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When calculating the PV of an interest rate derivative (IRD) that is linked to a rate index $-$ e.g. an interest rate swap $-$ we usually require the actual, or projected, index fixings in order to value all outstanding float cashflows, i.e. we need the actual or projected index fixing value $F_{t_{i}}$ with fixing date $t_{i}$ (not regarding settlement days). If the fixing date is in the past or today, $t_{i}\leq t$ , we look up historical fixings in the fixing history. If the fixing is in the future, $t_i>t$ ,we approximate it using the index' projection curve, i.e. by estimating a forward rate from $t_{i}$ to $t_{i+1}$ where the interim period equals the index' tenor ,e.g. 3M or 6M.

The question is how do we look up 'historical' fixings when valuing the IRD at a future point in time? As an example, say today is $t$ and we want to find the present value for a given valuation date $t+\mathrm{3M}$ in the future, for a floating cash flow indexed to 3M-EURIBOR whose underlying index would have been fixed at $t+\mathrm{1M}$ and that will pay at $t+\mathrm{4M}$. Clearly, the last available fixing in the market is observed at $t$ and we do not have any fixing history for dates $t+1D, t+2D\ldots$ Hence some sort of interpolation is required.

I'd argue that if we simulated the prices on a daily time grid, there would not be too much of a headache as we'd:

  1. calculate today's PV, write today's index fixing to the index's fixing history
  2. increase the valuation date by one business day
  3. if required, bootstrap curves / simulate a new curve
  4. either directly sample a fixing, or calculate a fixing from the curve from step 2.
  5. value the IRD.

What is the best way to go when the time grid is less granular, e.g. if we simulate in weekly, monthly, quarterly resolution? In this case, we will not have stored interim fixings in the history, making some sort of interpolation necessary. One idea would be to use the most recent (simulated) index fixing prior to some 'historical' date $t-\mathrm{3M}$ if no fixing for $t-\mathrm{3M}$ was available. I am wondering whether there exist an appropriate ansatz out there; a Brownian bridge maybe? Thanks for any pointers!

NB: I am aware that the actual PV effect of either ansatz will be negligible in most practical situations.

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  • $\begingroup$ Interpolation systematically lowers variance for interpolated point and that can influence pricing. $\endgroup$
    – emot
    Jul 2, 2021 at 12:44
  • $\begingroup$ @emot, thanks! Yep: that's the reason why I am asking for what people are commonly doing here. $\endgroup$ Jul 2, 2021 at 13:07
  • $\begingroup$ I'm curious: why can't you incorporate fixing days into the grid and just sample the grid points? That would be exact. Let's say you have values for month end but the fixing is in the mid month, you sample 2 days in a month. I am asking because I think this is the same number of calculations as with brownian bridge. With brownian bridge you would have to specify it in the way that it has mean and variance desired to the process modelled, but it would still be approximation. $\endgroup$
    – emot
    Jul 2, 2021 at 15:24
  • $\begingroup$ For one product, yes! But,commonly, you have multiple IRDs with various fixing dates - thence you will always ‘miss’ a couple of fixings I think $\endgroup$ Jul 2, 2021 at 15:51

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