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.
 8 Markovian projection.
 A. Markovian projection on displaced diffusion.
 B. Markovian projection on Heston model.
 9 Hamilton-Jacobi Equations.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Markovian projection on Heston model.

ollowing sections ( Markovian projection ) and ( Markovian projection on displaced diffusion ) we are developing a generic recipe for approximation of the process given by the SDE

 (MarkPr TargetEquation)
with some adapted diffusion process . We will be using Heston-type processes as the class of approximating processes.

The reference for this section is [Antonov2007] .

In line with the section ( Heston_equation_section ) we seek approximation in the class of the stochastic processes given by the equations

 (Heston approximation)
for variety of deterministic functions . The are standard Brownian motions. The refers to the difference . The analytical tractability of the equations ( Heston approximation ) was considered in the sections ( Heston equation ) and ( Displaced diffusion ).

Note that we may write the equation ( MarkPr TargetEquation ) in the equivalent form

 (MarkPr TargetEquations 2)
for some diffusion processes , , and To see that the equations ( MarkPr TargetEquations 2 ) are equivalent to ( MarkPr TargetEquation ) observe that we may introduce the process and claim that it a diffusion because is assumed to be a diffusion. Consequently, we introduce the diffusion process The process have correlated and uncorrelated components of the diffusion term, hence, we have the sum for some processes , . Finally, is positive, hence the multiplication by does not restrict the generality. The motivation behind such transformation is the aim to present the target process in the form similar to the form of the approximating process .

We will be applying the Gyongy's result ( Multidimensional Gyongy lemma ), hence, we put the target and approximating processes in the matrix form. We seek to approximate with given by the SDEs Note, that according to the lemma ( Multidimensional Gyongy lemma ) a general 2-dimensional process may be replicated by the local volatility process with the functions , given by the relationships We note that in our setting and . Following motivations of the previous section ( Markovian projection section ) the functions , minimize the following functionals: We substitute the expressions for : and simplify to

The rest of the calculation is a straightforward extension of the previous section ( Markovian projection on displaced diffusion ). We calculate the variations of the functional We minimize the functional : We calculate variations of the functional : We summarize the results of minimization: The functions are defined by (a) and (b). The absolute values is defined by (c). The and are defined by (d) and (e). The expectations are calculated by techniques outlined in the previous two sections.

 Notation. Index. Contents.