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.

Example of Markovian projection of a separable process on a displaced diffusion.

e aim to approximate the process given by the SDEs with the displaced diffusion process Here the , , , are numbers. are single dimensional stochastic processes.

We apply the recipes ( MarkPr1 Sigma ) and ( MarkPr1 Beta ). Such recipes require expressions for several expectations that we calculate below. The first term is zero because and We continue At this point we have expressions for the quantities of interest in terms of and . We apply the operation to the SDE for and obtain The quantities of interest are calculated above to be We substitute the expressions for and calculate the integrals: We substitute these expressions into ( MarkPr1 Sigma ) and ( MarkPr1 Beta ):

 Notation. Index. Contents.