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.

## No drift calculation.

e are calculating the expectation where the are given numbers, is a deterministic function and is the standard Brownian motion.

According to the Ito formula ( Ito_formula ) We integrate the above equality over the time interval and obtain We introduce the notations according to the relationship and write for some standard normal variable .

We proceed with evaluation of the quantity : where is the number defined by the relationship or We introduce the notation and continue

Summary

The expectation evaluates to

 (No drift Black Scholes)

 Notation. Index. Contents.