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## DETERMINISTIC VOLATILITY MODELS

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 $\int_0^t S_{\alpha}(-\kappa s^\alpha) ds since this is the reduction of a fractional Heston model removing the stochasticity. A simple method to check models of deterministic volatility is to compare the candidate deterministic volatility function with the empirical volatility function using Dupire's formula$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 $\sigma(t,x)$ the Black-Scholes PDE is

$v_t(t,x) + rxv_x(t,x) +\frac{1}{2}\sigma^2(t,x) x^2 v_{xx}(t,x) - rv=0$

with boundary conditions $v(t,L) = 0$ and $v(T,x) = (x-K)^+$ which we can solve numerically with python.