On martingale diffusions describing the `smile-effect' for implied volatilities (Q2756663)

Diffusion models describing the `smile-effect' of implied volatilities for option prices are considered. These models describe the stock price by martingale diffusions of the form \(dX(t) = b(X(t)) dW_t\), where \(W_t\) is a Brownian motion, \(b\) is a certain explicitly deduced smooth function of one variable that does not explicitly depend on time parameter \(t\). The value at time \(t\) of a unit volume European call option with maturity \(T\) is then calculated by the conditional expectation \(C(x, t, K, T) = E ((X(T)-K)^+\mid X(t) =x).\) Under some assumptions on the time-homogeneous diffusion process \(X(t)\) the author proves that despite other restrictions the following necessary simple symmetry condition holds for the resulting function \(C\): \(C(x, t, K, T) = x -K + C(K, t, x, T).\) The author shows that for certain examples of empirically observed option prices which are reported in the literature this functional equation does not hold.