Simulating Interest Rate Index Fixings on Coarse Time Grids
Summary
The document explains how interest rate derivatives use historical index fixings for dates already observed and projection curves to estimate future fixings. It then considers valuing a derivative at a simulated future date, when the fixing history available today does not contain observations for the intervening dates. This creates a gap when simulations use weekly, monthly, or quarterly steps rather than daily steps.
A daily simulation could record each fixing as it occurs and use it in later valuations. For coarser grids, the author considers carrying forward the most recent simulated fixing or using an interpolation approach such as a Brownian bridge. No method is established or tested, and the author notes that the present value impact may often be small. The discussion therefore frames a modeling choice rather than providing comparative evidence or a definitive solution.
Key ideas
- Interest rate derivative valuations use observed fixings for past dates and projection curves for future dates.
- A future valuation date can require fixings that are not available in today’s historical fixing record.
- Daily simulations can store intermediate fixings, while coarse grids leave gaps that require an approximation.
- The document suggests carrying forward recent fixings or considering a Brownian bridge, but does not evaluate either method.
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# How to simulate interest rate index fixings?
# How to simulate interest rate index fixings?
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:
- calculate today's PV, write today's index fixing to the index's fixing history
- increase the valuation date by one business day
- if required, bootstrap curves / simulate a new curve
- either directly sample a fixing, or calculate a fixing from the curve from step 2.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.