Content of present website is being moved to . Registration of will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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.

e saw in the section ( Gyongy lemma ) the possibility to replicate one dimensional distributions of a complex process $X_{t}$ of the generic form MATH with the local volatility process $Y_{t}$ of much simpler form MATH provided that the function MATH is given by the relationship

MATH (Gyongy 2)
The section ( Time series forecasting ) contains a generic proof that the conditional expectation minimizes unconditional expectation of the square difference. Speaking in terms of this section, the function MATH should be guessed to be a solution of the following minimization problem:
MATH (Markovian projection)
where the expectation is unconditional and is taken at the observation time. We note that the expectation is taken with respect to the original process $X_{t}$ and not with respect to $Y_{t}$ : we do not evaluate MATH . This point is not obvious from considerations of the section ( Time series forecasting ). Hence, we proceed with derivation of the ( Markovian projection ).

We define the functional MATH for some stochastic process $U_{t}$ and consider the problem MATH The solution $b$ has to satisfy MATH for any smooth deterministic function MATH . We calculate the derivative: MATH Next, we use the operation MATH consistent with the foundation formulas ( Chain rule ) and ( Total probability rule ). We slice the event space by the values of $U_{t}$ : MATH Hence, it suffices to have MATH Comparing the last relationship with the ( Gyongy 2 ) we see that indeed, the recipes ( Gyongy 2 ) and ( Markovian projection ) are equivalent.

It is important to note that the class of functions $\left\{ b\right\} $ in the minimization problem ( Markovian projection ) has to be restricted only by existence requirements for the process $Y_{t}$ . It is under such freedom that the conditional expectation and the minimization problem are equivalent. For general $b$ the process $Y_{t}$ is not always analytically tractable. We will be restricting our attention to some subclasses of functions $b$ that provide analytical tractability. Thus, we will be sacrificing equivalence to gain analytical tractability.

We present several examples of this technique in the following sections.

A. Markovian projection on displaced diffusion.
B. Markovian projection on Heston model.

Notation. Index. Contents.

Copyright 2007