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.
IV. Data Analysis.
V. Implementation tools.
1. Finite differences.
2. Gauss-Hermite Integration.
3. Asymptotic expansions.
4. Monte-Carlo.
A. Generation of random samples.
B. Acceleration of convergence.
C. Longstaff-Schwartz technique.
D. Calculation of sensitivities.
a. Pathwise differentiation.
b. Calculation of sensitivities for Monte-Carlo with optimal control.
5. Convex Analysis.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Pathwise differentiation.


e consider the original SDE MATH and a perturbed SDE MATH We introduce the difference MATH The difference $\delta X_{t}$ is given by the following SDE MATH We assume that the functions $a$ and $b$ are such that the solution of the original SDE $X_{t}$ changes slightly when the perturbation of the initial condition is slight. Hence, the $\delta X_{t}$ is small if $\delta x$ is small. Consequently, MATH We substitute these approximations into the SDE for $\delta X_{t}$ MATH and note that such SDE scales with the magnitude of $\delta x$ . We introduce the process $Y_{t}$ : MATH We arrive to the following system of SDEs: MATH The derivative of interest may be expressed in terms of $X_{t}$ and $Y_{t}$ : MATH





Notation. Index. Contents.


















Copyright 2007