Quantitative Analysis.
Trading Platform.
Python for Excel.
Author.

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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.
Bibliography.
Forum Notation Index Contents

Gyongy's lemma.

uppose the process $X_{t}$ is given by the SDE

MATH(martingaleX)
where the $\beta_{t}$ is some adapted stochastic process and $W_{t}$ is the standard Brownian motion. We are looking for a function MATH such that the process $Y_{t}$ given by the SDEMATH would have the same distributionsMATH for every $t.$

We assume that the $X$-SDE has a solution. Hence, there are MATH for all $t,K.$ We omit the condition $|X_{t_{0}}=x)$ from notation and calculateMATH where the $\theta$ is the step function and $\delta$ is the Dirac's delta function. We used the ( Ito formula) and the martingale property of $X$ ( martingale X). Similarly,MATH Therefore, if we set MATH then according to the results of the previous sectionMATH and consequentlyMATH

Summary

Let $X_{t}$ be given by the SDEMATH that is solvable for some adapted process $\beta_{t}$ thenMATH with the $b$ given byMATH has a solution and MATH for all $x$ and $t$.




a.Multidimensional Gyongy's lemma.

Forum Notation Index Contents


















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