Valuing an Overnight Index Swap’s Floating Leg

# PV of the Floating Side of an "Overnight Index Swap" (at the fixing Date)


# PV of the Floating Side of an "Overnight Index Swap" (at the fixing Date)












I have a mathematical / theoretical question regarding the PV of an Overnight Index Swap (Floating Side) at the time of fixing.

Starting from this question:

How to compute Overnight Index Swap (OIS) fixed rate?

--> At each Fixing Date of the Floating Cash Flows will be the Floater PV at par? Similar to a Floating Rate Note? If I look at the math I would guess that you don't have the same effect (that counter and denominator are the same).

EDIT: To be more precise: A Floating Rate Note will be at time of fixing pricing at par (Cash Flow Term and Discount Term cancel each other out of the equation). The same effect doesn't hold for an Overnight Index Swap - Floating Side (because calculation method of the interest is in arrears and a compounded geometrical mean)?

Thank you very much in advance.

Best Regards.

## Answer by ir7 (score 1)

https://quant.stackexchange.com/a/57205

Let's assume that the collateral rate on cash equals the overnight rate, that we have a schematic (lined/tiled up accrual periods and payments dates) strip of dates/times $T_0<T_1<\ldots <T_n$, accrual factor $\tau_t := \tau(t-1,t)$, and $c_t$ collateral rate at $t$ (overnight $t-1$ to $t$).

The floating coupon is then:

$$ \prod_{s=T_{i-1}}^{T_i}\left(1+\tau_sc_s \right) -1. $$

Let's further assume that we can live with approximating daily compounding by continuous compounding:

$$ \prod_{s=T_{i-1}}^{T_i}\left(1+\tau_sc_s \right) -1 \approx \mathrm{e}^{\int_{T{i-1}}^{T_i}c_sds} -1. $$

Then the time-$0$ present value of this strip of floating coupons is:

$$\sum_{i=1}^n \mathbf{E}^Q\left[\mathrm{e}^{-\int_{0}^{T_i}c_sds} \left(\mathrm{e}^{\int_{T{i-1}}^{T_i}c_sds} -1 \right)\right] = \sum_{i=1}^n \left( \mathbf{E}^Q\left[\mathrm{e}^{\int_{0}^{T_{i-1}}c_sds}\right] - \mathbf{E}^Q\left[\mathrm{e}^{\int_{0}^{T_{i}}c_sds}\right]\right) $$ $$ = \mathbf{E}^Q\left[\mathrm{e}^{\int_{0}^{T_{0}}c_sds}\right] - \mathbf{E}^Q\left[\mathrm{e}^{\int_{0}^{T_{n}}c_sds}\right], $$ that is, the difference of collateralized discount factors at stub time and last payment time (under assumptions made, we do have the 'telescopic' effect that makes FRN's 'at par').

Note: Let current time be $T_j$ (we are inside the strip timeline, not before it; $j\geq 1$). Under assumptions above, $T_j$ is also the fixing date (or rather the publishing date of the compounded index based on already fixed overnight rates) of the value of the $j$-th floating coupon. The current PV of the residual floating coupon strip will be:

$$\sum_{i=j+1}^n \mathbf{E}^Q\left[\mathrm{e}^{-\int_{T_j}^{T_i}c_sds} \left(\mathrm{e}^{\int_{T{i-1}}^{T_i}c_sds} -1 \right)\right] = \sum_{i=j+1}^n \left( \mathbf{E}^Q\left[\mathrm{e}^{\int_{T_j}^{T_{i-1}}c_sds}\right] - \mathbf{E}^Q\left[\mathrm{e}^{\int_{T_j}^{T_{i}}c_sds}\right]\right) $$ $$ = 1 - \mathbf{E}^Q\left[\mathrm{e}^{\int_{T_j}^{T_{n}}c_sds}\right]. $$

Note 2: If this strip of floating coupons were part of an FRN, we would add one extra cash flow to it at $T_n$ consisting of the reimbursement of the principal (set to $1$ here) of the note. So the PV of the extended strip would then show the strip being 'at par': $$ 1 - \mathbf{E}^Q\left[\mathrm{e}^{\int_{T_j}^{T_{n}}c_sds}\right] + \mathbf{E}^Q\left[\mathrm{e}^{\int_{T_j}^{T_{n}}c_sds} \cdot 1\right] =1. $$

Note 3: Under the same assumptions, time-$0$ par swap rate is then:

$$ K = \frac{P^{ois}(0,T_0) - P^{ois}(0,T_n)}{\sum_{i=1}^n \delta_i P^{ois}(0,T_i)},$$

where $P^{ois}(0,T):= \mathbf{E}^Q\left[\mathrm{e}^{-\int_{0}^{T}c_sds}\right]$, $\delta_i=\tau(T_{i-1},T_i)$.