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.

## Affine equation approach to integration of Heston equations.

he Heston equations are of the affine type. Hence, we proceed as discussed in the section ( section about Affine equation ).

We transform the equations to the -form ( Affine equation ). Let , hence In particular, According to the summary ( Affine characteristic function 1 )-( Affine boundary conditions ) the function is given by the expression where the functions and must satisfy the following system of ODEs: We transform the above relationships: The are to satisfy the final conditions: The next task is to solve the equation We introduce the convenience notation and write We perform the change of the unknown function as follows We perform the transformation as follows: In addition, Hence, and, consequently, We arrived to a linear equation. We look for solutions of the form It suffices to have We mean to integrate over because this is the argument of the characteristic function and we want to take the inverse Fourier or Laplace transform. Note that when the expression under the square root is positive. For large it is negative. Hence, we would rather change for real . Then the square root is never zero. Indeed, the real part would be , where the correlation is not greater then 1.

We perform the backward substitutions We introduce the quantity according to the relationship and transform the expression for to emphasize that the requirement uniquely identifies the .

We now consider the equation for : where the is a constant and the has just been calculated: We have hence It remains to evaluate the integral Since satisfies the final condition we have

 Notation. Index. Contents.