Options are derivatives. They derive their value from how the underlying actually moves as well as the market’s perception of how much they will move. So there’s a realized and implied component to the value of an option. When people start using options they are usually attracted to them as an inherently levered way to hedge or speculate on a stock. In other words, they are interested in options as a bet* on direction*.

In the course of trading options, directional players tend to acquire a solid understanding of delta and gamma.

- Delta is the option’s sensitivity to the stock price

The more OTM the option is the less sensitive it is to the stock price. Simply the option’s “moneyness” drives it’s delta

- Gamma is the option delta’s sensitivity to the stock price

Formally, gamma is the second derivative the option’s value with respect to the stock price. Intuitively, it is a measure of how the option’s delta changes as the stock’s moneyness changes. If a call is 20% OTM it may have a low delta. Say 5%. But if the stock suddenly surged 20%, the call would be ATM or 50% delta. It is now highly sensitive to the stock price. That increased sensitivity was the visible effect of gamma.

The movement of stocks is driving the value of options by pushing a giant universe of options in and out of the money at all times. The directional players are getting their fix whether its hedging or punting. This creates a robust ecosystem of buyers and sellers who are able to make very specific bets targeting the size and timing stock moves.

**Enter Vol Traders**

As the directional traders sling option prices around based on their outlooks for stocks, a much smaller segment of traders will notice a relative lack of attention on the other significant driver of an option’s price — it’s implied volatility. The perception of how much a stock will move in the future sets the price of the option today. In fact, it’s the largest unknown in an option price. With continuous bid/ask prices for stocks, a highly liquid interest rate market, and well-estimated dividends the other inputs into listed option prices are trivial.

So this smaller group of professional traders is actually buying and selling levels of a derived value — implied volatility. This is not strange. A stock trader is dealing in implied levels too — forward earnings. They are converting those to multiples to make comparisons of risk/reward across stocks. Option dealers do the same. They compare levels of volatility to find bargains or to sell overpriced perceptions of future volatility.

So in addition to delta and gamma, option pros are focused on vega — the change in an option price due to changes in implied volatility.

**A Basic Example of Vega**

Vol traders sometimes think in terms of straddles. To be long a straddle is to own the call and put on the same strike. If you own an ATM straddle you are agnostic on direction, you are just rooting for a large move. Straddles, since they are just the sum of a call and put, can be used to bet on implied volatility changing as well.

If a straddle is worth $10.00 and has a vega of $.30 then we can say for every 1 point move in the vol the straddle will change by $.30. So if vol increases 1 point, the straddle will appreciate to $10.30

This should remind you of delta. If the straddle had a delta of 30% then for a $1 increase in the stock the straddle would increase by $.30, also to $10.30. Delta describes how an option responds to the stock while the vega describes how the option responds to the vol.

**Vega As An Exposure**

If you are a stock trader you will measure your risk in how many dollars you are long or short. This allows you to answer what your p/l is for say a 1% move. Long $1,000,000 worth of stock and it goes up 5% you make $50,000.

We can do the same with vega. If I own 100 of those $10 straddles then I can say I’m long 3,000 vega:

100 x $.30 x 100 = 3,000

# of straddles x vega per straddle x option multiplier = 3,000

So if volatility increases by 1 point, the straddles increase to $10.30. Since I owned 100 straddles the value of my position goes from $100,000 ^{1} to $103,000 for a net p/l of $3,000.

If vol started at 20% and actually increased 10 points to 30%, the p/l would have been $30,000:

3000 vega x 10 vol points

**Vega Influences Everything**

When a directional stock trader sizes their position, the volatility of the stock is a key input. So it follows that one of an option trader’s primary concerns is *how volatile the implied volatility is*. As we saw above, a shift in perceptions of future volatility can lead to significant gains or losses due to vega exposure.

There’s more.

While vega measures a direct exposure, namely the p/l of an option position due strictly to changes in implied vol, it also acts as an indicator that a position has additional sensitivities to implied volatility. Recall that delta and gamma are driven by moneyness (aka how far in or out of the money) a position is.

*But moneyness depends on volatility!*

If a stock is trading for $50 we might compute that the 1 year $60 strike call which is 20% OTM is 1 standard deviation away for a given vol. Perhaps it has a 15% delta. What if it’s volatility quadruples?

That strike which is $10 away in fixed dollar space is now much closer is standard deviation space. In fact, it’s delta could now be 50%! ^{2} So the size of the position’s vega indicates the potential for change in the entire portfolio’s Greeks.

An option portfolio’s dynamic properties can lead to very complex exposures quickly as sensitivity to volatility propagates thru every line item in a portfolio. Professionals will use many more types of Greek’s to measure individual sensitivities. How does your delta change as time passes? How does your delta change as implied vol declines?

You can get as nerdy and esoteric as you want in trying to flatten your exposure to every greek. In practice, nobody does this (well maybe the French), but because portfolios have complex dependencies on implied volatility it’s handy to remember that large gross and net vegas point to the possibility that a portfolio’s risk is quite dynamic under the hood.

**Next Step: Vol convexity**

The importance of vega warrants further discussion. In the next part, I will cover the “gamma of vol” — how vega itself responds to changes in volatility. Just like a directional trader might use option gamma to acquire convexity with respect to the stock price, the option trader is looking to acquire convexity with respect to volatility.

Pro’s call it *volga*, but in the spirit of Moontower we will build an intuition without calculus. Stay tuned…