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 with a linear combination of the functions defined in the previous section. Using the results ( Recovery of implied distribution ) we write for some number . We perform integration by parts to move the derivatives to the function : To simplify the boundary terms we derive the put-call parity for function and : where the is the forward price of . We also note that we separated the values of and use the out-of-money contracts so that the extreme boundary terms would vanish. We simplify the boundary terms as follows:    We conclude (static replication formula)
This result is model-independent.

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