A Unified Model of Derivative Securities

Algebras of Sets

Let \Omega be a sample space of all possible outcomes. An algebra of sets on the set \Omega is a collection of subsets of \Omega close under complement and set union. By De Morgan’s laws they are also closed under set intersection. Elements of the algebra are events and we can reason about ‘not A’ (\Omega\setminus A), ‘A or B’ (A\cup B), and ‘A and B’ (A\cap B) for events A, B.

Exercise. If \mathcal{A} is an algebra then so is \mathcal{A}\cup\{\emptyset,\Omega\}.

We assume all algebras contain the empty set, hence also \Omega.

For \omega\in\Omega the atom containing \omega, A_\omega = \cap\{A\in\mathcal{A}\mid\omega\in A\}, also belongs to the algebra.

Exercise. Show \{A_\omega\mid\omega\in\Omega\} is a partition of \Omega.

Hint. Show \Omega = \cup_{\omega\in\Omega} A_\omega and either A_\omega = A_{\omega'} or A_\omega\cap A_{\omega'} = \emptyset for \omega, \omega'\in\Omega.

Exercise. Every algebra is generated by its atoms.

Hint: Show the smallest algebra containing the atoms of the algebra \mathcal{A} is \mathcal{A}.

This shows we can identify an algebra of sets with its atoms.

A function f\colon\Omega\to\mathbf{R} is \mathcal{A}-measurable if f^{-1}((-\infty, x]) = \{\omega\in\Omega\mid f(x)\le x\}\in\mathcal{A} for every x\in\mathbf{R}.

Exercise. Show f\colon\Omega\to\mathbf{R} is \mathcal{A}-measurable if and only if f is constant on the atoms of \mathcal{A}.

This shows \mathcal{A}-measurable functions are just functions on the atoms of \mathcal{A} so we write f\colon\mathcal{A}\to\mathbf{R} to indicate f is \mathcal{A}-measurable.

Measures

Let B(S) be the Banach space of bounded functions on the set S where the norm is \|f\| = \sup_{s\in S} |f(s)|, f\in B(S). The dual of B(S) is the set of all bounded linear functionals L\colon B(S)\to\mathbf{R}. The dual pairing is \langle f, L\rangle = Lf. Given E\subseteq S define \lambda(E) = L1_E where 1_E\colon S\to\mathbf{R} is the indicator function of E, 1_E(s) = 1 if s\in E and 1_E(s) = 0 if s\not\in E.

Since 1_\emptyset = 0 and 1_{E\cup F} = 1_E + 1_F - 1_{E\cap F} we have \lambda(\emptyset) = 0 and \lambda(E\cup F) = \lambda(E) + \lambda(F) - \lambda({E\cap F)} so \lambda is a (finitely additive) measure on S. The set of such measures is denoted ba(S) and is also a Banach space with \|\lambda\| = \sup_{\|f\|\le1} \int_S f\,d\lambda where \int_S f\,d\lambda = Lf. The linear map L\mapsto\lambda is an isometric isomorphism.

If \mathcal{B}\subseteq\mathcal{A} are algebras and \lambda\in ba(\mathcal{A}) we write \lambda|_\mathcal{B} to denote the restriction of \lambda to \mathcal{B}. If (\mathcal{A}_t)_{t\in T} satisfy \mathcal{A}_t \subseteq \mathcal{A}_u, t \le u, and (M_t)_{t\in T} are measures with M_t = M_u|_{\mathcal{A}_u}, t \le u, we say (M_t) is a martingale measure.

For g\in B(S) define M_g\colon B(S)\to B(S) by M_g f = fg, f\in B(S), to be the linear operator of multiplication by g, Its adjoint M_g^*\colon ba(S)\to ba(S) is defined by \langle M_gf,\lambda \rangle = \langle f,M_g^*\lambda \rangle, f, g\in B(S), \lambda\in ba(S). In terms of integrals \int_S M_gf\,d\lambda = \int_S f\,dM_g^*\lambda, or simply \int_S (fg)\,d\lambda = \int_S f\,d(g\lambda).

Market Model

Let T be the set of trading times, \Omega the sample space of possible outcomes, and I be the set of market instruments. Partial information is represented by partitions \mathcal{A}_t, t\in T, where \mathcal{A}_t\subseteq\mathcal{A}_u, t\le u. The prices of the market instruments at time t is the vector X_t\in\mathbf{R}^I. Instruments can be bought or sold in any amount at the given price. The cash flows of the market instruments at time t is C_t\in\mathbf{R}^I. Stocks have dividends, bonds have coupons, and futures have daily margin adjustments as cash flow. The price of a futures is always 0.

A market model specifies prices and cash flows based on available information \begin{aligned} X_t&\colon\mathcal{A}_t\to\mathbf{R}^I \\ C_t&\colon\mathcal{A}_t\to\mathbf{R}^I \\ \end{aligned}

Trading Strategy

A trading strategy is a finite collection of increasing stopping times (\tau_j) and functions \Gamma_j\colon\mathcal{A}_{\tau_j}\to\mathbf{R}^I. At time \tau_j, \Gamma_j shares are purchased in each instrument. Share accumulate to positions \Delta_t = \sum_{\tau_j < t} \Gamma_j. Note the strict inequality; it takes some time for a trade to settle. Let \Gamma_t(\omega) = \Gamma_j(\omega) 1(t = \tau_j(\omega)) so \Gamma_t = \sum_{s<t} \Gamma_s.

The value, or mark-to-market, at time t is V_t = (\Delta_t + \Gamma_t)\cdot X_t. It is the amount that would be obtained by selling the existing positions and the trades just done at prevailing market prices. The impact on the trading account at time t is A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t. You receive cash flows proportional to to the existing positions and pay for the trades executed at time t.

The value and account associated with a trading strategy are \begin{aligned} V_t&\colon\mathcal{A}_t\to\mathbf{R}^I \\ A_t&\colon\mathcal{A}_t\to\mathbf{R}^I \\ \end{aligned}

Arbitrage

We say arbitrage exists for a market model if there is a trading strategy with \sum_j \Gamma_j = 0, A_{\tau_0} < 0, and A_t \le 0 for t > \tau_0. The initial trade makes money and subsequent trades never lose money.

Theorem. (Fundamental Theorem of Asset Pricing) A model has no arbitrage if and only if there exist positive measures (D_t)_{t\in T} with X_t D_t = (X_u D_u + \sum_{t < s \le u} C_s D_s)|_{\mathcal{A}_t}.

If a money market account exists with price R_t (and no cash flows) the trading strategy \Gamma_t = 1, \Gamma_{t+\Delta t} implies R_t D_t = (R_{t+\Delta t}D_{t+\Delta t}.

Lemma. For any arbitrage-free model V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\mathcal{A}_t}. Note how value and account correspond to prices and cash flows. A trading strategy creates a synthetic instrument.

Lemma. If (D_t)_{t\in T} are positive measures and (M_t)_{t\in T} is an \mathbf{R}^I-valued martingale measure then X_t D_t = M_t - \sum_{s\le t} C_s D_s is an arbitrage-free model.