Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.
 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.

## Local volatility.

uppose a stock follows the SDE in the risk neutral world. The is a deterministic function of time. The is the moment of observation. We aim to express the volatility as a function of and its derivatives with respect to strike.

We use the representation in terms of distribution density We calculate the -derivative and substitute the equation ( Forward_Kolmogorov ):

We evaluate each integral via integration by parts and with help of results ( Distribution density via Call ).

Summary

(Differentiating call with respect to maturity 1)Assume then

We now switch to a process of the form where the function is still given by Thus we cannot simply do a -change of variables to reduce to the previous case. We repeat our calculations. The equation ( Forward_Kolmogorov ) now has the form We integrate by parts and use the formulas ( Distribution density via Call ):

Summary

(Differentiating call with respect to maturity 2) Assume then

 Notation. Index. Contents.