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Vega, Volga and Vanna

What is Vega?

Vega is the change in the value of the option with respect to change in volatility.

Within the Greeks Vega’s importance rises given how misunderstood the behaviour of volatility is and the impact changes in volatility have on option prices. In earlier chapters we have seen:

a) Implied volatility is not constant.

b) Deep out of money options react very differently to changes in implied volatility and

c) Volatility ends up behaving as a function of time to expiry and money-ness. We used implied volatility surfaces to plot the behaviour of volatility across these two dimensions.

In this chapter we will take a deeper look at Vega and its two associated derivatives as well as examine Vega’s relationship with Gamma. As part of this exploration process we will introduce the concept of Shadow Gamma and Vanna – both instances of what we could call cross Greeks. Since by now we have spent sufficient time with the concept of surface plots, we will also add a new dimension, the underlying asset price, to our surface plots.

Calculating Vega

The equation for calculating Vega is given by:

Since assume no dividends, the formula simplifies to:

We can use either of the two equations to calculate Vega. Similar to Gamma, the value of Vega is the same for both call and put options.

Vanna – Volatility’s cross Greek

Vanna, a second order cross Greek, can be defined as:

Calculating Vanna

In the Black Scholes model, Vanna is calculated using the following equation:

Volga – Volatility Gamma

Volga or Volatility Gamma determines the rate of change in Vega on account of a unit change in volatility. The same relationship convexity has with duration and gamma has with delta.

It is also possible to express both Vanna and Volga in terms of Vega.

We know that Vega is given by:

The formula for Vega, Vanna & Volga above indicate a direct linkage with time. Unlike Gamma where Gamma peaks with a reduction in time for at the money option, for Vega, Volga and Vanna, it is increasing time that give volatility an opportunity to impact option value. The Vega Greeks will decline as time to expiry comes closer to zero. This creates different choices that need to be balanced when we try to hedge Gamma and Vega together.

Plotting Vega and Gamma

The plot below calculates value of Vega and Gamma for an option against changing level of strike prices. In this specific case the current spot price lies between 270 and 280 which is where (at and near money) where Vega peaks. Despite the fact that fact that we have a different scale for measuring Vega and Gamma, the interesting thing in the above graph is the similarity in shape for the two Greeks.

Figure 1 Vega and Gamma against spot

Vega and Gamma against time

It is when we plot Vega against changing expiry for deep out of money options and at money options that we see a difference emerging in the relationship between Vega and Gamma. For deep out of money option reducing time to maturity reduces both Vega and Gamma.

Figure 2 Vega and Gamma against spot – deep out of money options

For at the money option, the impact of time on Vega and Gamma is the exact opposite. Vega rises as we increase time to expiry. Gamma rises as we decrease time to expiry.

Figure 3 Vega and Gamma against time – at the money options