Research

The Black-Scholes/Merton model assumes continuous time trading. This is an unrealistic artifact of the mathematical theory of Ito processes. Traders can only make a finite number of transactions.

See Simple Unified Model for details.

Sell side traders need to know when and how much to hedge in order to minimize risk.

We assume market instruments I are available to trade at price X_t\colon\mathcal{A}_t\to\boldsymbol{{R}}^I and have no cash flows.

A (cash settled) derivative specifies payments \hat{A}_j at stopping times \hat{\tau}_j. We wish to find trades \Gamma_t\colon\mathcal{A}_{t}\to\boldsymbol{{R}}^I resulting in amounts A_t = \hat{A}_j when t = \hat{\tau}_j and A_t = 0 otherwise in order to satisfy the contract. If a money-market account is available then we can satisfy A_t = 0 when t\not=\tau_j, but it is not obvious a self-financing strategy is optimal.

Following Markowitz we define an efficent trading strategy as one that makes A_t - \hat{A}_{\tau_j} white noise with minimum variance.