Mar 6, 2026
Trading is buying and selling market instruments that have prices and cash flows associated with instrument ownership, then accounting for the values and amounts involved. This note only considers the simple case where there is perfect liquidity and prices can be any real number. In actual trading there is a bid-ask spread, depending on whether you are selling or buying, that tends to increases along with the size of the trade. Prices can only be integral multiples of the minimum tick size of the instrument. We completely ignore credit considerations of the counterparties involved.
Instruments I have \boldsymbol{{R}}^I = \{x\colon I\to\boldsymbol{{R}}\}1 valued prices (X_t)_{t\in T} and cash flows (C_t)_{t\in T} at trading times T. Only a finite number of cash flows (e.g., stock dividends, bond coupons, futures margin adjustments) are not zero.
A trading strategy is a finite number of increasing stopping times \tau_0 < \cdots < \tau_n and \boldsymbol{{R}}^I valued trades \Gamma_j done at \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. It takes some time for trades to settle into a position.
The value (or mark-to-market) of a trading account is the cash value of liquidating existing positions and trades just done at current market prices: V_t = (\Delta_t + \Gamma_t)\cdot X_t. The amount appearing at time t\in T in the trading account is all cash flows proportional to existing position minus the cost of the trades just executed: A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t.
We consider the case of a money market instrument and stock X_t = (R_t, S_t). For simplicity we assume cash flows are zero. The trade \Gamma_t = (M_t, N_t) is how much of the money market and stock is purchased at time t. We write \Delta_t = (\backslash M_t, \backslash N_t) where backslash is read as partial sum or scan. The adjacent difference of (x_0,\ldots, x_n) is (x_0, x_1 - x_0, \ldots x_n - x_{n-1}).
Excercise. Show the partial sum of adjacent differences is the identity function.
An elementary trade is buying \Gamma shares at \tau_0 and selling \Gamma shares at \tau_1 > \tau_0. For a bond and a stock \Gamma_0 = (M, N) and \Gamma_1 = -\Gamma_0 = (-M, -N). Money market accounts never have cash flows. We assume the stock has no dividends so C_t = (0,0). In this case A_t = -\Gamma_t\cdot X_t.
| X | \Gamma | \Delta | (\Delta + \Gamma)\cdot X | -\Gamma\cdot X | ||||
| t | R | S | M | N | \backslash M | \backslash N | V | A |
| [0, \tau_0) | R_t | S_t | 0 | 0 | 0 | 0 | 0 | 0 |
| \tau_0 | R_0 | S_0 | M | N | 0 | 0 | MR_0 + NS_0 | -MR_0 - NS_0 |
| (\tau_0, \tau_1) | R_t | S_t | 0 | 0 | M | N | MR_t + NS_t | 0 |
| \tau_1 | R_1 | S_1 | -M | -N | M | N | 0 | MR_1 + NS_1 |
| (\tau_1, \infty) | R_t | S_t | 0 | 0 | 0 | 0 | 0 | 0 |
There is no activity prior to the first trade. The initial trade of M in the money market and N in the stock costs MR_0 + NS_0 and is equal to the value of the position. When the trades settle the position becomes \Delta = (M,N) and its value is MR_t + NS_t until the position is closed out by trading \Gamma = (-M, -N).
We can eliminate all but the \tau_j rows by replacing the position \Delta column with \Delta + \Gamma, the position that occurs after the trade has settled.
| X | \Gamma | \Delta + \Gamma | (\Delta + \Gamma)\cdot X | -\Gamma\cdot X | ||||
| t | R | S | M | N | \backslash M | \backslash N | V | A |
| \tau_0 | R_0 | S_0 | M | N | M | N | MR_0 + NS_0 | -MR_0 - NS_0 |
| \tau_1 | R_1 | S_1 | -M | -N | 0 | 0 | 0 | MR_1 + NS_1 |
A forward contact on the stock pays S_1 - f at \tau_1. To replicate this using the bond and stock we require A_1 = S_1 - f = MR_1 + NS_1. This is satisfied by taking N = 1 and MR_1 = -f. The cost of setting up this hedge is A_0 = fR_0/R_1 - S_0. The initial cost is zero when f = S_0R_1/R_0, which is the standard cost-of-carry formula.
