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.
 A. Recovering implied distribution.
 B. Local volatility.
 C. Gyongy's lemma.
 a. Multidimensional Gyongy's lemma.
 D. Static hedging of European claim.
 E. Variance swap pricing.
 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.

## Multidimensional Gyongy's lemma. laim

Suppose the process taking values in is given by the SDE where the is a matrix valued adapted stochastic process, is a column-valued adapted process and the is the column of the standard Brownian motions. The process taking values in and given by the SDE has the same component-wise distributions as if the deterministic functions and are given by the relationships The is a -th row of the matrix and the multiplication above is the scalar product.

 Notation. Index. Contents.