I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 A. Definition of normal variable.
 B. Linear transformation of random variables.
 C. Multivariate normal distribution. Choleski decomposition.
 D. Calculus of normal variables.
 E. Central limit theorem (CLT).
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

Central limit theorem (CLT).

he closeness of normal variables with respect to the linear transformation may be traced to the Central Limit theorem that we introduce in the present section.

Proposition

(Elementary CLT) If is a collection of iid (independent identically distributed) random variables with a common finite mean and a common finite variance then the distribution of converges pointwise to the distribution of as the sample size approaches infinity:

 (CLT)
for some standard normal random variable .

Better versions of CLT are presented in the section ( Central limit theorem (CLT) II ).

To see why CLT is true, consider the following calculation.

Suppose the iid variables have the common density . We calculate the Fourier transform (=characteristic function) We use independence of . We expand the in powers of . We use , ,... We apply the transformation . We utilize , = , , . For a normal distribution we calculate We use . The Fourier transform is an isometry in . By comparing (*) with (**) we conclude It still remains to derive the pointwise convergence. We postpone such issues until the section ( Vague convergence ). Advanced versions of CLT are presented in the section ( Central limit theorem (CLT) II ).

 Notation. Index. Contents.