August 22, 2025
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 the capital markets, then there exists a (not generally unique) valuation operator, L.
The not generally unique valuation operators are determined by deflators: positive measures that depend only on the information available at each trading time. Ross’s theory applies to all instruments, not just a bond, stock, and option that Black and Scholes (Black and Scholes 1973) and Merton (Merton 1973) considered. His insight was to realize option valuation had nothing to do with probability and used the Hahn-Banach theorem to show prices are constrained by the geometry determined by 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). We also make that unrealistic assumption.
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 (Harrison and Kreps 1979) is impossible to carry out. 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. If Brownian motion hits a at time \tau then it hits a infinitely often in the interval [\tau, \tau + \epsilon] for any \epsilon > 0 since Brownian motion has unbounded variation on any finite interval. See (Karatzas and Shreve 1991) Section 3.2.
A derivative instrument is a contract. The seller is obligated to to satisfy cash flows specified in the contract the buyer agrees to. The fundamental problem every sell side trader faces the first day on the job is when and how many market instruments to buy in order to manage risk.
This note provides a simple universal model that provides a foundation for making advances on this important and poorly understood problem.
Let \Omega be a sample space representing everything that can happen. Information at time t is modeled 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 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) to denote the space of bounded functions on \Omega. Recall its vector space dual can be identified with the space of finitely additive measures on \Omega denoted ba(\Omega). For any f\in B(\Omega) define the linear operator of multiplication by f, M_f\colon B(\Omega)\to B(\Omega), by M_fg = fg. Its adjoint is the linear operator M_f^*\colon ba(\Omega)\to ba(\Omega). This defines multiplying a measure by a function f\phi = M_f^*\phi for f\in B(\Omega) and \phi\in ba(\Omega).
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 holding 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. 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 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 \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.
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: 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. It follows \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 is established 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.
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} = (\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 proof of the contrapositive involves the Hahn-Banach theorem, but there is no need for that. Every arbitrage-free model used in practice has the above parameterization.
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.
If we choose a strategy trading at every T = \epsilon\boldsymbol{{N}} then as \epsilon goes to zero this can be recognized as a generalization of the classical B-S/M model where \Delta is delta and \Gamma is gamma.
A feature of the SUM is there is no canonical choice for the trading times (\tau_j). 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 consider this question that has yet have a definitive answer.
Static hedging (Nazir, Peter)
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-Musiela Market model(Brace, Gatarek, and Musiela 1997) corresponds to using the canonical deflator built from futures instead of repos. The Heath-Jarrow-Morton model[HeaJarMor1992] corresponds to using the continuous time forward rates.
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 singletons \{s\}, s < t, and the atom [t,\infty). We assume if default occurs at time t then we learn about it later. It is common to assume recovery is a constant times the value at default of the zero coupon bond.
A risky zero coupon bond maturing at u pays in full at u if \tau > u and \rho D_\tau(u) at \tau if \tau\le u. By equation (2) its value satisfies V_t D_t = (\rho D_\tau(u)1(t\le\tau\le u) D_\tau + 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 V_t D_t = (\rho D_\tau(u)1(t\le\tau\le u) D_\tau + 1(\tau >u)D_u)|_{\mathcal{A}_t}. ???
If rates are zero then D_t(u) = 1. Writing V_t as D_t^{\tau,\rho}(u) we have D_t^{\tau,\rho}(u) = (\rho 1(t\le\tau\le u) + 1(\tau >u))|_{\mathcal{A}_t}
The Black-Scholes(Black and Scholes 1973) and Merton(Merton 1973) model has \Omega = C[0,\infty), the set of continuous functions on the non-negative real numbers, and T = [0,\infty) with instruments a bond and a stock having no associated cash flows. The martingale measure is M_t = (r, se^{\sigma B_t - \sigma^2t/2})P where (B_t) is standard Brownian motion and P is Wiener measure. The deflator is D_t = e^{-\rho t}P.
We can generalize this to any Levy 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 Levy process. ???cite
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 \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(Kol?) kernel. Not 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 get a Poisson distribution with parameter \lambda = ???.
A European option on underlying S pays \nu(S_t) for some function \nu at expiration t. For a put \nu(x) = \max\{k - x, 0\} and for a call \nu(x) = \max\{x - k, 0\} where k is the strike. If we have a money-market instrument R with price R_t = 1/D_t and a self-financing strategy exists then the cost of setting that up at time 0 is determined by {V_0 D_0 = (\nu(S_t) D_t)|_{\mathcal{A}_0}}. Using the “probability” measure P = D_0 this can be written V_0 = E[\nu(S_t)D_t].
The Black model(Black 1976) sidesteps involving iterest rates by computing the forward value of an option on the forward E[\nu(F_t)] where F_t = S_t at expiration.
F_t = fe^{\sigma B_t - \sigma^2t/2}, where f = se^{\rho t}. The forward value is E[\nu(F_t)]. If a European option pays \nu(F) shares then E[F\nu(F)] = fE[e^{sX - \kappa(s)}\nu(F)] = fE^s[\nu(F)] where E^s is share measure. The Radon-Nikodym derivative of the correponding “probability” measures is dP^s/dP = e^{sX - \kappa(s)}. Note e^{sX - \kappa(s)} > 0 and E[e^{sX - \kappa(s)}] = 1 so P_s is a “probability” measure.
The general Black model is F = fe^{sX - \kappa(s)} where X has mean zero and variance one. Every positive random variables with finite mean and log variance has this form. f = se^{\rho t}, and s = \sigma\sqrt{t}.
The forward put value with strike k is p(k) = E[\max\{k - F, 0\}] = E[(k - F)1(F\le k)] so p(k) = P(F\le k) - fP^s(F\le k). Put prices determine the distribution of F, as Breeden and Litzenberg (Breeden and Litzenberger 1978) showed, since \partial_k p = E[1(F\le k)] = P(F\le k).
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))].
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.