When Black–Scholes Option Vega Is Positive
Summary
The discussion distinguishes standard European calls and puts from options with more general payoffs. Under the Black–Scholes assumptions described, call and put values increase with volatility, so their vega is positive. The replies point to the familiar vega formula as a direct way to confirm this result, although the original question does not work through the derivation in detail.
For other payoffs, volatility sensitivity need not be positive: binary options are given as a counterexample. A reply sketches why convex payoffs have nonnegative value changes as volatility rises, using a coupling in which the higher-volatility terminal value is the lower-volatility value times an independent, mean-one factor, followed by conditional Jensen’s inequality. This supports defining implied volatility for convex payoffs. The discussion does not give a complete classification of payoffs, and its conclusions depend on the stated Black–Scholes setting; it also notes that smile-sensitive payoffs may not be well represented by that model.
Key ideas
- European calls and puts in the Black–Scholes model have positive vega.
- Vega need not be positive for arbitrary payoffs, with binary options cited as a counterexample.
- A coupling and conditional Jensen’s inequality can establish nonnegative volatility sensitivity for convex payoffs.
- The conclusions rely on model assumptions and may not capture volatility-smile effects.
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Full text
# Is vega of Black-Scholes European type option always positive?
# 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?
## Answer by Christian Fries (score 6, accepted)
https://quant.stackexchange.com/a/7903
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...
## Answer by Brian B (score 3)
https://quant.stackexchange.com/a/7917
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.
## Answer by Lost1 (score 2)
https://quant.stackexchange.com/a/82196
For a convex function. this can be proven by using some coupling argument. Let $f$ be the payoff, we can write the payoff as
$f(cX)$, where $X$ is a log-normal distribution with some parameter, c adjust for the drift and forward price.. Now consider $f(cY)$, where $Y$ is another log-normal distribution with some parameters corresponding to BS model with a higher volatility parameter.
It is possible to couple these on the same probability space such that
$Y = XZ$ where $Z$ has mean of 1 and independent of $X$.
The result then follows from conditional Jensen inequality.
This is why implied volatility can be defined for any convex payoff.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.