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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.
5. Heston equations.
A. Affine equation approach to integration of Heston equations.
B. PDE approach to integration of 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.

PDE approach to integration of Heston equations.

e are still investigating the equations ( Affine equations ), ( Heston equations ) and aiming to recover the expression for MATH were the process MATH is given by the equations MATH and the MATH are constants, MATH are increments of independent standard Brownian motions. According to the general theory of the Backward Kolmogorov's equation, see the section ( Backward equation section ), we have the following PDE and initial condition: MATH MATH We look for a solution of the form MATH We substitute such representation into the PDE: MATH MATH To transform the boundary condition we use the inverse Fourier transform: MATH The expression of the form MATH is Dirac's delta function. Indeed, for any smooth quickly decaying $f\left( x\right) $ and Fourier transform MATH MATH Hence, MATH

We continue with investigation of the equation MATH We seek a solution of the form MATH We have MATH hence MATH The last equation should be satisfied for every $v$ . Hence, we separate powers of $v$ : MATH The above equations are subject to the final conditions MATH The expressions for $\alpha$ , $\beta$ may be obtained with the technique described in the section on the Ricatti equation ( Ricatti equation ).

We perform the transform back to the $\psi$ : MATH

Notation. Index. Contents.

Copyright 2007