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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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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.
A. Analytical tractability of displaced Heston equations.
B. Displaced Heston equations with term structure.
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.

Displaced Heston equations.


his chapter follows [Piterbarg2003a] .

The SDEs ( Heston equations ) MATH were covered in the section Heston equation . We will be considering the quantity $S_{t}$ in its martingale measure $\mu=0$ . We also perform a slight modification of the first equation:

MATH (Displaced Heston equations)
where the $b$ is some number $b\in R$ . Such transformation preserves analytical tractability and allows for modelling of skew in volatility smile for short maturities. The term structure of volatility smile prompts introduction of time dependency of the parameter $b$ . Such dependency is incompatible with the analytical tractability that was established in the section ( Heston equation ).




A. Analytical tractability of displaced Heston equations.
B. Displaced Heston equations with term structure.

Notation. Index. Contents.


















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