How Volga Creates Asymmetry in Option Price Responses to Volatility Shocks
Summary
The document explains why an option’s price response to upward and downward implied-volatility shocks may be similar in magnitude without being exact opposites. A Taylor expansion expresses the price change through vega, the first derivative with respect to volatility, and volga (also called vomma), the second derivative. Vega contributes with opposite signs for equal positive and negative shocks, while volga contributes the same sign in both cases because the shock is squared.
Consequently, the two impacts are exact opposites when volga is zero; otherwise, the second-order term makes them asymmetric. The response points to a vanilla option volga curve as an illustration and says the value varies with forward moneyness, with no volga at the money in that illustration. This is a local approximation, and the explanation does not provide numerical estimates or address all effects that may matter for a large stress. The answer is useful for interpreting stress results, but the approximation’s accuracy depends on the size of the volatility bump and the instrument’s nonlinear exposures.
Key ideas
- Vega makes the first-order price response reverse sign between equal upward and downward volatility shocks.
- Volga contributes with the same sign for shocks of either direction because its term depends on the squared bump.
- Equal and opposite price impacts follow when volga is zero.
- Option volga varies with forward moneyness, and the cited illustration has zero volga at the money.
Tags
Full text
# Should price impact be the same for positive/negative implied volatility shocks?
# Should price impact be the same for positive/negative implied volatility shocks?
I am using a vendor system to stress a portfolio which contains (among others) derivatives with implied volatility exposure.
The issue is that when using a 1000 bps implied volatility stress upwards and downwards the result is really close in both cases (with an opposite sign obviously)
Is this expected?
## Answer by Quantuple (score 2, accepted)
https://quant.stackexchange.com/a/32712
It may help you to notice that, for a bump in implied volatility $\delta \sigma$, the impact on the price of the derivative $V$ is given by: $$ \delta V = \underbrace{\frac{\partial V}{\partial \sigma}}_{\text{Vega}} \delta \sigma + \frac{1}{2} \underbrace{\frac{\partial^2 V}{\partial \sigma^2}}_{\text{Volga, Vomma}} (\delta \sigma)^2 + o((\delta \sigma)^2) $$
Hence, the positive ($\delta V^P$) and negative ($\delta V^N$) price impacts for respective bumps $\delta \sigma^P= \vert\delta\sigma\vert$ and $\delta\sigma^N = - \vert\delta\sigma\vert$:
$$ \delta V^P = \frac{\partial V}{\partial \sigma} \mid \delta \sigma \mid + \frac{1}{2} \frac{\partial^2 V}{\partial \sigma^2} (\delta \sigma)^2 $$ $$\delta V^N = -\frac{\partial V}{\partial \sigma} \mid \delta \sigma \mid + \frac{1}{2} \frac{\partial^2 V}{\partial \sigma^2} (\delta \sigma)^2 $$ hence $$ \delta V^P = -\delta V^N + \frac{\partial^2 V}{\partial \sigma^2} (\delta \sigma)^2 + o((\delta \sigma)^3) $$ and when no Volga (also called Vomma): $$ \delta V^P = - \delta V^N $$
For illustration purpose here is the Volga curve of a vanilla option of time to maturity $\tau$ as a function of forward moneyness $m=K/F(0,\tau)$. Observe how an ATM option has no Volga and how this changes as you move away from the money.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.