This can be extended to any closed out trading strategy (\tau_j, \Gamma_j) with \sum_j \Gamma_j = 0.
| t | R | S | M | N | \backslash M | \backslash N | V | A |
|---|---|---|---|---|---|---|---|---|
| \tau_0 | R_0 | S_0 | M_0 | N_0 | M_0 | N_0 | R_0 M_0 + N_0 S_0 | -R_0 M_0 - N_0 S_0 |
| \tau_1 | R_1 | S_1 | M_1 | N_1 | M_0 + M_1 | N_0 + N_1 | (M_0 + M_1)R_0 + (N_0 + N_1)S_1 | -M_1 R_1 - N_1 S_1 |
| \cdots | \cdots | \cdots | \cdots | \cdots | \cdots | \cdots | \cdots | \cdots |
| \tau_n | R_n | S_n | -\sum_j M_j | -\sum_j N_j | 0 | 0 | 0 | R_n\sum_j M_j + S_n\sum_j N_j |
A futures contracts always have price zero. The contract is specifed by an expiration T on an underlying S and is marked (usually daily) at times 0 = t_0 < t_1 < \cdots < t_n = T. the quote \Phi_T = S_T at expiration. Prior to expiration the quote \Phi_t, t < T, is determined by the market. Futures have cash flows that are differences of quotes: C_{t_j} = \Phi_{t_j} - \Phi_{t_{j-1}}, j > 0. In an arbitrage-free model futures quotes are a naturally occurring martingale.
A derivative instrument is a contract that pays amounts \hat{A}_j at times \hat{\tau}_j. It can be hedged with the money market account and stock if we can find trades \Gamma_j = (M_j, N_j) with A_j = \hat{A}_j, 0\le j\le n. This is usually not possible. The fundamental problem faced by traders and portfolio managers is when and how much to trade to minimize risk.
Since V_j = (\Delta_j + \Gamma_j)\cdot X_j a reasonable guess for the positions should be \Delta_j + \Gamma_j = dV_j/dX_j, the usual delta hedge. The trades are the adjacent difference of positions.
The Fundamental Theorem of Asset Pricing allows us to compute V_t in terms of a linear pricing measure identified by (Ross 1978).
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.
Ross showed a market model is arbitrage free if there exist positive measures (D_t) with \tag{1} X_t D_t = (\sum_{t < u \le v} C_u D_u + X_v D_v)|_{\mathcal{A}_t} where \mathcal{A}_t is the information available at time t. Using the definitions of V_t and A_t we have \tag{2} V_t D_t = (\sum_{t < u \le v} A_u D_u + V_v D_v)|_{\mathcal{A}_t}. The proof starts with V_t D_t = (\Delta_t + \Gamma_t)X_t D_t and using equation (1) to replace X_t D_t. Note how value V_t corresponds to price X_t and amount A_t corresponds to cash flow C_t. This shows every trading strategy creates a synthetic market instrument.
Every arbitrage-free model is parameterized by a martingale measure (M_t) taking values in \boldsymbol{{R}}^I and positive measures (D_t). Given cash flows (C_t) the prices (X_t) satisfy X_t D_t = X_0 M_t - \sum_{s\le t}C_s D_s. For example, the Black-Scholes/Merton model is parameterized by the martingale measure M_t = (1, e^{\sigma B_t - \sigma^2 t/2})P and positive measures D_t = e^{-\rho t}P where P is Wiener measure and B_t is standard Brownian motion. Ross showed there is no need for the partial differential equations arising from Ito’s formula and generalized their result to any collection of instruments, not just a bond, stock, and option.
A European option has only one cash flow at expiration. In this case we need A_j = 0 for 0 < j < n and A_n equal to the option payoff. The amount for setting up the initial hedge is A_0 = -V_0. Since V_0 = M_0 R_0 + N_0 S_0 we have M_0 = (V_0 - N_0 S_0)/R_0. Given stock trades N_j, 0\le j < n the condition A_j = 0 holds when M_j = -N_j S_j/R_j, 0 < j < n. Since the positions close out we have A_n = -M_n R_n - N_n S_n = (\sum_{j=0}^{n-1} M_j)R_n + (\sum_{j=0}^{n-1} N_j)S_n.