1.4.3 Vega

Vega is the first derivative of the option price with respect to IV. It is important to conceptualize that IV represents the expected future move of the underlying stock, and so is the expected settlement value of the option. Then Vega is a measure of how much the price of an option will change when the expected settlement value of the option changes. Or put another way, it is the change in the option price with respect to changes in expected future range of the underlying stock price. Options with high vega (typically far out of the money options) are primarily sensitive to changes in IV.

It is possible to create a portfolio of options on a single entity that is only sensitive to changes in IV and not to spot price by purchasing a decreasing amount of calls and increasing amount of puts as you move away from spot. This is called a synthetic variance swap and represents the expected future value of a variance swap on a single entity. The VIX index is the value of a synthetic variance swap on the SPX roughly 30 days to expiration.

Summary of Terms

$V$ = Contract Value

$\nu$ = Contract Vega

$S$ = Spot Price

$K$ = Strike Price

$\sigma$ = Implied Volatility

$\tau$ = Years to Expiration

$r$ = Risk Free Rate

$q$ = dividend yield

Calculation

\nu = \frac{\partial V}{\partial \sigma}

\nu = {Se^{-q\tau} \over \sqrt{2\pi}}e^{-{1 \over 2}d_+^2}\sqrt{\tau}

d_+ = {1 \over \sigma \sqrt{\tau}}\bigg[\ln\bigg({S \over K}\bigg) + \bigg(r - q + {\sigma^2 \over 2}\bigg)\tau\bigg]

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