What is the formula behind the standard deviation of an option strike?
What is the formula behind the standard deviation of an option strike?
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Hakim · External communityPost link
External question — Quantitative Finance Stack Exchange
Author: Hakim
Original post: https://quant.stackexchange.com/questions/85351
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I am using Interactive Brokers as my broker, and I see in the option chain that each strike has a specific standard deviation. What is the formula? The AI suggested to me that it is derived from the Black-Scholes model and implied volatility, but I need the actual data so I can plot them against the formula that yields that standard deviation. This is for a 0DTE iron condor strategy where I choose the wings at 2 standard deviations and set a stop loss for a high-risk, high-probability win rate.
Thank you for your help.
Hakim Kabissa
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D Stanley · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: D Stanley
Original post: https://quant.stackexchange.com/a/85354
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There is no empirical "standard deviation" for a strike. Based on the
linked question you answered
, IB is most likely showing you how many "standard deviations" a strike is from the spot price, based on the assumption that the future returns of the underlying stock are normally distributed, which makes the math easier. The variance used to calculate the z-score is very likely based on implied volatilities that also assume that returns are normally distributed. History has shown, however, that more extreme changes are more likely that what the normal distribution assumes ("fat tails" in statistical parlance). The market knows this, and places higher probabilities on more extreme movements in either direction, which is one reason that "implied volatility" is not constant across option strikes. So the z-score could be calculated from option prices for that strike, prices of at-the-money options, or a combination of many prices.
You could also look at historical returns to determine a confidence interval, but those returns could be biased either way - if there was a recent large market move, they could be biased upward, or biased downward if the market has been calm but market participants expect large movements in the near future.
All the "SD" tells you is a probability that the underlying will move within a certain interval based on some model (+/- 34% for normally distributed returns). There's no way to know what the
actual
probability is - either it happens or it doesn't.
All that to say if your goal is to estimate a "2 SD" move for the wings of your Iron Condor, then using either IB's "standard deviation" or a "standard deviation" that is based on implied volatility is appropriate. There's not one "right" answer. Plus since option strikes are discrete, there's going to be some amount or rounding error anyway.
It's like asking what a "2 standard deviation" change in the air temperature in Paris over the next day is. Do you base it on history? What the weather models predict? The Farmer's Almanac? Any of those is just an
estimate
of future events.
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João · External communityPost link
External answer — Quantitative Finance Stack Exchange
Author: João
Original post: https://quant.stackexchange.com/a/85355
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From
IBKR
website Q&A:
The standard deviation for the options chain is calculated based on the implied volatility. Specifically:
The implied volatility is estimated for the 8 options on the 4 closest to market strikes in each expiry.
These implied volatilities are fit to a parabola as a function of the strike price for each expiry.
The at-the-market implied volatility for an expiry is the value of the fit parabola at the expected future price.
A linear interpolation (or extrapolation) of the variance is done based on the squares of the at-market volatilities.
The standard deviation is then the square root of this estimated variance.
So in summary, the standard deviation is derived from the implied volatilities of the options prices using a model that fits a curve to the volatilities and interpolates to find the at-the-money volatility.
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Quoted from Forex.com.bd-Editorial External answer — Quantitative Finance Stack Exchange Author: D Stanley Source score (net votes, not local likes): 1 Original post: https://quant.stackexchange.com/a/85354 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. There is no empirical "standard deviation" for a strike. Based on the linked question you answered , IB is most likely showing you how many "standard deviations" a strike is from the spot price, based on the assumption that the future returns of the underlying stock are normally distributed, which makes the math easier. The variance used to calculate the z-score is very likely based on implied volatilities that also assume that returns are normally distributed. History has shown, however, that more extreme changes are more likely that what the normal distribution assumes ("fat tails" in statistical parlance). The market knows this, and places higher probabilities on more extreme movements in either direction, which is one reason that "implied volatility" is not constant across option strikes. So the z-score could be calculated from option prices for that strike, prices of at-the-money options, or a combination of many prices. You could also look at historical returns to determine a confidence interval, but those returns could be biased either way - if there was a recent large market move, they could be biased upward, or biased downward if the market has been calm but market participants expect large movements in the near future. All the "SD" tells you is a probability that the underlying will move within a certain interval based on some model (+/- 34% for normally distributed returns). There's no way to know what the actual probability is - either it happens or it doesn't. All that to say if your goal is to estimate a "2 SD" move for the wings of your Iron Condor, then using either IB's "standard deviation" or a "standard deviation" that is based on implied volatility is appropriate. There's not one "right" answer. Plus since option strikes are discrete, there's going to be some amount or rounding error anyway. It's like asking what a "2 standard deviation" change in the air temperature in Paris over the next day is. Do you base it on history? What the weather models predict? The Farmer's Almanac? Any of those is just an estimate of future events.
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