_sum.md

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Simple Unified Model

Market

$X_t$ - prices at time $t$ indexed by instruments
$C_t$ - discrete cash flows at time $t$ indexed by instruments
E.g., stock dividends, bond coupons, futures margin adjustments.

Trading

$(\tau_j, \Gamma_j)$ - times and trades indexed by instruments with $\tau_0 < \cdots < \tau_n$
$\Delta_t$ - trades accumulate to a position $\Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s$
It takes some time for a trade to settle into a position.

Accounting

$V_t = (\Delta_t + \Gamma_t)\cdot X_t$ - value (mark-to-market) at time $t$
$A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t$ - amount deposited in trading account at time $t$

Arbitrage

Trades $(\tau_j, \Gamma_j)$ with $A_{\tau_0} > 0$, $A_t \ge 0$ for $t > \tau_0$, and $\sum_j \Gamma_j = 0$.

Fundamental Theorem of Asset Pricing

No arbitrage iff there exist positive, adapted measures $D_t$ with $$ \tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u} C_u D_s)|_{\mathcal{A}_t}, $$ where $\mathcal{A}_t$ is information at time $t$.

With $V_t$ and $A_t$ as above, $$ \tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u} A_s D_s)|_{\mathcal{A}_t}. $$ Values $V_t$ corresponds to prices $X_t$, amounts $A_t$ correspond to cash flows $C_t$.
Trading strategies create synthetic market instruments.

Every arbitrage-free model is parametrized by adapted vector-valued measures $M_t$ where $M_t = M_u|_{\mathcal{A}t}$, $t \le u$ and $$ \tag{3} X_t D_t = X_0 M_t - \sum{s \le t} C_s D_s. $$