Quantitative Analysis
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Numerical Analysis
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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.
9. Hamilton-Jacobi Equations.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Mean reverting equation.


e study properties of a process $X_{t}$ given by the SDE MATH where MATH and $a\left( t\right) $ are some regular deterministic functions and $dW_{t}$ is a standard Brownian motion. We introduce a process $Y_{t}$ : MATH then MATH We substitute (*) and obtain MATH or MATH We integrate the last relationship for $t\in\lbrack0,T]$ and obtain MATH After multiplication by MATH we conclude MATH

To explain relevance of the mean reverting equation let us consider an equation MATH frequently used as a first-approximation simplistic model for a forward curve MATH The $t$ is the observation time and the $T$ is the expiration time. The front end of the curve is most volatile. The volatility decreases as $T-t$ increases. This is a simple but realistic model of propagation of new information through forward curve.

We integrate the above equation as follows: MATH MATH MATH We introduce the "spot" price MATH and compute the SDE for the process $x_{t}=\log p_{t}.$ We have MATH where the arrow marks application of the $d_{t}-$ operation. MATH MATH MATH MATH Hence, MATH MATH We conclude that the SDE for the $x_{t}$ is mean reverting with the parameters MATH given by MATH MATH

Note that $\lambda$ is the mean-reversion parameter and the slope of the forward volatility curve.





Notation. Index. Contents.


















Copyright 2007