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.

Gyongy's lemma.

uppose the process is given by the SDE

 (martingaleX)
where the is some adapted stochastic process and is the standard Brownian motion. We are looking for a function such that the process given by the SDE would have the same distributions for every

We assume that the -SDE has a solution. Hence, there are for all We omit the condition from notation and calculate where the is the step function and is the Dirac's delta function. We used the ( Ito formula ) and the martingale property of ( martingale X ). Similarly, We set and apply the summary ( Differentiating call with respect to maturity 2 ) with , then with the same final conditions for and . Then Consequently

Summary

Let be given by the SDE that is solvable for some adapted process then with the given by has a solution and for all and .

 a. Multidimensional Gyongy's lemma.
 Notation. Index. Contents.