Hello,Is the following correct ? For example, I take options between maturity 30 and 60 days and moneyness between 0.9 and 1.1, I use their market price to obtain only one implied vol and then I use this implied vol in BS formula for an out of sample comparison.

What do you want to do exactly? If you want to summarize the entire B&S implied vol surface with only one value in order to price out the money options, you will be in trouble

Not only one value, it was just for the example.I have divided my options into 20 groups by interavals of moneyness and maturities : DITM, ITM, ATM, OTM, DOTM and 0-30-60-90-120.For example, I take the group ATM with 30<=maturity<60, I fit only one IV value for the whole group which I use for another group ATM with 30<=maturity<60 to do the out of sample comparison and the same for other moneyness and maturities.

If you compute the implied volatility in B&S model for all strikes and maturity, you will see that it is a pretty smooth surface. By grouping options in buckets of moneyness and volatility, you are just approximating this surface by some kind of stairs. B&S is a convenient model to express the value of an option in ?B&S volatility points?. For example, you are building a more complex model, you calibrate it on the options available, and then you calculate prices of out of the sample options. Theses prices can be inverted with the B&S model providing you the entire implied vol surface of the B&S model continuously in strike and maturity.

Thank you for your answer RentMe,I understand, so what I'am trying to do is not correct ?I would like to compute adhoc BS prices but my sample don't allow me to do so because I don't get sufficient maturities (at the same day only one maturity is observed with several strikes), so is the adhoc BS usable here ? if it is usable only with strikes which day (maturity) should I use ?

Think about the B&S model. Why the volatility (that should be a unique property of the underlying) depends on the strike and the maturity? After that, the only thing that is possible to do is to build arbitrage free vol surface using B&S model and conditions of non arbitrage. But B&S cannot invent information you do not have. For example if you want to price a very long maturity option and you have only short maturity options for calibration, you can just ?extrapolate?, which means that you are choosing (as a trader does) the model parameter that is not too far away from the reality. As I said before, B&S is only convenient to express the result of more complex models in terms of B&S implied volatility points. More complex models create a dynamic of the vol surface and are not only calibrated on vanilla.After that, if you need to stay in the B&S world, maybe you could interpolate volatility between strikes and maturities, nevertheless this does not seem very satisfactory (but it?s really used by some traders)

What you are doing is interpolating the volatility surface with a piecewise constant function. To get a smoother surface, you could try bilinear interpolation or splines.

Ok thank you for your answers, I'll try to smouth the IV by linear regression.

Maybe this article can help and at least provides some examples.