What does it mean to be Long Vega?

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The terms long and short here refer to the same relationship pattern when speaking of being long or short a stock or an option. That is, if volatility rises and you are short volatility, you will experience losses, ceteris paribus, and if volatility falls, you will have immediate unrealized gains. Likewise, if you are long volatility when implied volatility rises, you will experience unrealized gains, while if it falls, losses will be the result (again, ceteris paribus).(For more on these factors see, Getting to Know The "Greeks".) 

-	Vega Sign	Rise in IV	Fall in IV
Long call	Positive	Gain	Lose
Short call	Negative	Lose	Gain
Long put	Positive	Gain	Lose
Short put	Negative	Lose	Gain

Figure 9: Outright options positions, Vega signs and profit and loss (ceteris paribus).

The long call and the long put have positive Vega (are long volatility) and the short call and short put positions have a negative Vega (are short volatility). To understand why this is, recall that volatility is an input into the pricing model - the higher the volatility, the greater the price because the probability of the stock moving greater distances in the life of the option increases and with it the probability of success for the buyer. This results in option prices gaining in value to incorporate the new risk-reward. Think of the seller of the option - he or she would want to charge more if the seller's risk increased with the rise in volatility (likelihood of larger price moves in the future). 

Therefore, if volatility declines, prices should be lower. When you own a call or a put (meaning you bought the option) and volatility declines, the price of the option will decline. This is clearly not beneficial and, as seen in Figure 9, results in a loss for long calls and puts. On the other hand, short call and short put traders would experience a gain from the decline in volatility. Volatility will have an immediate impact, and the size of the price decline or gains will depend on the size of Vega. So far we have only spoken of the sign (negative or positive), but the magnitude of Vega will determine the amount of gain and loss.