# Passage from dates ranges to real numbers in modelling : which market practice?

Let's say I model a 6M forward Libor rate as a process $(L^1_t)_t$ that's a diffusion, with in view a Monte-Carlo (MC) pricing of some product. At some point I will have real life dates $T_i$'s that I will have to convert to real numbers : more precisely, at some point I will have a $dt$ that will represent the time between $T_i$ and $T_{i+1}$, and I will have to look a the year fraction $y$ that the period $[T_i, T_{i+1}]$, and this $y$ is going to be my $dt$, that I will plug in my MC sample path generation.

My question is : which convention practitioners use to calculate the year fraction represented by the period $[T_i, T_{i+1}]$ ? Do they use the actual/$365.25$ so called "quants" convention ? Or do they rather use the market convention for the concerned Libor (forward) rate $L^1$ ?

What happens in the case I have another forward Libor rate $L^2$ (with its own market convention) possibly living in another market place, and that I am now modelling the vector $(L^1_t, L^2_t)_t$ with a quadratic covariation $\langle L^1, L^2\rangle_t = \rho dt$ ? In this case the $dt$ concerns $L^1$ and $L^2$ that have possibly different conventions. For sure, the same convention should be used for all $dt$'s (those in front of terms concerning $L^1$, those in front of terms concerning $L^1_t$ as well as those in front of "mixed" terms). Which convention do the practiotioners choose ? Actual/$365.25$ ? Or do they use the convention related to the numéraire they are simulating under ?

Remarks. 1) Of course my questions concern anything simulable, not only "Xibor" rates.

2) For sure the differences should be really small between an MC price (all other parameters remaining equal) calculated with actual/$365.25$ and an MC price calculated with the $30/360$ convention, but am I really interested in what the practitioners do. I have seen both used : actual/$365.25$ as well as numéraire convention, but I'd like to know if there's any consensus for this modelling choice.

## 1 Answer

For reporting outputs, you should certainly use the proper market convention (e.g., 30/360 and simple interest for USD Libor). However, internal model convention is a different matter. Many quants use Actual/365.25 or Actual/365, combined with continuous compounding. The day count convention only needs to ensure 1-to-1 mapping between date and rate/discount factor (incidentally, the 30/360 convention would NOT satisfy this requirement). Continuous compounding makes math much easier.

Once you have constructed your model, you can convert outputs to any convention needed to satisfy final reporting requirements.