Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.