I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 1 Black-Scholes formula.
 2 Change of variables for Kolmogorov equation.
 A. One dimensional Black equation.
 B. Two dimensional Black 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.

## Two dimensional Black equation.

e compute the quantity where the are volatilities of the assets (positive real numbers), is correlation (real number, ), is an integrable function and , are independent standard Brownian motions.

According to the multidimensional version of the backward Kolmogorov's equation ( Backward equation section ) the function is a solution of the problem Similarly to the previous section ( one-dim case ) we introduce the process and observe In addition and Therefore, we define the process Conversely, Hence, for the function may be represented as with being the solution of

 Notation. Index. Contents.