Black–Scholes Vega Limits and Volatility Sensitivity
Summary
The note examines how a European call or put’s Black–Scholes vega changes as volatility approaches zero or becomes very large. It gives the vega formula and the expression for d₁, then uses their limiting behavior to argue that vega tends to zero at both extremes. The reasoning applies to positive maturity and the usual Black–Scholes inputs; the note does not explore special boundary cases in detail.
For whether vega is monotonic or convex in volatility, the answer points to volga, the derivative of vega with respect to volatility, as a starting point. It does not carry out that analysis or establish general monotonicity or convexity properties. Thus, the response supplies endpoint intuition but only a route toward answering the shape question. It cites an external reference for a volga formula, without reproducing or verifying it.
Key ideas
- Black–Scholes vega is proportional to the standard normal density evaluated at d₁.
- The answer argues that vega tends to zero as volatility grows without bound.
- It also argues that vega tends to zero as volatility approaches zero under the stated setup.
- Volga, the volatility derivative of vega, can be used to study monotonicity and curvature.
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Full text
# How does Vega of a call/put behave under the Black-Scholes model?
# How does Vega of a call/put behave under the Black-Scholes model?
I have two questions. I would prefer a reference if possible.
- Is the value of vega bounded for $\sigma\in [0,\infty)$? (I assume so, I imagine it goes to 0 as $\sigma$ go to infinity.)
- Are there any well established properties of vega? (e.g. convexity/monotonicity? How fast does it go to 0, as $\sigma\rightarrow\infty$? How does it behave as $\sigma\rightarrow 0$?)
A partial answer would be more than welcome.
## Answer by Richi Wa (score 1, accepted)
https://quant.stackexchange.com/a/10122
if we write down the formula for vega we get $$ \text{vega} = S \sqrt{T} e^{-q T} \frac{1}{\sqrt{2\pi}} \exp(- \frac{d_1^2}2) $$ where $$ d_1 = \frac{\log(S/K) + (r-q +\sigma^2/2)T}{\sigma \sqrt{T}}. $$ We have $$ \lim_{\sigma \rightarrow \infty} d_1 = \infty, $$ as $\sigma^2$ grows quicker than $\sigma$ for $\sigma \rightarrow \infty$ and therefore $$ \lim_{\sigma \rightarrow \infty} \text{vega} = 0, $$ due the definition above and the continuous functions involved.
Furthermore $$ \lim_{\sigma \rightarrow 0} d_1 \text{ is unbounded}, $$ because $$ \lim_{\sigma \rightarrow 0} d_1 = \lim_{\sigma \rightarrow 0} \frac{\log(S/K)}{\sigma \sqrt{T}} + \frac{(r-q) \sqrt{T}}{\sigma} +\sigma/2 \sqrt{T} $$ and the first two summands are unbounded for $\sigma \rightarrow 0$ and the third tends to zero. Thus depending on $\log(S/K)$ - the moneyness - and $r-q$ we get $$ \lim_{\sigma \rightarrow 0} d_1 = \pm \infty $$ and in any case $$ \lim_{\sigma \rightarrow 0} \text{vega} = 0. $$
For the second part of convexity/monotonicity you need $\frac{\partial \text{vega}}{\partial \sigma}$ this is called Volga.
Wilmot has the formula for Volga here but I did not check it. You can analyze it in order to analyze montonicity. You need its derivative for convexity ...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.