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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)=\...

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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, ...

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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 ...

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