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.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Mean reverting equation.

e study properties of a process given by the SDE where and are some regular deterministic functions and is a standard Brownian motion. We introduce a process : then We substitute (*) and obtain or We integrate the last relationship for and obtain After multiplication by we conclude

To explain relevance of the mean reverting equation let us consider an equation frequently used as a first-approximation simplistic model for a forward curve The is the observation time and the is the expiration time. The front end of the curve is most volatile. The volatility decreases as increases. This is a simple but realistic model of propagation of new information through forward curve.

We integrate the above equation as follows: We introduce the "spot" price and compute the SDE for the process We have where the arrow marks application of the operation. Hence, We conclude that the SDE for the is mean reverting with the parameters given by

Note that is the mean-reversion parameter and the slope of the forward volatility curve.

 Notation. Index. Contents.