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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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Author
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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.
6. Displaced Heston equations.
7. Stochastic volatility.
8. Markovian projection.
A. Markovian projection on displaced diffusion.
a. Example of Markovian projection of a separable process on a 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 displaced diffusion.


e are operating with the motivation of the section ( Markovian projection ). The reference for this section is [Antonov2006] . The goal is to approximate the process $X_{t}$ of the form MATH using the process $Y_{t}$ of the form MATH Here the $\Lambda_{t}$ is some generic process (existence/uniqueness restrictions apply), MATH and MATH are deterministic functions. This corresponds to restricting our attention to minimization among the class of linear functions. We seek MATH where the MATH refers to the scalar product in two dimensions. Hence, we will be evaluating variations with respect to $\beta$ and $\sigma$ and equating them to zero. We seek $\sigma$ and $\beta$ such that MATH MATH where the $\delta\beta$ and $\delta\sigma$ are any smooth deterministic functions. We proceed with calculation of the derivatives: MATH MATH where we introduced the notation MATH . Hence, the functions $\beta$ and $\sigma$ are solved from the two equations $\delta_{\beta}F=0$ and $\delta_{\sigma}F=0$ in terms of the expectations MATH , $p=0,2$ , $k=0,1,2,3,4$ . The $E_{kp}$ is significantly easier to calculate because $E_{kp}$ may be represented as a solution of a system of ODEs. We will see examples of such technique immediately below and in few following sections. Moreover, we are already operating in the approximation mode, hence we may expand in series of the volatility scale (assuming that the general magnitude of volatility is much less then 1). We establish that $\Delta X$ is of magnitude $\Lambda$ : MATH

MATH (Variance of target process)
and consider leading terms of the equations $\delta_{\beta}F=0$ and $\delta_{\sigma}F=0$ : MATH The leading terms of the second equation are MATH hence
MATH (MarkPr1 Sigma)
The leading terms of the first equation are MATH hence
MATH (MarkPr1 Beta)

Similarly, to ( Variance of target process ) we apply $\int d_{t}$ operation under the expectation sign and produce an ODE problem for all the interesting expectations.




a. Example of Markovian projection of a separable process on a displaced diffusion.

Notation. Index. Contents.


















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