Long/Short Vega and Option Positions

Why do you get long vega when you buy an option and short vega when you sell an option?

I would have thought that for both buying and selling options the vega would change according to whether the option was ITM or OTM. This would be because as an option participant an increase/decrease in volatility would change the chance of expiring at a financially beneficial position. However this not appear to explain the first statement (which I believe is correct).

Answer

The risk exposures/sensitivities of long and short positions always have different signs. This has to hold since derivatives are zero sum games.

Vega is always positive for a long position in a European plain vanilla option (or any convex payoff in general). This is true even when the option is already in-the-money. As volatility increases, the probability of very positive and very negative returns increases. As the holder of the option, you are protected against moves in one direction but participate in the other.

You can construct a very simple binomial example to illustrate this. Consider a one period setting. You are long a call with strike 90. The current stock price is 100 and there are no rates or dividends.

1. First consider a "low volatility" scenario, where the stock either goes up to 105 or down to 95. The payoff is either 15 or 5 and the initial price is 10 with a time value of 0.


2. Now consider a "high volatility" scenario, where the stock either goes up to 120 or down to 80. They payoff is either 30 or 0 and the initial price is 15 with a time value of 5.

You see that due to the convexity of the payoff, a higher volatility is advantageous even when the option is already in-the-money since losses are limited. European vanilla options derive their time value (ignoring rates, dividends, ...) from the possibility of crossing the strike, no matter whether they are already in-the-money or not.