Aug 12, 2026
Define a simple mathematical model of the financial world that can be implemented with existing technology to provide real-time valuation, hedging, and risk management tools across all asset classes.
As is customary in mathematical finance we ignore certain salient real-world features. Instrument prices and not real numbers, they are integer multiples of tick size. Likewise for trading amounts. The amount being bought or sold affects the price. The price can also depend on credit considerations of the counterparties involved.
Historically, companies set up departments for each instrument class: equities, fixed income, foreign exchange, commodities, etc. Ad hoc approaches have had limited success in modeling the joint distributions between these. We describe a systematic approach amenable to machine learning.
Financial companies need mathematical models that can be implemented in software by people who are not experts in business or mathematics. See Programming as Theory Building
Replace the (Black and Scholes 1973) and (Merton 1973) model with a modern version of the (Ross 1978) model. Ross extended their theory from a bond, stock, and option to any collection of instruments and showed instrument valuation involves only geometry. There is no need for probability, much less Ito’s lemma or partial differential equations. The B-S/M model is a special case of Ross’s model.
Ross defined a cash flow as a jump in stock price. Adding explicit notation for cash flows results in a more expressive and simpler theory.
Let T be a set of totally ordered trading times, I the set of tradeable instruments, \Omega the set of possible outcomes, and (\mathcal{A}_t)_{t\in T} partitions of \Omega indicating the information available at time t\in T. See Mathematical Prerequisites.
Prices and cash flows are bounded functions X_t,C_t\colon\mathcal{A}_t\to\boldsymbol{R}^I where C_t = 0 except when there is a cash flow.
A trading strategy is finite collection increasing stopping times (\tau_j)_{j=0}^n and trading amounts \Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{R}^I of the number of shares of each instrument purchased at time \tau_j.
Shares accumulate to a position \Delta_t = \sum_{\tau_j < t} \Gamma_j. Note the strict inequality – it takes time for a trade to settle into a position. We write this pointwise on \Omega as \Delta_t = \sum_{s < t} \Gamma_s where \Gamma_s = \Gamma_j when s = \tau_j and is zero otherwise.
The value, or mark-to-market, of a trading strategy is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It is the amount you would make if it were possible to unwind your existing position and trades just executed at current market prices.
The amount that shows up in your brokerage account or trade blotter is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t – you get cash flows associated with the instruments you own proportional to your existing position and have to pay for the trades just executed.
We ignore the fact prices and trading amounts are integral increments of minimal increments determined by the markets they trade in. We also ignore the fact that the amount being traded has an affect on price. We totally ignore the fact that the counterparties involved in the trade will be quoted different prices based on their credit rating, among other things.
Arbitrage exists if there is a trading strategy with A_{\tau_0} > 0, A_t\ge0 for 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 out.
Every arbitrage-free model is parameterized by positive adapted measures, deflators, (D_t)_{t\in T} and a vector-valued martingale measures (M_t) on \Omega indexed by instruments satisfying \tag{1} X_t D_t = X_0 M_t - \sum_{s\le t} C_s D_s where M_t = M_u|\mathcal{A}_t for t \le u. For example, the Black-Scholes/Merton model (with no dividends) is {D_t = e^{-\rho t}P} and {M_t = (1, e^{\sigma B_t - \sigma^2t/2})P} where P is Wiener measure and B_t is standard Brownian motion.
If the model includes repurchase agreements then deflators are the stochastic discount.
An immediate consequence is \tag{2} X_t D_t = (X_u D_u + \sum_{t < s\le u} C_s D_s)|\mathcal{A}_t Using the definition of value and amount \tag{3} V_t D_t = (V_u D_u + \sum_{t < s\le u} A_s D_s)|\mathcal{A}_t
Trading strategies create synthetic market instruments.
This is the skeleton key to understanding derivatives.
A derivative is a synthetic market instrument. Buy side firms ask sell side firms to quote a price in order to hedge their future cash flow risks. Quants use math that gets turned into software to tell traders the cost of setting up a hedge to do that. Traders figure out how much vigorish they can get away with without losing the trade.
A (cash settled) derivative is a contract to pay amounts (\hat{A}_j) at stopping times (\hat{\tau}_j), 0\le j\le n. A perfect hedge is a trading strategy (\tau_j,\Gamma_j) with A_t = \hat{A}_j when t = \hat{\tau_j} and is zero otherwise.
If a perfect hedge exists (it almost never does) then by (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 depends only on the contract specifications and deflator (D_t).
Since V_t = (\Delta_t + \Gamma_t)\cdot X_t its Fréchet derivative is D_{X_t} V_t = \Delta_t + \Gamma_t. There is no position at \tau_0 = 0 so \Gamma_0 = D_{X_{\tau_0}} V_{\tau_0} is a candidate for the initial hedge.
