How Libor Fixing Dates Affect Interest Rate Derivative Valuation
Summary
The document clarifies the distinction among a Libor fixing date, the start of the accrual period, and its end. In many markets, the rate is fixed shortly before the accrual period begins; it is a forward rate for the period from start to end. This means the start date remains relevant even though the rate is observed earlier. The response notes a market convention exception for GBP.
The timing distinction affects payoff valuation differently depending on payoff shape. For linear products such as swaps, the fixing lead time does not change the valuation in the way described. For nonlinear products such as caps, floors, and range accruals, volatility should run only until the fixing date, when the rate becomes known, rather than until the accrual start date. The explanation is a general timing principle; actual valuation still depends on the instrument’s terms and applicable market conventions.
Key ideas
- Libor may be fixed before the accrual period starts, while applying to the full period from its start to its end.
- The fixing date is when the forward rate for that accrual period becomes known.
- The lead time between fixing and accrual start does not affect linear swap payoffs in the same way as nonlinear payoffs.
- For nonlinear rate derivatives, the volatility horizon ends at fixing rather than at the accrual start.
- Market conventions differ, with GBP noted as an exception to the common two-day lead.
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Full text
# Fixing date, start date, end date in interest rate derivative valuation?
# Fixing date, start date, end date in interest rate derivative valuation?
I was reading a technical report by Hagan, which can be downloaded here on the valuation of accrual swaps and range notes.
It caught my attention that in the valuation he comments this:
Consider the $jth$ period of a coupon leg, and suppose the underlying indice is k-month Libor. Let $L(τ_{st})$ be the k-month Libor rate which is fixed for the period starting on date $τ_{st}$ and ending on $τ_{end(τ st)} = τ_{st}+k$ months. The Libor rate will be fixed on a date $τ_{fix}$, which is on or a few days before $τ_{st}$, depending on currency. On this date, the value of the contibution from day $τ_{st}$ is clearly:
\begin{align*} V(τ_{fix},τ_{st})=Z(τ_{fix},t_{j})\Bigg\{\frac{\alpha_jR_{fix}}{M_j} && if &&R_{min} \le L(τ_{st}) \le R_{max} \end{align*}
And $0$ if it's not in the interval.
My confusion here is that he defines and uses three dates in the valuation: fixing date, start date, and end date.
In the prospectus I've used for the valuation of this kind of instruments (range notes in particular) these three dates are stated (fixing date, start coupon date, ending coupon date). But for the valuation, I usually assume that the fixing date equals the start date $t_{fix}=t_{st}$ and discount my payoffs from the ending date. So the start date doesn't play a role in my valuation.
Is it common to have a fixing date prior to the start date and use both dates in the valuation? Would this mean that in each fixing date you use the forward rate that covers the period from the start date to the end date, rather than the actual rate observed on that date?
Much help appreciated
## Answer by Antoine Conze (score 1, accepted)
https://quant.stackexchange.com/a/37627
On most markets (GBP being a notable exception) the Libor fixes 2 days before its start date, so the Libor rate is actually a forward rate computed on $t_{\text{fix}}$ that covers the period $t_{\text{start}}$ to $t_{\text{end}}$.
It has no impact on valuation of products with linear payoffs (such as swaps), but it does have an impact on the valuation of derivatives with non linear payoffs (such as caps and floors, range accrual, etc.) because volatility should be applied only up to the fixing date, so that volatility terms in pricing formulas are $\sigma \sqrt{t_{\text{fix}} - t_0}$ instead of $\sigma \sqrt{t_{\text{start}} - t_0}$.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.