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Was going through how Implied Volatility is used by option traders and in delta hedging. Correct me if I am wrong, doesn't IV consider a standard deviation of the stock price over say the past 1 year? Now as far as I remember, we do not consider standard deviation for non stationery time series. If this is so, isn't IV flawed? Or there is a more elegant method out there?
Thanks and Cheers!! Cryptex
No.
Implied volatility isn't a historical measure of standard deviation. Implied volatility is used to relate a market price to some model, be that Black-Scholes or something more sophisticated.
Another way to phrase it, implied vol is that single vol input into a model, such that the model reproduces the market prices. Different models will have different implied vols.
And even in the Black-Scholes model, the volatility isn't a measure of the standard deviation of the stock price. It's a measure of the standard deviation of the log-return of price.
Consider Geometric Brownian Motion for a stock price: $dS_t = \mu S_t dt + \sigma S_t dW_t $. The distribution of $\ln \frac{S_t}{S_0}$ is $N(\mu-\frac{\sigma^2}{2}, \sigma^2 t)$, where $N(.)$ is the Normal distribution. In other words, over $t=1$ year, the standard deviation of the log-return of the stock is $\sigma$.
In contrast, the standard deviation of the stock price over 1 year is given by $S_0 e^\mu \sqrt{e^{\sigma^2} -1 }$. This quantity looks nothing like the implied vol you are deriving from market option prices.
A final comment: nothing about implied vol calculation is dependent on "the past 1 year". It's strictly a forward-looking concept. Look up Markov property on Wikipedia.
A pithy way to put it is "implied volatility is the wrong number to put in the wrong formula to get the right price." That is, implied volatility is by definition the parameter $\sigma$ to plug into the Black-Scholes option pricing formula to get the market price of a vanilla option. This is called "volatility," but in reality it isn't the same as the result of any historical volatility calculation; hence its the "wrong number." The Black-Scholes model is a vastly simplified model that does not reflect the true complexities of the market; hence "wrong formula." But everyone knows the conventions, so implied volatility gives a correct way to quote option prices.
The variables going into IV are not reliable, and in Black-Scholes there are so many flaws that anything derived from it, including IV, is unreliable as well. This is well-known as a flaw. IV is only an estimate. However, it is based on the fixed underlying and option values today, and these will change in the future. So IV does not measure future volatility in any sense. The commonly held belief is that "volatility leads price" but in fact, it is the other way around: "Price leads volatility. Thus, historical volatility is the only reliable method for understanding the evolving risk in a particular option contract.