Given \tau_1 > \tau_0 we have \Gamma_1 = D_{X_{\tau_1}} V_{\tau_1} - \Gamma_0. This can be repeated with successive increasing stopping times.
There is no canonical way of choosing trading times.
Choosing \tau_j = j\Delta t and letting \Delta t go to zero results in the B-S/M model where \Delta is delta and \Gamma is gamma.
A better choice might be to specify a \Delta X and only trade when the underlying moves by that amount. If futures are available on X this can reduce transaction costs.
Going back to (Markowitz 1952) and (Roy 1952) one might want to find a trading strategy making A_t - \hat{A}_t white noise with minimum variance.
In this section we identify instruments by their prices and cash flows.
A repurchase agreement, or repo, is specified by and effective date t, a time interval \Delta t, and a rate f. Its price at time t is 1 and has cash flow of e^{f\Delta t} \approx 1 + f\Delta t at time t + \Delta t. By (2) we have 1D_t = e^{f\Delta t}D_{t + \Delta t}|\mathcal{A}_t If we assume D_{t + \Delta t} is known at time t then D_t = e^{f\Delta t}D_{t + \Delta t}.
Given increasing times (t_j) and repo rates (f_j) known at time t_j we have D_{t_0} = e^{f_0 \Delta t_0}D_{t_1} = \cdots = e^{\sum_{0 \le j < k} f_j \Delta t_j}D_{t_k} where \Delta t_j = t_{j+1} - t_j. This provides the canonical deflator {D_{t_k} = e^{-\sum_{0 \le j < k} f_j \Delta t_j}D_{t_0}}. The continuous time version of this is the usual stochastic discount {D_t = e^{-\int_0^t f_s\,ds}D_0} where f_t is the continuously compounded instantaneous forward rate at time t.
The zero coupon bond D(u) pays a unit cash flow at maturity u. Given deflators (D_t) it satisfies X^{D(u)}_t D_t = D_u|\mathcal{A}_t, u \ge t. Writing X^{D(u)}_t = D_t(u) we have the price of a zero coupon bond at time t maturing at u is the Radon-Nykodym derivative D_t(u) = d(D_u|\mathcal{A}_t)/dD_t,
Suppose a bond can default at stopping time \rho and has recovery \rho as a fraction of the value at default.
We need deflators (D_t) and a vector-valued martingale measure (M_t) indexed by market instruments. If repurchase agreements exist then their is a canonical deflator. The difficult problem is coming up with partitions representing partial infomation and martingale measures that can be tuned to market data.
We assume all sets are finite since every computer implementation is finite. Continuous time results can be recovered by taking appropriate limits.
Every finite dimensional vector space is isomorphic to \boldsymbol{R}^n for some positive integer n but it is useful to keep track of where they come from. If S is finite and B(S) are (necessarily) bounded functions from S and is isomorphic to \boldsymbol{R}^n where n is the cardinality of S. The space of (finitely-additive) measures ba(S) on S is also isomorphic to \\R^n where n is the cardinality of S.
Recall the vector space dual of B(S) is isometrically isomorphic to ba(S) for any set S. These are distict mathematical types even though both are isomorphic to \boldsymbol{R}^n. If L\in B(S)^* is a linear functional define the measure \lambda\in ba(S) by \lambda(A) = L(1_A) for A\subseteq S where 1_A(s) = 1 if s\in A and 1_A(s) = 0 if s\not\in A. It is a measure since {\lambda(A\cup B) = \lambda(A) + \lambda(B) - \lambda(A\cap B)} follows from 1_{A\cup B} = 1_A + 1_B - 1_{A\cap B} and \lambda(\emptyset) = 0 since 1_\emptyset = 0. We define multiplication of a bounded function g and a finitely additive measure \lambda by \langle f,g\lambda\rangle = \langle fg,\lambda\rangle. See (Dunford and Schwartz 1958) for the details.
If \mathcal{A} is a finite algebra of sets on \Omega then the atoms of \mathcal{A} form a partition of \Omega and the algebra is generated by the partition of its atoms.
Information is modeled by a partition of sets on the sample space \Omega of all possible outcomes. No information is the singleton partition \{\Omega\}. Complete information is the partition of singletons \{\{\omega\}\mid\omega\in\Omega\} Partial information is knowing which atom \omega belongs to.
A function on \Omega is measurable with respect to an algebra \mathcal{A} if and only if it is constant on atoms of the partition. In this case it is a function on the atoms and we write X\colon\mathcal{A}\to\boldsymbol{R}.
A stopping time is a function \tau\colon\Omega\to T where \{\omega\mid\tau(\omega) = t\} is in the algebra of sets generated by \mathcal{A}_t for all t\in T.
If F\colon X\to Y is a function between normed linear spaces the Frechet derivative DF\colon X\to\mathcal{L}(X,Y) is defined by F(x + h) = F(x) + DF(x)h + o(h), x,h\in X when it exists.