Mar 13, 2026
(Ross 1978) showed how to value uncertain future cash flows. His two main results are:
There is no need for probability, only geometry.
The theory applies to any portfolio of instruments.
Perhaps this was too astonishing for people to appreciate. There is no need for Brownian motion or Ito processes, much less partial differential equations. It applies to any portfolio of instruments, not just the bond, stock, and option (Black and Scholes 1973) and (Merton 1973) considered.
Ross’s definition of a cash flow was a jump in stock price. Stock prices jump between close and open but there is no dividend payment involved. Fixed Income bonds are defined by their cash flows. Futures always have price zero with cash flows the difference of their daily quotes. Adding a knob for cash flows to Ross’s theory leads to a model that maps more closely to practical implementation.
Every instrument has bounded price X_t and cash flow C_t at trading time t. The cash flow is zero except for a finite number of times. If I is the collection of all instruments then we can model them simultaneously if we let X_t and C_t be vectors indexed by I.
Ross showed if the there is no trading strategy that makes money on the first trade and never loses money until the trading strategy is closed out if and only if there exist positive measures D_t with \tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{\mathcal{A}_t} where \mathcal{A}_t is the partition of information available at time t.
Ross used the Hahn-Banach theorem to prove their existence but there is no need for that. Every arbitrage-free model has a simple parameterization.
Every arbitrage-free model is parameterized by a vector-valued measure M_t satisfying {M_t = M_u|_{\mathcal{A}_t}, t \le u} (a martingale measure) where \tag{2} X_t D_t = X_0 M_t - \sum_{s\le t} C_s D_s.
Exercise. Show equation (2) implies equation (1).
Hint: Replace t by u in equation (2) and plug that into the right-hand side of equation (1).
This is the so-called “easy direction” but why bother with the contrapositive when we have a ready supply of arbitrage-free models? The fun part is coming up with martingales that can fit market data.
The Black-Scholes/Merton model for the bond and stock 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 and B_t is standard Brownian motion.
Exercise. Show X_t = (R_t, S_t) where R_t = r e^{\rho t} and S_t = s e^{\sigma B_t + (\rho - \sigma^2/2)t}.
Hint: Use equation (2) with C_t = (0, 0).
A trading strategy is a finite number of increasing stopping times \tau_0 < \cdots < \tau_n and trades \Gamma_j indexed by instruments that depend only on information available at time \tau_j. Trades accumulate to positions \Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s where \Gamma_s = \Gamma_j when s = \tau_j and is zero otherwise. Note the strict inequality. Trades take time to settle into a position. A trading strategy is closed out if \sum_j \Gamma_j = 0.
Trading involves accounting. The value (or mark-to-market) is V_t = (\Delta_t + \Gamma_t)\cdot X_. It is the putative value of liquidating the existing position and trades just executed assuming that can be done at current market prices. The trades just executed are not yet in the position, but they soon will be, \Delta_t + \Gamma_t = \Delta_{t+\epsilon}.
The amounts involved in trading are A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t. Cash flows proportional to the existing position are credited to the trading account and trades just executed are debited at the current market prices.
Equation (1) and the definition of V_t and A_t result in \tag{3} V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\mathcal{A}_t}. The proof starts with V_t D_t = (\Delta_t + \Gamma_t)\cdot X_t D_t and using equation (1).
Note how value and amount in equation (3) correspond to price and cash flow in equation (1).
Trading strategies create synthetic market instruments.
This is the skeleton key to valuing, hedging, and managing the risk of derivative instruments.
A (cash settled) derivative is a contract where the buyer will pay the seller to cover amounts \hat{A}_k at times \hat{\tau}_k. The buyer is willing to pay the seller a premium to take the risk off their hands. This short note provides nothing to suggest what premium, vigorish, or baksheesh should be charged for that service.
The job of a quant is to find a trading strategy (\tau_j, \Gamma_j) with A_{\hat{\tau}_k} = \hat{A}_{k} and A_t = 0 otherwise. Unless you belong to a Pythagorean cult that believes in the mathematical absurdity of continuous time trading, this is generally not possible. A start at solving this difficult and not well-understood problem is to note V_t = (\Delta_t + \Gamma_t)\cdot X_t and V_t can be calculated using the contract terms \hat{A}_t by equation (3).
The Fréchet derivative of V_t with respect to X_t is D_{X_t}V_t = \Delta_t + \Gamma_t. At time 0 the position is 0 so this gives us the initial trade \Gamma_0. At any time after that we have \Gamma_t = D_{X_t}V_t - \Delta_t. The position \Delta_t is known at time t so this can be used to come up with candidate trading strategies. What this theory does not tell you is when you should trade. If you let \tau_j = j\Delta t for some time increment \Delta t then this results in the usual Black-Scholes/Merton delta hedge as the 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 |X_{\tau_{j+1}} - X_{\tau_j}| > \Delta X.