Sunday, May 8, 2011

Black-Scholes is perfectly fine for volatility inversion

Inverting a given option premium to an implied vol by BS is uncontroversial and unaffected by volatility skew/smile. If risk-free rate, spot price, time to expiry (aka TTL) are all /unanimously/ observed, then according to BS equation there's one-to-one mapping between option premium and implied vol. It's like converting kilograms to pounds. Therefore exchanges often quote premiums in vol, forcing everyone to use BS to back out the dollar values.

That doesn't mean we agree to all the BS assumptions, including the constant-vol assumption.

BS is perfectly fine for inversion + ... perhaps ... greek calculations....

We can safely use the original BS equation for inversion, and then completely discard the BS model after that. We can use a different model (usually related to BS) to describe or model the dataset of implied vol numbers. Some common models include

local-vol model
SABR model
(I guess both are stochastic vol models??)

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