Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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.
A. Recovering implied distribution.
B. Local volatility.
C. Gyongy's lemma.
D. Static hedging of European claim.
E. Variance swap pricing.
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.

Local volatility.

uppose a stock $S_{t}$ follows the SDE MATH MATH in the risk neutral world. The MATH is a deterministic function of time. The $t_{0}$ is the moment of observation. We aim to express the volatility MATH as a function of MATH and its derivatives with respect to strike.

We use the representation MATH in terms of distribution density MATH We calculate the $T$ -derivative MATH and substitute the equation ( Forward_Kolmogorov ): MATH MATH

We evaluate each integral via integration by parts and with help of results ( Distribution density via Call ). MATH MATH


(Differentiating call with respect to maturity 1)Assume MATH then MATH

We now switch to a process of the form MATH MATH where the function $C$ is still given by MATH Thus we cannot simply do a $\log$ -change of variables to reduce to the previous case. We repeat our calculations. MATH MATH The equation ( Forward_Kolmogorov ) now has the form MATH MATH We integrate by parts and use the formulas ( Distribution density via Call ): MATH MATH


(Differentiating call with respect to maturity 2) Assume MATH then MATH

Notation. Index. Contents.

Copyright 2007