Numeric Adjustment to Black–Scholes for the smile.

Hi,

This is basically a hunch that I have. Needs further investigation and verification.

Only even moments are risk measures of a returns distribution under risk neutral assumptions.

Volatility (vol), Volatility of Volatility (vvol), Volatility of Volatility of Volatility (v2vol), … all are proxy to the even moments of a returns distribution

Vvol means the tails may become fatter and … v2vol they can become even more fatter.

An option price can be decomposed to basket of options having vol determined using moment matching techniques.

Likewise a option can be de composed to a basket of option vol, vvol, …

The vol skew can be calibrated to be series of vivol.

Let
f1(x; μ1, σ1) = 1 / (σ1 * sqrt(2 * π)) * [1 - exp( - (x - μ1) ^ 2 / (2 * σ1 ^ 2) )]

or
f1(x; μ1, β1) = 1 / (2 * β1) * [1 - exp( - x - μ1 / β1)]

Using this I think you can calibrate Implied Vol (IV)

IV = ∑ ki * fi * vivol

Such that

σ1 or β1 = 0

σ1 or β1 < σ2 or β2 < …

by calibrating for Ki

Any opinions?

Best regards, Suminda Sirinath Salpitikorala Dharmasena