Pricing a Vanilla Swap with a Forward Start Date

Summary

The document derives the par rate for a vanilla EURIBOR swap whose start date coincides with its trade date. It expresses the swap rate as the floating leg’s present value divided by the fixed leg annuity. The floating leg is represented using forward rates for each accrual period and discount factors for the corresponding payment dates; the fixed leg is represented by its accrual fractions and discounted cash flows. The derivation relies on the stated martingale property of each forward rate under its payment-date forward measure.

It then asks how this calculation changes when the swap begins after the trade date, as in a convention where the start date is two business days later. The document does not provide that adjustment or discuss the treatment of the intervening period. Its formula and assumptions are limited to the stated setup, so the forward-start case remains an open question rather than a completed derivation.

Key ideas

• For a swap starting on the trade date, the par rate equals floating leg present value divided by fixed leg annuity.
• The fixed leg annuity sums discounted accrual fractions across payment dates.
• The floating leg valuation uses forward rates and discount factors for its payment dates.
• The stated argument relies on forward rates being martingales under the relevant forward measures.
• The document poses but does not solve the adjustment for a swap that starts after the trade date.

Tags

# How is swap rate calculated for a vanilla swap when there is a lag between the trade date and the start date


# How is swap rate calculated for a vanilla swap when there is a lag between the trade date and the start date












Take a vanilla EURIBOR swap and suppose that the start date of the swap is equal to the trade date. To compute the swap rate, you say that the value of the swap at the trade date must be zero, which finally gives you the following expression of the swap rate :

$$s_{T_{trade}} = \frac{\sum_{j=1}^N \delta_j^{fl} P_{T_j^{fl}}^{disc} L_{T_{trade}}^{[T_{j-1}^{fl}, T_{j}^{fl}]}}{\sum_{i=1}^M \delta_i^{fix} P_{T_i^{fix}}^{disc}}$$

where $P_t^{disc}$ is the value (at trade date $T_{trade}$) of the discount curve zero-coupon and where the $T_j^{fl}$ dates of the floating leg and the $T_i^{fix}$ dates of the fixed leg are computed from the trade date $T_{trade}$, such that $T_{trade} = T_0^{fl} = T_0^{fix}$. The computation is simple : the present value of the fixed leg is the swap strike time the annuity which is the denominator in the previous expression, while the present value of the floating leg is equal to the numerator of the previous expression, as each $L_{t}^{[T_{j-1}^{fl}, T_{j}^{fl}]}$ is a martingale under the $T_j^{fl}$-forward measure.

How is the swap rate computed and how does the previous computation change when the start date $T_{start}$ of the swap is equal to the trade date plus two business days (which is the convention for EURIBOR swaps) so that $T_{start} = T_0^{fl} = T_0^{fix} > T_{trade}$ ?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.