# 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

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 $\sigma \sqrt{t_{\text{start}} - t_0}$.
• Thanks, I get what you are saying. For linear payoffs the Libor rate would be a martingale under $Q_{tend}$ regardless of it being observed at $t_{fix}$ or at $t_{start}$. Also the period $t_{start}$ and $t_{end}$ doesn't changes, what changes is the observation time. And for non linear payoffs the volatility is applied only up to the fixing time since one would usually assume a model (e.g. log-normal) and at one point solve using an integral that would cover the period from $t_0$ to $t_{fix}$ which ends up affecting the volatility (bit of over-simplification here). – Aldo Shumway Jan 8 '18 at 23:19
• I think I've only got one question left about this. What happens when the fixing date comes? Is the Libor observed in the market for that day fixed for the calculation of the coupon? or is another forward rate calculated from $t_{start}$ to $t_{end}$ ? – Aldo Shumway Jan 8 '18 at 23:32