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.

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]

Vega market impact

The figure below shows the vega of a call or a put with a strike of $100. Note that the value of vega looks similar to a normal distribution centered around the contract strike. Someone who is interested in trading vega may thus conclude that they should buy contracts near spot to maximize the impact of vega. However, contracts near spot are also very sensitive to delta, and this sensitivity to spot can drown out the effect of vega.

vega

What the trader really wants is a contract with high vega RELATIVE to delta. The figure below shows the ratio of vega to delta for a contract with a $100 strike price as a function of changing spot price. When the option is in the money, the vega/delta tends to zero, and increases linearly as the contract goes further out of the money. Thus, the volatility trader would target far out of the money contracts to maximize sensitivity to IV relative to spot price. The limit on how far out of the money the trader can go is determined by the bid/ask spread of the contract prices deep out of the money, as well as their liquidity.

vega

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.