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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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 $X_{t}$ 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 SDE MATH would have the same distributions MATH 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 calculate MATH 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 We set MATH and apply the summary ( Differentiating call with respect to maturity 2 ) with $a=0$ , then MATH with the same final conditions for $C_{X}$ and $C_{Y}$ . Then MATH Consequently MATH


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

a. Multidimensional Gyongy's lemma.

Notation. Index. Contents.

Copyright 2007