How Implied Volatility Changes Put Backspread Gamma and Vega
Summary
The document explains how implied volatility can affect the Greeks of a put backspread, which holds one short put and two lower-strike long puts. It separates the effect on an individual option’s gamma from the net position’s vega. For gamma, the formula has a factor that falls as volatility rises, while another factor depends on moneyness, volatility, and time to maturity. That second factor can rise or fall, so the overall change is usually downward but is not guaranteed in every case, especially for deep out-of-the-money options.
For vega, the position’s response depends on where the underlying is relative to the strikes. Near entry, when the short put is at or near the money, it can have more vega than the two lower-strike puts combined, making the spread short vega. If the underlying falls, the long puts move closer to the money and can eventually have greater vega than the short put. The discussion is qualitative and offers plotting gamma across strikes and volatility as a way to investigate; it gives no numerical examples or universal threshold.
Key ideas
- Higher implied volatility lowers one component of an option’s gamma, but another component varies with strike, volatility, and maturity.
- The gamma of a put backspread can respond differently depending on how far out of the money its options are.
- A backspread near entry may be short vega when its short put is near the money and its long puts are farther out of the money.
- As the underlying falls, the long puts can gain vega relative to the short put and change the spread’s net vega.
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# How does high IV effect a put backspread?
# How does high IV effect a put backspread?
I have a hard time understanding how high IV effects the amount of gamma obtained via a put backspread. Is it via the angle on the payoff or via the ratio one gets i.e number of OTMs one can buy? or simply just that one has bigger numbers in the credit and debit for the position?
## Answer by Jan Stuller (score 2, accepted)
https://quant.stackexchange.com/a/55223
Below, Gamma is denoted by $\Gamma$, and $IV=\sigma$:
$$ \Gamma = \left(\frac{1}{S_0*\sigma*\sqrt{T}}\right)*\left(\frac{1}{\sqrt{2\pi}}e^{\frac{-d_1^2}{2}} \right) $$
The expression in the first bracket is inversely proportional to $\sigma$, so isolating just this first expression: higher $\sigma$ will trivially lower this first expression.
The second bracketed expression is a Standard Normal PDF, where:
$$d_1=\frac{ln\left(\frac{S_0}{Ke^{-rT}}\right)}{\sigma\sqrt{T}}+0.5\sigma\sqrt{T}$$.
For OTM put options, $ln \left( \frac{S_0}{Ke^{-rT}} \right)$ is positive, because $S_0>Ke^{-rT}$. So increaing $\sigma$ will make this log-term smaller, which will make $d_1$ closer to zero and therefore the PDF $\left(\frac{1}{\sqrt{2\pi}}e^{\frac{-d_1^2}{2}} \right)$ will move closer to its peak. At the same time, increasing $\sigma$ will make the expresison $0.5\sigma\sqrt{T}$ larger, moving the $d_1$ away from zero, and therefore moving the PDF away from its peak.
So in conclusion: increasing IV decreases the first bracketed expression, and it is not possible to say with certainty whether it increases or decreases the second bracketed expression: whether the second bracketed expression increases with increasing $\sigma$ will depend on:
(i) how much OTM the option is,
(ii) the current level of $\sigma$,
(iii) Time to maturity $T$.
In general, increasing IV decreases Gamma via the first bracketed expression, but there can be limited cases where deep OTM options will gain Gamma from rapid increases in IV. To get a clearer idea, the best way to go about this is to try to plot the gamma for different strikes and play around with the $\sigma$ parameter.
## Answer by Jan Stuller (score 2)
https://quant.stackexchange.com/a/55129
In a put back-spread, you are short one put and long two puts with a lower strike.
As far as Vega (the spread's price sensitivity to IV) is concerned, it depends on where you enter the spread. Usually, you'd enter the spread where the short put is at the money or near being at the money: this ATM put will have a high Vega, whist the two puts with lower strike will have a low vega to start with. So if the underlying price stays where it is, you're actually short Vega and increasing IV should increase the value of the short put more than the value of the two OTM puts. So in this scenario, increasing IV will lower the price of the spread.
As the underlying price moves lower and your short put becomes ITM, your two long puts will gradually come closer to being ATM. Since Vega is highest for ATM options, there will come a point where Vega on the two long puts will become higher than the Vega on the short put: then, increasing IV will increase the price of the spread.
Edit: picture below shows that deep OTM or ITM options have low sensitivity to IV (so if the options are on the SPX, the sensi to VIX for ITM or OTM options is much lower than for ATM options: that should answer your question as to "why the one ATM put benefits from higher VIX more than two OTM puts"):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.