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Valuing Swap Floating Payments with Forward Libor Rates

Article Quant Q&A · Author: Ramzy
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Summary

The document explains how a swap’s floating-leg value can be written using forward Libor rates rather than future realized fixings. Its method starts with the risk-neutral present value of a payment, changes to the forward measure associated with a zero-coupon bond maturing on the payment date, and uses the martingale property of the relevant forward rate. The expected future fixing can then be represented by its current forward value, with the payment discounted by that bond price.

The application sums the discounted expected floating coupons across reset periods. The measure and discount bond must correspond to each coupon’s payment date; the exposition’s generic notation is simplified, and its final displayed expression uses a discount factor whose maturity appears inconsistent with the payment date described just above it. The derivation is a pricing identity under its stated measure assumptions, not a trading signal or empirical result.

Key ideas

  • A future cash flow can be valued by discounting its expectation under the forward measure tied to its payment-date bond.
  • The corresponding forward rate is a martingale under that forward measure.
  • A floating coupon’s expected fixing can therefore be replaced by the current forward rate in the valuation.
  • Each coupon should be discounted to its actual payment date.

Tags

Fixed incomeDerivatives pricing
Full text
# How to change the Libor rate to Forward Libor rate in Swap?


# How to change the Libor rate to Forward Libor rate in Swap?












The realised PV of a swap (notional is 1 ) is :

$Swap(t)=\sum^n_{i=1} \tau_i \times D(t,Ti) \times (L(Ti, Ti, Ti+ \tau_i) - K)$

How do we get the expression with forward rate :

$Swap(t)=\sum^n_{i=1} \tau_i \times D(t,Ti) (L(t,Ti,Ti+ \tau_i) - K)$

I know we are supposed to used forward measure but I don't see how ?

Thank you !

## Answer by byouness (score 2, accepted)

https://quant.stackexchange.com/a/49788

#### General fact:

From a mathematical standpoint, we can write the PV of a flow to be received at $T$ as the value of its expectation under the $T$-forward measure (which is also the value of the forward at $t$: $F(t, T)$) discounted using the zero-coupon bond. We can show this by changing measures from the risk-neutral measure $\mathbb{Q}$ to the $T$-forward measure $\mathbb{Q}_T$ associated to the zero-coupon bond $P(t,T)$ as numéraire.

Indeed, using the numéraire change formula, we can write: $$ \begin{aligned} PV(t) &= \mathbb{E}_t^\mathbb{Q} \left[ e^{-\int_t^T r(u) du } X(T)\right] \\ &= \mathbb{E}_t^\mathbb{Q} \left[ e^{-\int_t^T r(u) du } F(T, T)\right] \\ &= P(t, T) \mathbb{E}_t^{\mathbb{Q}_T} \left[F(T, T) \right]\\ &= P(t, T) F(t, T)\\ \end{aligned} $$

In the last step, we used the fact that the forward price is a martingale under the $T$-forward measure: $$ F(t, T) = \mathbb{E}_t^{\mathbb{Q}_T} \left[F(T, T) \right] = \mathbb{E}_t^{\mathbb{Q}_T} \left[X(T) \right] $$

#### Application to the floating leg of the swap:

Let us apply this to the floating leg. The payment date of each flow is $T_i + \tau_i$, sowe switch from the risk-neutral measure $\mathbb{Q}$ to the $(T_i + \tau_i)$-forward measure $\mathbb{Q}_{T_i + \tau_i}$:

$$ \begin{aligned} FloatingLeg(t) &= \sum^n_{i=1} \tau_i \times P(t, T_i + \tau_i) \times \mathbb{E}_t^{\mathbb{Q}_{T_i + \tau_i}} \left[L(T_i, T_i, T_i + \tau_i)\right] \end{aligned} $$

Under $\mathbb{Q}_{T_i + \tau_i}$, the Libor forward $\left(L(t, T_i, T_i + \tau_i)\right)_t$ is a martingale, and as a result:

$$ FloatingLeg(t) = \sum^n_{i=1} \tau_i \times P(t, T_i + \tau_i) \times L(t, T_i, T_i + \tau_i) $$

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.