## DETERMINISTIC VOLATILITY MODELS

November 28, 2015 by zulfahmed

There is a history for deterministic volatility models to explain the smile with work by Bruno Dupire, Mark Rubinstein, Derman and Kani from 1994. My approch is to seek the form of volatility functions of the form latex \sigma(T,K)^2 = \sqrt{2\frac{\frac{\partial C}{\partial T} + rK\frac{\partial C}{\partial K}}{K^2 \frac{\partial^2 C}{\partial K^2}}$

For local volatility the Black-Scholes PDE is

with boundary conditions and which we can solve numerically with python.

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