Option Vega: Why Calls and Puts Gain Value with Volatility

# What does it mean to be "long or short in volatility"?


# What does it mean to be "long or short in volatility"?












I've heard a question regarding pricing of european calls. The question is:

> Is the call long or short in volatility when it is (deep) OTM? What is the profile of the implied volatility?

I know that in that case the answer is "long". Conversely the call would be short in vol if it was ITM.

I see a relation between long and the wish that volatility is high in order to go ITM if you hold the call. Also if you are ITM, I can see that your interest is to continue ATM. Therefore you want the volatility to be low.

I don't understand what exactly this term means neither where it comes from. I guess it is related to volatility trading/arbitrage.

Could someone please help me out and give me a precise definition for the term "long/short in volatility"?

## Answer by rhaskett (score 11)

https://quant.stackexchange.com/a/16307

I'll expand on Mark's and SRKX's answers which are both correct but brief. To be clear the words long and short have been generalized in finance. They used to mean that you owned a stock or had sold a stock short. Now they are often used to say you make money when a value goes up (long) or make money when some value goes down (short).

In this case whenever you own a call or a put you are "long" volatility. Meaning that as volatility increases the value of your position increases (holding everything else the same). How much added value that you get for a certain increase in volatility (called vega) depends on how in/out of the money the option is at currently among other things, but if you own the call/put it is always positive as more volatility means more possible upside.

When you sell calls or puts, then volatility decreases are good for your position so you are called "short" vol.

## Answer by Mark Joshi (score 9)

https://quant.stackexchange.com/a/16297

the vega of a call is always positive. The holder of a call option is therefore long volatility whatever the spot price.

## Answer by SRKX (score 5)

https://quant.stackexchange.com/a/16298

Mark Joshi's answer is absolutely right, but just to elaborate a bit:

The Vega of an option is the sensitivity of its value with respect to volatility $\nu = \frac{\partial V}{\partial \sigma}$.

For calls, it makes sense that the Vega is always positive, not matter the level of the underlying.

If you take the Black-Sholes model, you can find the theoretical value of Vega here.

## Answer by AKdemy (score 0)

https://quant.stackexchange.com/a/64271

There are a number of ways to look at this:



- Put-call parity: $Vega_{put}=Ke^{-r\tau}\phi(d_2)\sqrt{\tau}$ which is the same for as for the call (not immediately obvious but easy to test if you plug in the values (or just trust existing derivations).



The images below have varying spot on the horizontal axes. The first one is for short time and low vol. You can see how vega is quickly becoming zero for deep ITM and OTM calls. Increasing time and vol has largely the same effect, it increases the area where vega is greater than zero and adds time value on top of the intrinsic value.

Or similarly, in a 3D graph, you can also see how more time, or more vol increases the value of the call (vega is never negative).

- The time value argument can be taken a step further: It can be shown that early exercise of a non dividend paying American call option is never optimal. Check here for dividend paying stocks. Here more vol means that spot could be even higher or also substantially lower. If it falls below strike (however unlikely that may be for deep ITM), it would have been unwise to exercise and pay more than the actual value at expiry. If a sharp decline is expected, an investor will still always be better off to sell the option rather than to exercise it (which would mean the time value is lost and only the intrinsic value is gained), or even keep the option and short the stock. In any case, the argument is the same as with the insurance logic provided before. Once you exercise, you have no insurance anymore.

- Really deep ITM (or OTM) options will be unaffected by vol changes. The first point shows this mathematically, but think of it intuitively. If you have the right to buy an iphone for 3 million pounds, would you use that option or rather buy it online for a fraction? Volatility in the price of the iphone will not change that, as it is reasonable to assume it will be well below 3 million in the foreseeable future. Hence, your option is worthless and does not change no matter the vol (ignoring Weimar republic inflation scenarios).