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 where the are numbers, are deterministic functions and is a standard Brownian motion. We use the notation and results of ( No drift Black Scholes ). Note that hence, We introduce the notation and assume at a particular time moment . Then and We introduce the notation It is conventional to introduce the quantities

Summary

The expectation evaluates to

The expressions in the above summary are not very convenient for calculations. Hence, we perform another transformation. With the notation the expressions for take the form

 (Black Scholes formula)

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

Summary

Let be the one dimensional process given by the SDE where the functions are deterministic and is the standard Brownian motion. The expectation is given by the expressions where we use the notation If we do not express as a function of and formally calculate the partial derivatives of then we obtain

 (Black Scholes property 1)

Notation

We will use the following notation

 (BlackScholesUndiscountedCall)

 Notation. Index. Contents.