Forward FX Pricing with Curves Anchored at Future Dates
Summary
The document asks how to interpolate forward FX values when the valuation date and the curves’ anchor date are moved into the future. It uses USD/MXN zero-rate curves to compare a forward calculated from today to a later expiry with one recalculated after moving the anchor date to an earlier expiry and setting spot to the previously observed forward. The author expects the two calculations to agree and reports seeing this behavior in commercial systems.
The proposed pricing relationship expresses the forward as spot multiplied by the foreign discount factor and divided by the domestic discount factor. The central issue is that ordinary price-curve interpolation may not preserve this relationship when the anchor date shifts: the short-period rate must produce the same terminal forward as the longer-period calculation. The document poses the interpolation problem but does not supply a solution, derivation, or empirical validation beyond the author’s software observations. Its example assumes valid market dates and does not discuss curve construction, day counts, or conventions.
Key ideas
- A forward FX value depends on spot and the ratio of foreign to domestic discount factors.
- Shifting the curve anchor date should preserve consistency between forwards calculated over adjacent horizons.
- Interpolating discount rates can produce different results from interpolating outright prices.
- The document identifies the consistency problem but leaves the interpolation method unresolved.
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Full text
# Any resources or literature on interpolation schemes for future dates?
# Any resources or literature on interpolation schemes for future dates?
I have a whole stack of the popular option trading/modelling books (Natenburg, Sinclair, Hull, etc.) None of them however address the idea of pricing or modelling values at a point in the "future". That is moving curve anchor dates or horizon dates forward along the various curves (vol, rates, forwards, etc.)
For instance suppose I have the zero rate curves for USDMXN and I want to run this experiment (assume all dates are valid market dates):
- Today, Nov 16, determine the forward outright point for a Dec 15 expiry and call that X.
- Today, Nov 16, determine the forward outright point for a Jan 14 expiry call that Y.
- Today, Nov 16, set anchor date for curves to Dec 15 and move current spot to X, (some systems seem to require that you do this, other's do not, effectively we want to move spot to the place on the forward curve that we previously witnessed it top be in step 1, such that when we interpolate to Jan 14 our starting point is equivalent.)
- Today, Nov 16, determine the current forward outright point for Jan 14 AS IF the current date was Dec 15 as given by the future anchor date and call that Z
My expectation is that Y == Z, and this is in fact what I see from a various of off the shelf products (including BBG).
Simple interpolation schemes work fine for standard price curves, but they don't work for discount curves, given that you need to find the rate value that allows appreciation in a shorter period (Dec to Jan) but with the same target (in this case Y) as the longer period (Nov to Jan).
edit:
My objective is this: Have an interpolation scheme for forward points that calculates the forward correctly, from zero rate curves, irrespective of where the anchor t is.
$$F_{t,T} = S_t\frac{D^f_{t,T}}{D^d_{t,T}} $$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.