Why Long Vanilla Options Have Positive Vega, Including In-the-Money Options
Summary
The document explains why a long position in a standard European vanilla option generally has positive vega, even when the option is already in the money. Vega measures how an option’s value responds to a change in volatility. The key idea is that a vanilla option has a convex payoff: greater volatility can increase favorable outcomes while the holder’s loss remains limited to the option premium.
A one-period binomial illustration compares a call under lower and higher volatility, showing how the more dispersed outcomes increase the option’s value and time value in the example. A second explanation uses put-call parity: a deep in-the-money call can be viewed as stock plus a deep out-of-the-money put, and the stock contributes no vega. The discussion concerns standard vanilla payoffs; it distinguishes them from digital options, whose fixed payout makes moneyness probability more central. It does not cover how vega varies in magnitude across strikes, maturities, or market conditions.
Key ideas
- A long European vanilla option has positive vega, including when it is in the money.
- Convex payoffs can benefit from increased dispersion because downside losses are limited.
- The binomial example illustrates how wider possible outcomes can add option time value.
- Put-call parity links the vega of a deep in-the-money call to that of an out-of-the-money put.
- The explanation does not apply in the same way to fixed-payout digital options.
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# Long/Short Vega and Option Positions # 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 by LocalVolatility (score 5, accepted) https://quant.stackexchange.com/a/30191 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. - 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. - 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. ## Answer by dm63 (score 4) https://quant.stackexchange.com/a/30193 No, you are incorrect. A deep in the money option is long vega. It's not just about the probability of being in the money, it's about how far in the money it is. Your reasoning is correct if we are talking about digital options which pay a fixed amount if the option expires in the money, but incorrect for regular options. One way to prove this explicitly: a deep in the money call is equal (by put call parity) to a long position in the underlying + a deep out of the money put. Since the position in the underlying has no vega, the deep ITM call has the same vega as the deep OTM put.
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