator built from futures instead of repos. The (Heath, Jarrow, and Morton 1992) model corresponds to using the continuous time forward rates. Every stochastic interest rate model is a special case of these. They just have specific parameterizations to make calculations more convenient.
Fixed income instruments pay fixed cash flows c_j at times t_j. From equation (2), the value of a fixed income instrument at time t is determined by V_t D_t = (\sum_{t_j > t} c_j D_{t_j})|_{\mathcal{A}_t} so V_t = \sum_{t_j > t} c_j D_t(t_j).
Bonds can default and the holder recovers only a fraction of the remaining value. Let \tau be the time the bond defaults and \rho be the fraction of the value recovered. These are both random variables. The original sample space must be augmented by the cartesion product with [0,\infty)\times [0,1] indicating the default time and recovery. At time t the information we have about the default time is either \tau < t and we know exactly when default happened, otherwise we only know \tau\ge t. This corresponds to the partition of [0,\infty) consisting of singleton atoms \{s\}, s < t, and the atom [t,\infty). We assume if default occurs at time t then we learn about it later.
A risky zero coupon bond maturing at u pays in full at u if \tau > u and \rho D_\tau(u) at \tau if t\le\tau\le u. By equation (2) its value D_t^{\tau,\rho}(u) satisfies D_t^{\tau,\rho}(u) D_t = (\rho D_\tau(u)1(t\le\tau\le u) D_\tau + 1(\tau >u)D_u)|_{\mathcal{A}_t}. Since D_\tau(u)D_\tau|_{\mathcal{A}_t} = D_u|_{\mathcal{A}_t} for t\le\tau\le u this simplifies to D_t^{\tau,\rho}(u) D_t = ((\rho 1(t\le\tau\le u) + 1(\tau >u))D_u)|_{\mathcal{A}_t}. If we make the usual unrealistic assumptions that \tau is independent of all (D_t)_{t\in T} and \rho is constant then D_t^{\tau,\rho}(u) D_t = (\rho P(t\le\tau\le u) + P(\tau > u))D_t(u). If \rho = 1 or \tau = \infty we have D_t^{\tau,\rho}(u) = D_t(u).
The credit spread s_t^{\tau,\rho}(u) is defined by D_t^{\tau,\rho}(u) = \exp(-s_t^{\tau,\rho}(u)) D_t(u) so s_t^{\tau,\rho}(u) = -\log (\rho P(t\le\tau\le u) + P(\tau > u)) If \tau is exponentially distributed with hazard rate \lambda then for small \lambda we have the back-of-the-envelope approximation s_0^{\tau,\rho}(u) \approx \lambda u(1 - \rho) If there is full recovery, \rho = 1, or no default, \lambda = 0, then s_0^{\tau,rho} = 0.
!!!revisit
The Black-Scholes and Merton models have trading times T = [0,\infty), sample space \Omega = C[0,\infty), the set of continuous functions on the non-negative real numbers, and deflator D_t = e^{-\rho t}P where P is Wiener measure. The martingale measure is M_t = (r, se^{\sigma B_t - \sigma^2t/2})P where (B_t) is standard Brownian motion.
We can generalize this to any Lévy process (X_t) with X_1 having mean zero and variance one. In this case M_t = (r, se^{\sigma X_t - t\kappa(\sigma)})P where \kappa(\sigma) = \log E[\exp(\sigma X_1)] is the cumulant of X_1. Independence and stationarity imply the cumulant of X_t is t\kappa(\sigma) and the distribution of X_1 determines the Lévy process. See (Sato 1999) for basic facts about Lévy processes.
Since X_1 is infinitely divisible ???cite and has finite variance its cumulant can be characterized by \gamma\in\boldsymbol{{R}} and a non-decreasing function \Gamma\colon\boldsymbol{{R}}\to\boldsymbol{{R}} with \lim_{x\to -\infty}\Gamma(x) = 0 and \lim_{x\to\infty}\Gamma(x) = 1 where \kappa(s) = \log E[e^{sX_1}] = \gamma s + \int_{-\infty}^\infty K_s(x)\,dG(x) and K_s(x) = (e^{sx} - 1 - sx)/x^2 = \sum_{k=2}^\infty s^n x^{n-2}/n! is the Kolomogorov(Kolmogorov 1931) kernel. Note G(x) = 1_[0,\infty) gives the normal distribution with mean \gamma and variance one. If G(x) = 1_[a,\infty), a\not=0, we have \kappa(s) = \gamma s + (e^{as} - 1 - as)/a^2 = (\gamma - 1/a)s + (1/a^2)(e^{as} - 1). Recall the cumulant of a Posson distribution with parameter \lambda is \lambda(e^s - 1) and the cumulant of b + cX is bs + \kappa(cs) where \kappa is the cumulant of X. This implies \lambda = 1/a^2, a = c, and \gamma - 1/a = b so every infinitely divisible distribution can be approximated by a normal plus a sum of independent Poisson distributions.
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: {(ax)(i) = ax(i)} and {(x + y)(i) = x(i) + y(i)}, a\in\boldsymbol{{R}}, {x,y\in\boldsymbol{{R}}^I}.
A real valued function takes an element of a set to a real number. A real valued measure takes a subset of a set to a real number. Measures do not count things twice and the measure of nothing is zero. If B(S) is the normed vector space of bounded functions on S where the norm of f\in B(S) is {\|f\| = \sup\{|f(s)|\mid s\in S\}} then its dual B(S)^* can be identified with the normed vector space of bounded finitely additive measures on S, ba(S). The dual pairing {\langle f,\phi\rangle = \int_\Omega f\,d\phi} is defined for simple functions, a finite linear sum {f = \sum_i a_i 1_{A_i}}, A_i\subseteq\Omega, by \int_\Omega f\,d\phi = \sum_i a_i\phi(A_i). Dunford and Schwartz(Dunford and Schwartz 1958) show this is well-defined and an isometry so it can be extended to all of B(\Omega).
If \Phi\colon B(S)\to\boldsymbol{{R}} is linear define \phi(E) = \Phi 1_E where 1_E(s) = 1 if s\in E and 1_E(s) = 0 if s\not\in S. Since 1_{E\cup F} = 1_E + 1_F - 1_{E\cap F} we have \phi(E\cup F) = \phi(E) + \phi(F) - \phi(E\cap F). Measures do not count things twice. Since 1_\emptyset = 0, \phi(1_\emptyset) = 0.
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. An atom B\in\mathcal{A} satisfyies A\subseteq B and A\in\mathcal{A} implies A=\emptyset or A = B. In this case a bounded function X\colon\Omega\to\boldsymbol{{R}} is \mathcal{A}-measurable if and only if X is constant on atoms of \mathcal{A}. That makes it a bounded function on the atoms of \mathcal{A} so it belongs to 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.
Every arbitrage-free model of prices (X_t) and cash flows (C_t) indexed my market instruments
has the form
X_t D_t = X_0 M_t - \sum_{s\le t} C_s D_s
where (D_t) are positive
adapted measures and (M_t) is a
martingale measure indexed by market instruments. If repos are available
they determine a canonical deflator so the problem reduces to finding
martingale measures.
If we know the distribution of an individual instrument at time t it is trivial that M_s = (X_t/X_0\,D_0)|\mathcal{A}_s is a martingale measure. If options are available with expiration t then