April 25, 2024
Market instruments can be bought or sold at a price and ownership entails cash flows. Shares of instruments can be traded based on available information and accumulate to positions. The mark-to-market value and amounts involved with trading correspond to price and cash flows respectively. The Unified Model demonstrates the connection between dynamic trading and how to value, hedge, and manage the risk of any portfolio.
Every arbitrage-free model of prices and cash flows is parameterized by a vector-valued martingale whose components are indexed by market instruments and positive measures.
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canonical deflator is the reciprocal of the money market account.A derivative security is a contract between two parties: I will give you this on these dates if you will give me that on those dates. Derivatives must have existed since before recorded history. The Nobel prize winning breakthrough of Black, Scholes, and Merton showed how to synthesize derivatives by dynamically trading market instruments based on the borrowing cost used to fund the hedge. Estimating the growth rate of the underlying securities was not neccesary.
This paper provides a unified model for valuing, hedging, and managing the risk of any derivative security. It shows how they can be synthesized by trading market instruments and turns the spotlight on what may be the next Nobel prize winning problem: how should you hedge if you can’t do it continuously?
The Unified Model can be used for any portfolio of stocks, bonds, currencies, commodities, and even other derivatives in the portfolio. Academic literature tends to focus on prices, but cash flows should be placed on equal footing. The fact is that derivative contracts are specified by their cash flows. No arbitrage places constraints on their price dynamics.
Every instrument has a price X_t and a cash flow C_t at any trading time t\in T. Instruments are assumed to be perfectly liquid: they can be bought or sold at the given price in any amount. Cash flows are associated with owning an instrument and are almost always 0; stocks have dividends, bonds have coupons, currencies have no cash flows, commodities have storage costs. European options have exactly one cash flow at expiration.
Let T be a totally ordered set of trading times, I the set of all market instruments, \Omega the sample space of possible outcomes, and {\mathcal{A}}_t an algebra of sets modeling information available at time t\in T. Prices and cash flows are bounded {\mathcal{A}}_t-measurable functions {X_t, C_t\colon\Omega\to\boldsymbol{R}^I}, {t\in T}.
We write X\in B({\mathcal{A}}) if X\colon{\mathcal{A}}\to\boldsymbol{R} is a bounded {\mathcal{A}}-measurable function. Note if {\mathcal{A}} is finite then its atoms are a partition of \Omega and being {\mathcal{A}}-measurable is equivalent to X being constant on atoms. In this case X\colon{\mathcal{A}}\to\boldsymbol{R} is standard mathematical notation for a function when {\mathcal{A}} is identifed with its atoms.
Traders buy and sell shares based on information available at each trading time. They trade a finite number of times and eventually close out all postions.
A trading strategy is a finite collection of strictly increasing stopping times (τ_j) and trades (\Gamma_j) where \Gamma_j:{\mathcal{A}}_{τ_j}\to \boldsymbol{R}^{I} indicating the number of shares to trade in each instrument at time τ_j. If \tau is a stopping time then {\mathcal{A}}_\tau = \{A\in{\mathcal{A}}_t\mid t\in T, t\le\tau\}.
Trades accumulate to a position \Delta_t = \sum_{τ_j < t}\Gamma_j = \sum_{s < t}\Gamma_s where \Gamma_s(\omega) = \Gamma_j(\omega) when s = τ_j(\omega), \omega\in\Omega. A trade at time t is not included in the position at time t; it takes some time for trades to settle.
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The value, or mark-to-market is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It is the amount that would result from closing out the entire position at current market prices, assuming that is possible. The trading account is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t; cash flows propostional to existing positions are credited and the cost of the trade at time t is debited.
Arbitrage exists if there is a trading stragegy (\Gamma_t)_{t\in T} with A_0 < 0, A_t \ge 0, t > 0, and \sum_{t} \Gamma_t = 0.
Theorem. (Funamental Theorem of Asset Pricing) There is no arbitrage if and only if there exist positive measures (D_t)_{t\in T} on \Omega with X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{{\mathcal{A}}_t}
Lemma. For any arbitrage free model and any trading strategy V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{{\mathcal{A}}_t}
Lemma. If X_t D_t = M_t - \sum_{s\le t} C_s D_s where M_t = M_u|_{{\mathcal{A}}_t}, t \le u then there is no arbitrage.