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 were the process is given by the equations and the are constants, 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: We look for a solution of the form We substitute such representation into the PDE: To transform the boundary condition we use the inverse Fourier transform: The expression of the form is Dirac's delta function. Indeed, for any smooth quickly decaying and Fourier transform Hence,

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

We perform the transform back to the :

 Notation. Index. Contents.