Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 1 Black-Scholes formula.
 2 Change of variables for Kolmogorov equation.
 3 Mean reverting equation.
 4 Affine SDE.
 5 Heston equations.
 6 Displaced Heston equations.
 7 Stochastic volatility.
 8 Markovian projection.
 9 Hamilton-Jacobi Equations.
 A. Characteristics.
 B. Hamilton equations.
 C. Lagrangian.
 D. Connection between Hamiltonian and Lagrangian.
 E. Lagrangian for heat equation.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Characteristics. e would like to extend a solution of the equation ( First order PDE ) from a surface of boundary condition along some family of curves We seek an effective way to define such a family. We hope to reduce the non linear PDE ( First order PDE ) to a system of ODEs.

Notation

We introduce the notations  We differentiate the definition (*) along the parameter : We also differentiate the ( First order PDE ) along for every : We would like to get rid of the derivative , hence, we differentiate the definition (**) of with respect to : Therefore, if we would require we would easily remove the from (***): We collect our results as follows.

Definition

The functions that satisfy the system of ODE are called characteristics of the PDE ( First order PDE ).

If a function solves the PDE and solves (c) with and then p and z solve (a) and (b). One may always attempt to go in reverse: extend a solution of PDE from some (consistent) boundary condition using a family of solutions of (a),(b),(c).

 Notation. Index. Contents.