When working on calibration for LMM model, we need to have Initial Libor quotes and Swaptions Black vol quotes on the market data. We have data provided by Bloomberg. However, before performing the calibration, we wish to check the coherence of the market data, e.i if we use the right Libor/Discount Rate for the swaption volatility cube (VCUB).
The problems is i can not have the coherence of these data. The ATM swaps has the strike which is the swap rate interest at the present moment. The swap rate is calculable directly from the Libor quotes (deduced from the Discount Rate).
$$ L(t,T_i,T_{i+1}) = \frac{1}{T_{i+1}-T_{i}}(\frac{P(t,T_{i})}{P(t,T_{i+1})} - 1) $$
In the SWPM tab of bloomberg, they said they build swaption VCUB data from EUR 45, which have built data yield curve, zero rate, forward rate ... for calculating swap/swaptions values.
I firstly used these Interest rate curve and recompute swap rate by myselft, I can not reproduce the ATM strike quoted on ATM swap strike matrix.
I secondly use the quoted zero rate curve to recompute the discount curve, $$ P(t,T_i) = \frac{1}{1+r_i T_i} $$ Same thing, i can not reproduce the same values as Bloomberg quotes.
However, these values computed and quoted are approximative equal (relative error around 1% max). So i thing that could be due to the difference of day count, or somethings else. Ideally, i hope check every data are perfectly coherent with theoretical formulas, in the way that the calibration result is not biased due to any kind of these error.
My first question is so does somebody has the idea of why that happens like that, is this normal? How to fixed these data coherence. (Error might come from my computations)
My second question is : is this a big issu if we perform a volatility calibration with these incoherence exist in our data?
Many thanks
