The Black–Scholes Relationship Between Vega and Gamma

Summary

The document addresses how option vega relates to gamma and explains a formula attributed to Nassim Taleb’s Dynamic Hedging. Under Black–Scholes, gamma is expressed using the asset price, volatility, time to expiration, and the normal density at d1; vega uses the same density along with asset price and square root of time. Comparing these expressions gives vega as gamma multiplied by asset price squared, volatility, and time remaining.

A second response offers an intuition: expected profit from gamma rebalancing corresponds to an option’s value at a given volatility, so comparing those values across volatility assumptions connects the idea to vega. The derivation supports the stated formula within the Black–Scholes model. The intuitive description is brief and does not specify a general integration procedure or explain how to choose volatility levels, so the result should not be treated as a model-independent identity.

Key ideas

• Within Black–Scholes, vega equals gamma multiplied by the square of the asset price, volatility, and time to expiration.
• The formula follows by comparing the model’s standard expressions for gamma and vega.
• The document links expected gamma rebalancing profit to option value at a given volatility.
• The stated formula is model-specific, and the intuitive integral explanation is not fully developed.

Tags

# Link between Vega and Gamma


# Link between Vega and Gamma












> "The vega is the integral of the gamma profits ( ie expected gamma rebalancing P/L) over the duration of the option at one volatility minus the same integral at a different volatility...Mathematically, it is: $$\text{Vega} = \sigma t S^2 \text{Gamma}$$ where $S$ is the asset price, $t$ the time left to expiration and $\sigma$ the volatility.

This is again from Dynamic Hedging by Taleb. I cannot understand the first sentence because it gives no indication of which volatilities to pick nor what the integrand of the integral would be.

Shortly after Taleb states the formula above, again no justification as to where it came from.

Please could someone explain this better.

## Answer by Gordon (score 15, accepted)

https://quant.stackexchange.com/a/24913

Under the Black-Scholes model, \begin{align*} Gamma &= \frac{N'(d_1)}{S \sigma \sqrt{T-t}}\\ Vega &= SN'(d_1) \sqrt{T-t}. \end{align*} Then, it is easy to see that \begin{align*} Vega = S^2 \sigma (T-t) Gamma. \end{align*}

## Answer by XXXXXXX (score 1)

https://quant.stackexchange.com/a/78548

I believe in its most fundamental form it is best to internalize that the expected gamma rebalancing P/L = Option price at one volatility. Thus the difference of the option prices at different volatilities by this interpretation shall be the vega, or the sensitivity of an options price to a change in implied vol.

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.