April 25, 2024
T — totally ordered set of trading times.
I — market instruments.
\Omega — all possible outcomes.
\mathcal{A}_t — a partition of \Omega indicating the information available at time t\in T.
X_t\colon\mathcal{A}_t\to\boldsymbol{R}^I — prices at time t of market instruments.
C_t\colon\mathcal{A}_t\to\boldsymbol{R}^I — cash flows, usually 0, at time t of market instruments.
(\tau_j, \Gamma_j) — a finite trading strategy of strictly increasing stopping times \tau_j and corresponding trades \Gamma_j\colon\mathcal{A}_{\tau_j}\to\boldsymbol{R}^I with \sum_j \Gamma_j = 0.
\Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s where \Gamma_s = \Gamma_j 1(\tau_j = s) — the position resulting from trading.
V_t = (\Delta_t + \Gamma_t)\cdot X_t — the value, or mark-to-market, of the strategy.
A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t — the amount showing up in the trading account.
Arbitrage exists if there is a trading strategy with A_{\tau_0} > 0 and A_t \ge0, t > \tau_0.
Fundamental Theorem of Asset Pricing. There is no arbitrage if there exist positive measures D_t on \mathcal{A}_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}, t\le u.
Lemma. With the above notation V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\mathcal{A}_t}, t\le u.
Trading strategies create synthetic instruments where price corresponds to value and cash flow corresponds to account.
Every arbitrage free model has the form X_t D_t = M_t - \sum_{s\le t} C_s D_s where M_t = M_u|_{\mathcal{A}_t} is a martingale measure.
X,C\colon\sum_{j=0}^n B(\mathcal{A}_j, \boldsymbol{R}^I)
A\colon\sum_{j=0}^n B(\mathcal{A}_j, \boldsymbol{R}^I)\to\sum_{j=0}^n B(\mathcal{A}_j) where A(\oplus \Gamma_j) = \oplus \Delta_j\cdot C_j - \Gamma_j\cdot X_j.
\mathcal{G}_0 = \{\oplus \Gamma_j\mid \sum_j \Gamma_j = 0\}.
\mathcal{P} = \{\oplus A_j\mid A_0 > 0, A_j\ge 0\}.
Arbitrage if there exists \Gamma\in\mathcal{G}_0 with A(\Gamma)\in\mathcal{P}.
A^*\colon \sum_{j=0}^n ba(\mathcal{A}_j)\to\sum_{j=0}^n ba(\mathcal{A}_j, \boldsymbol{R}^I)