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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
Author
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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.
a. Example: European put-call parity.
b. Example: Log contract.
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.

Static hedging of European claim.


e are seeking static replication of the function MATH with a linear combination of the functions MATH defined in the previous section. Using the results ( Recovery of implied distribution ) we write MATH for some number $K_{0}$ . We perform integration by parts to move the derivatives to the function $h$ : MATH To simplify the boundary terms we derive the put-call parity for function $C$ and $P$ : MATH where the MATH is the forward price of $X_{t}$ . We also note that we separated the values of $k$ and use the out-of-money contracts so that the extreme boundary terms would vanish. We simplify the boundary terms as follows: MATH MATH

MATH MATH We conclude

MATH (static replication formula)
This result is model-independent.




a. Example: European put-call parity.
b. Example: Log contract.

Notation. Index. Contents.


















Copyright 2007