Why Call Option Vega Is Positive as Volatility Rises
Summary
The document addresses why a European call’s value rises with volatility even when it is already in the money. The questioner focuses on the increased chance of finishing out of the money, which seems to imply a lower price. The answers explain that this probability change is only part of the effect: greater volatility also changes the size of potential payoffs. A call has limited downside at expiration and gains from larger upward moves, so the favorable outcomes can outweigh the added chance of finishing out of the money.
One answer suggests using a one-step binomial model to build intuition. The discussion gives a qualitative explanation rather than a formal proof or numerical example. Its conclusion concerns European calls in the Black–Scholes setting; it does not explore how other contract features, pricing assumptions, or market frictions might affect option sensitivity.
Key ideas
- A call’s payoff is limited on the downside and can grow with the underlying price.
- Higher volatility changes both the probability and size of possible expiration payoffs.
- For a European call in the Black–Scholes model, the document states that vega is positive.
- A one-step binomial model can help illustrate the payoff asymmetry behind positive vega.
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Full text
# Why is the Vega positive? # Why is the Vega positive? We know that in the Black-Scholes model, the Vega of a European call option is always positive. This can be proved easily, so my question is not really about the result per se. My problem is that I find this result somehow counterintuitive. This is my argument: if an option is out of the money and the volatility rises, then the probability that the option end in the money also grows, so that the price grows, and I find it convincing that the vega should be positive in this case. But think now of an option which is in the money (maybe not "too much" in the money). In this case if the volatility grows, doesn't the same argument show that the probability that the option finishes out of the money in this case increases, and consequently the price of the option should actually decrease, not increase. But it is a fact that the vega is always positive, also for out of the money options. So, where is the fallacy in my intuition?? Thanks for your answers. ## Answer by RRG (score 2, accepted) https://quant.stackexchange.com/a/30753 The payout from a call option is non-linear with limited downside and unlimited upside. So while volatility can increase or decrease the value of the underlying, the payoff is greater on an up-move than the loss on a down-move, hence the positive vega. ## Answer by M. Jeunesse (score 2) https://quant.stackexchange.com/a/30744 If volatility increases, then not only probability to be in the money increases but also the money you get when you are in the money will increase too. If you want to convince yourself, try it on the one-step binomial model.
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