# How do you deal with Inflation lag in a MC simulation?

Consider the UK RPI index. This index is published every month around the 15th (give or take a few days). The publication refers to the RPI index of the month before, so there is a lag of a few weeks between when the RPI is "set" and when it is published. How do I deal with this lag?

To be more specific, I'm performing a Monte Carlo simulation of the RPI index using the Jarrow-Yildirim model. The inflation process is determined through a geometric Brownian motion:

$$I(T) = I(t) \exp\left( \int_t^T(n(t) - r(t))dt - \frac{1}{2} \sigma^2 (T-t) + \sigma_I (W_I(T) -W_I(t))\right)$$

where $n(t)$ and $r(t)$ are the nominal and real rates. These follow a Hull-White process in the JY model.

So suppose I gather all my data on June 1st, i.e. discount curves and perhaps other data to calibrate the JY model. This sets the drift term of the real and nominal process. But on June 1st the "last known RPI" is that of April. How do I account for this lag?

My view: I simply perform a simulation starting from April. I interpret all data that I have as if it was gathered in April as well. In some sense I "roll back the clock" on my data, so I can make it match with the RPI index. Does that make sense?

My problem with this is is that 1) any valuations using this MC simulation give prices that are valid "in April", but I like to have a "June 1st" price. 2) I'm not sure if the data of June 1st can simply be rewind to April. Can prices of ZC inflation bonds, YoY swaps and caps be interpreted like that (i.e. simply shift all the paydates)?

In inflation world, the deal payoff is always based on a certain lag convention. That is, the value $I(T)$ always refers to a published index level several months ago or is interpolated based on those published index levels.
For your simulation, except for your model parameters, you also need the spot level $I(t_0)$ on the valuation date $t_0$, which can be referred or interpolated from the published index levels as above based on the respective lag convention.