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.
 A. Markovian projection on displaced diffusion.
 a. Example of Markovian projection of a separable process on a displaced diffusion.
 B. Markovian projection on Heston model.
 9 Hamilton-Jacobi Equations.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Markovian projection on displaced diffusion.

e are operating with the motivation of the section ( Markovian projection ). The reference for this section is [Antonov2006] . The goal is to approximate the process of the form using the process of the form Here the is some generic process (existence/uniqueness restrictions apply), and are deterministic functions. This corresponds to restricting our attention to minimization among the class of linear functions. We seek where the refers to the scalar product in two dimensions. Hence, we will be evaluating variations with respect to and and equating them to zero. We seek and such that where the and are any smooth deterministic functions. We proceed with calculation of the derivatives: where we introduced the notation . Hence, the functions and are solved from the two equations and in terms of the expectations , , . The is significantly easier to calculate because may be represented as a solution of a system of ODEs. We will see examples of such technique immediately below and in few following sections. Moreover, we are already operating in the approximation mode, hence we may expand in series of the volatility scale (assuming that the general magnitude of volatility is much less then 1). We establish that is of magnitude :

 (Variance of target process)
and consider leading terms of the equations and : The leading terms of the second equation are hence
 (MarkPr1 Sigma)
The leading terms of the first equation are hence
 (MarkPr1 Beta)

Similarly, to ( Variance of target process ) we apply operation under the expectation sign and produce an ODE problem for all the interesting expectations.

 a. Example of Markovian projection of a separable process on a displaced diffusion.
 Notation. Index. Contents.