Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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.
A. Analytical tractability of displaced Heston equations.
B. Displaced Heston equations with term structure.
a. Parameter averaging.
b. Parameter averaging applied to displaced diffusion.
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.

Parameter averaging.


e approximate the solution $X_{t}$ of the SDEs MATH with the solution $Y_{t}$ of the SDEs MATH The function MATH is restricted to allow for existence of the solution $X_{t}$ . We scale the deterministic function MATH so that MATH for some $x_{0}$ . The $g\left( x\right) $ is to be chosen to have the property MATH and to be optimal in the sense of the following proposition.

Proposition

Define MATH for MATH and consider solutions $X_{\varepsilon,t}$ and $Y_{\varepsilon,t}$ of the following SDEs MATH The functional MATH has the properties MATH The particular function $g\left( x\right) $ that minimizes MATH has the property

MATH (Parameter averaging)

Proof

We calculate the MATH directly MATH MATH Hence, MATH Observe that MATH MATH Therefore, MATH Note that $X_{0,t}=Y_{0,t}$ (by the form of SDEs for $X$ and $Y$ ), hence MATH The above expression for MATH should be minimized with respect to $g$ . Hence, we differentiate with respect to the MATH and equate to 0: MATH Finally, MATH and the claim follows.





Notation. Index. Contents.


















Copyright 2007