# Is vega of Black-Scholes European type option always positive?

We assume we work in the risk-neural measure with a stock which pays no dividend and a continuous discount rate.

For PUT and CALL only: can someone please clarify if what I said is correct?

The intuitive answer is yes, because bigger volatility you are more likely to end up in a region that that is

I looked it up the wikipedia for the formula, but I am a bit lazy trying to prove it is positive

For a GENERAL PAY-OFF FUNCTION:

when is this still true? I would think it would be true for a monotone function or maybe a convex function? Does anyone know any exisiting literature on this?

• Please re-read you question, there is a part that doesn't make sense. What do you mean by "a general payoff function"?
– SRKX
May 4, 2013 at 22:59

If you modify your question to "European Call and Put under a Black-Scholes Model" the answer is: yes.

It's trivial to verify it from the formula $S e^{d_1} \sqrt{T-t}$.

For a general payoff the question is more difficult to answer. In general vega will not be positive. I believe that you can derive some conditions on the payoff assuming a Black-Scholes Model, but I believe that these conditions are "almost useless", since such a general payoff (like a call spread) would depend on the volatility smile and would not be valued using a Black-Scholes Model...

• hence my question has two parts. one regarding put and call, the other regarding to a general pay off, which we may just assume to be a positive fuction May 4, 2013 at 15:22
• I added a remark on general payoffs to my answer. With respect to your comment: assuming that the pay off is positive does not help. Vega is not about level or slope, it is about convexity. It would depend on WHERE your payoff is convex and WHERE it is concave. And how strong convextiy is depending on the underlying. May 6, 2013 at 17:34
• I am thinking about this question again. Let the pay off function be dependent on $S_t$ only, i.e. $g(S_t)$. I believe we can show if $g$ is convex, then so must the option price be. Put and calls are obvious applications for this. May 17, 2013 at 12:52

As Christian notes, under the Black-Scholes model standard european options have prices that are monotonic in volatility.

You can see that binary options do not share this property but I suspect you are correct about convex payoffs.