Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
A. Ricatti equation.
B. Evaluation of option price.
C. Laplace transform.
D. Example: CDFX model.
a. Definition of CDFX model.
b. The martingale normalization (CDFX).
c. Fourier transform (CDFX).
d. Calculation of Fourier transform (CDFX).
e. Calculation of Premium Leg of CDS.
f. Calculation of the protection leg of the CDS.
5. Heston equations.
6. Displaced Heston equations.
7. Stochastic volatility.
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.

Calculation of Fourier transform (CDFX).

e summarize the situation as follows. We have a 3-dimentional process $\Theta_{t}$ given by the equation MATH where we use the notation MATH

We calculate the increment MATH For $dZ_{t}$ to be a martingale, we must have MATH We introduce the notations MATH as follows: MATH MATH MATH Consequently, we have MATH We separate terms with every power and coordinate of $\Theta$ : MATH

We now recover the expressions for $g,h,p,q$ before we substitute it into the above ODEs for $\beta$ and $\alpha$ . By comparing definitions MATH we see MATH Similarly for $b$ we have definitions MATH hence MATH Also, MATH

We now recover the ODEs for $\beta$ : MATH

We calculate the expression MATH under assumption that $\rho_{xv}=\rho$ is the only non-zero correlation: MATH

Hence, MATH Equivalently, MATH

The equation for $\alpha$ resolves to MATH

We summarize the results up to this point as follows MATH

We would like to compare these results with the results of the paper [CarrWu2006a] . Hence, we perform additional transformations.

We calculated earlier MATH Hence, MATH where the MATH is the notation of the paper. We change variable from $t$ to $\tau=T-t$ , hence, MATH The last expressions compare exactly with the paper [CarrWu2006a] .

Notation. Index. Contents.

Copyright 2007