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.
 A. Analytical tractability of displaced Heston equations.
 B. Displaced Heston equations with term structure.
 a. Parameter averaging.
 b. Parameter averaging applied to displaced diffusion.
 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.

## Parameter averaging.

e approximate the solution of the SDEs with the solution of the SDEs The function is restricted to allow for existence of the solution . We scale the deterministic function so that for some . The is to be chosen to have the property and to be optimal in the sense of the following proposition.

Proposition

Define for and consider solutions and of the following SDEs The functional has the properties The particular function that minimizes has the property

 (Parameter averaging)

Proof

We calculate the directly Hence, Observe that Therefore, Note that (by the form of SDEs for and ), hence The above expression for should be minimized with respect to . Hence, we differentiate with respect to the and equate to 0: Finally, and the claim follows.

 Notation. Index. Contents.