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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?

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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.

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  • $\begingroup$ Thanks, I did some calculations on 10 % OTM montly puts and gamma inceased going from vol 25 to 30. Dunno weather that is considered far OTM, I am not really a experienced options trader. $\endgroup$
    – user123124
    Jun 25 '20 at 18:30
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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"):

enter image description here

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  • $\begingroup$ sorry this is not what I ment, I ment in terms when you put it on not how the price develops. In other words what difference does a high reading on the VIX make when you put on or buy a back spread in contrast to a low reading on the VIX. Dont erase the answer tho, the above was also nice to read. $\endgroup$
    – user123124
    Jun 22 '20 at 18:02
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    $\begingroup$ So if you put on a back-spread on the SPX 500 then the VIX reading will directly affect the price of your spread in line with what I wrote. Imagine you enter the spread where the first option is ATM and the two lower-strike options are OTM. Then, higher VIX decreases the value of your spread, whilst lower VIX increases the value of your spread. If the SPX tanks though and the lower-strike options get closer to being ATM, then the relationship will change: higher VIX will increase the price of your spread and lower VIX will decrease it. The key concept is that your Vega changes with SPX price. $\endgroup$ Jun 22 '20 at 18:10
  • $\begingroup$ wouldnt a high VIX, which as far as I understand increase option prices, force on to go further OTM to get the two options matching up in price to the ATM? Since they are two and the other one is just "1". Its not the Vega but the amount of gamma I get that I am intrested in. $\endgroup$
    – user123124
    Jun 23 '20 at 5:27
  • $\begingroup$ Gamma is second order sensitivity to the underlying price. So if the options are on the SPX 500, Gamma would be the second order sensitivity to SPX price, NOT the VIX. Your only sensitivity to VIX is Vega (and Vomma, but let's not worry about Vomma for now). The two OTM options don't increase in price much when VIX goes up, because their Vega is very low, compared to the one ATM options. I added a picture to my answer that might help understand this graphically. $\endgroup$ Jun 23 '20 at 6:41
  • $\begingroup$ and so you are saying with a higher VIX I should be able to put on a 3:1 ratio instead of a 2:1 all else being equal? anyway, high iv should also affect gamma right i.e it flattens it. $\endgroup$
    – user123124
    Jun 23 '20 at 7:23

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