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They would only have been equal (up to the usual MC accuracy and bias) should the black-box model had assumed a GBM dynamics as in the classic Black-Scholes framework. $D_{MC}$ and $D_{BS}$ will indeed differ in general because digital options are sensitive to the implied volatility skew, which is inexistent in a Black-Scholes world where $\sigma (K,T)=\... 2 In practice, an implied volatility always refers to the volatility that you need to plug into the Black-Scholes', or Black's, pricing formula to obtain the market price. You may have a different model (e.g., a Heston style stochastic variance model), for which you are able to calibrate the parameters by matching your model price to the market price. However, ... 1 Page 3 of this document ad-co.com/analytics_docs/ALevin_QP_2012.pdf shows the result, originally given in Risk Magazine by Blyth and Uglum. The intuition for the formula is given in my comment above. The original motivation for such a formula was for interest rate options in the 1990s. Everyone had a lognormal pricing model, but traders understood that ... 1 Indeed parameters are selected so that the quoted option prices are as close as possible to the model option prices. Alternatively, quoted and model implied volatilities can be used instead of prices.The first category are those that minimize the error between quoted and model. The second category,are those that minimize the error between quoted and model ... 1 It might be worth noting that in the standard Black-Scholes model the implied volatility$\sigma$is assumed to be a positive non-zero constant. If this were the case then we could simply look at the stock$S\$, look at the historical data, and then compute the log-returns on some arbitrary time scale, and then compute the standard deviation, and we would ...

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you can find from the CBOE paper above mentioned that the value is pretty much that of a strip of vanillas, weighted by 1/K^2. Typically if spx spot goes down, then realized vol increases. Together with the increase of realized vol, implied vol gets "re-evaluated" and typically marked higher with a steeper skew etc. the remark of the implied vol surface ...

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