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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.
A. No drift calculation.
B. Calculation with drift.
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.

Calculation with drift.


e are evaluating the expectation MATH where the $t,T,y,K$ are numbers, MATH are deterministic functions and $W_{t}$ is a standard Brownian motion. We use the notation and results of ( No drift Black Scholes ). Note that MATH hence, MATH We introduce the notation MATH and assume MATH at a particular time moment $t$ . Then MATH and MATH We introduce the notation MATH It is conventional to introduce the quantities MATH

Summary

The expectation MATH evaluates to MATH

The expressions in the above summary are not very convenient for calculations. Hence, we perform another transformation. With the notation MATH the expressions for $d_{1},d_{2},C$ take the form

MATH (Black Scholes formula)

One notable property of the formula ( Black Scholes formula ) is revealed by the following calculation MATH

In addition, MATH

Summary

Let $Y_{t}$ be the one dimensional process given by the SDE MATH where the functions MATH are deterministic and $W_{t}$ is the standard Brownian motion. The expectation MATH is given by the expressions MATH where we use the notation MATH If we do not express $K$ as a function of $\kappa$ and formally calculate the partial derivatives of MATH then we obtain

MATH (Black Scholes property 1)

Notation

We will use the following notation

MATH (BlackScholesUndiscountedCall)





Notation. Index. Contents.


















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