January 26, 2025
In “A Simple Approach to the Valuation of Risky Streams” Stephen Ross(Ross 1978) showed
If there are no arbitrage opportunities in the capital markets, then there exists a (not generally unique) valuation operator, L.
As shown in the Unified Model, Ross’s linear valuation operators correspond to deflators: adapted, positive, finitely-additive measures indexed by trading time.
Market instruments have prices and associated cash flows. Stocks have dividends, bonds have coupons, futures have daily margin adjustments. The price of a futures is always zero. A market model consist of vector-values prices (X_t) and cash flows (C_t) indexed by market instruments. Prices and cash flows depend only on information available at time t. This is modeled by algebras of sets \mathcal{A}_t at each trading time t and requiring prices and cash flows to be measurable with respect to the algebras.
A model is arbitrage free if and only if there exist a deflators (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 | indicates restriction of measure. Recall a function times a measure is a measure and the conditional expectation {Y = E[X|\mathcal{A}]} if and only if Y(P|_{\mathcal{A}}) = (XP)|_{\mathcal{A}}, where P is a probability measure.
If (M_t) is s vector-valued martingale measure 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. For example, the Black-Scholes/Merton model with no dividends corresponds to X_t = (r\exp(\rho t), s\exp(\rho t + \sigma B_t - \sigma^2 t/2)), C_t = (0, 0) and {D_t = \exp(-\rho t)P|_{\mathcal{A}_t}} where (B_t) is Brownian motion, P is Wiener measure, and {\mathcal{A}_t} is the standard filtration.
A trading strategy is a finite sequence (\tau_j) of increasing stopping times and vector-valued (\Gamma_j), depending on information available at time \tau_j, indicating the number of shares purchased at time \tau_j. Let \Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s be the (settled) position at time t, where \Gamma_s = \Gamma_j if s = \tau_j and is zero otherwise.
The amounts A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t show up in the brokerage account during trading: you receive cash flows proportional to existing positions and pay for trades just executed at the prevailing market prices. The mark-to-market of the trading strategy at time t is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It the the value of unwinding existing positions and the last trades at prevailing market prices. A simple consequence of these definitions is \tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u}A_s D_s)|_{\mathcal{A}_t} Note how value V_t in (2) corresponds to prices X_t in (1), likewise amount A_t to cash flows C_t.
Trading strategies create synthetic market instruments.
A derivative is a synthetic market instrument. Its contract specifies a finite sequence of increasing stopping times \overline{\tau}_j and amounts \overline{A}_j paid at these times. A European option has a single constant stopping time \bar{\tau} with payoff {\overline{A} = \phi(X_{\overline{\tau}})} for some function \phi.
Suppose we could find a a trading strategy (\tau_j), (\Gamma_j) with \sum_j \Gamma_j = 0, A_t = \overline{A}_j when t = \overline{\tau}_j and is zero (self-financing) otherwise. The condition {\sum_j \Gamma_j = 0} requires the hedge to be eventually closed. This is a perfect hedge and the value of the derivative at time t would be determined by equation (2): {V_t D_t = (\sum_{\overline{\tau}_j > t} \overline{A}_j D_{\overline{\tau}_j})|_{\mathcal{A}_t}}.
Perfect hedges do not exist in practice. A fundamental problem in mathematical finance is how to hedge a derivative when a perfect hedge does not exist. A first attempt at a solution is to assume a perfect hedge exists. The initial hedge at \tau_0 = 0 can be computed from V_0 = \Gamma_0\cdot X_0 and {V_0 D_0 = (\sum_{\overline{\tau}_j > t} \overline{A}_j D_{\overline{\tau}_j})|_{\mathcal{A}_0}}. \Gamma_0 D_0 = D_{X_0}(\sum_{\overline{\tau}_j > 0} \overline{A}_j D_{\overline{\tau}_j})|_{\mathcal{A}_0}, where D_{X_0} is the Fréchet derivative. Just as in the B-S/M theory, the (putative) initial hedge is the derivative of value with respect to current prices. Note that value can be computed using only the deflators and the contract specified amounts.
For \tau_1 = t > 0 we have V_t = (\Delta_t + \Gamma_t)\cdot X_t so (\Delta_t + \Gamma_t) D_t = D_{X_t}(\sum_{\overline{\tau}_j > t} \overline{A}_j D_{\overline{\tau}_j})|_{\mathcal{A}_0}. For t > 0 sufficiently small we have \Delta_t = \Gamma_0 so we can solve for \Gamma_t. This procedure does not specify what value of \tau_1 to choose.
The Unified Model does not provide an answer to when hedge, it only puts your nose directly in the problem of when and how much to hedge. The classical Black-Scholes/Merton theory gives the inapplicable answer that you should trade “continuously”.
If repurchase agreements exists then there is a canonical choice for the deflators. A repurchase agreement with rate f_t, known at time t, has price 1 at time t and pays \exp(f_t\,dt) at time t + dt. For any deflator (D_t) equation (1) gives {1 D_t = (\exp(f_t\,dt)D_{t + dt})|_{\mathcal{A}_t} = \exp(f_t\,dt)D_{t + dt}|_{\mathcal{A}_t}}. For a deflator with D_{t + dt} known at time t we have {D_t = \exp(f_t\,dt)D_{t + dt}}. The canonical deflator is {D_t = \exp(-\int_0^t f_s\,ds)D_0}.
The repurchase rates (f_t) are called the (continuously compounded) short rate process. Every interest rate model is just a specific parameterization of this. The deflators determine the prices of zero coupon bonds. If D_t(u) is the price of a zero coupon bond paying 1 unit at time u then equation (1) implies {D_t(u) D_t = D_u|_{\mathcal{A}_t}} so the price is the Radon-Nikodym derivative.
The price of zero coupon bonds determine the value of all risk-free fixed income instruments.