Valuing Floating-Rate Bonds Under Stochastic Interest Rates
Summary
The document explains how to value a floating-rate bond when both interest rates and future coupon amounts are uncertain. For a coupon period, it defines the rate using the price of a zero-coupon bond at the rate-setting date, then changes to a forward measure associated with the payment date to calculate the coupon’s present value.
The derivation shows that the value of one floating coupon is the difference between the current prices of zero-coupon bonds maturing at the period’s start and end. Adding these coupon values and the bond’s principal repayment gives the note’s value. The result is model-free under the stated floating-rate convention: the derivation does not require interest rates to be deterministic. It does not address credit risk, transaction costs, or alternative coupon conventions, so those features would require additional valuation inputs.
Key ideas
- A floating coupon can be defined from the zero-coupon bond price observed when the rate is set.
- Changing to the payment-date forward measure simplifies the expected discounted coupon calculation.
- The present value of one floating coupon equals the difference between two zero-coupon bond prices.
- A floating-rate note’s value combines its coupon payments with repayment of principal.
- The result assumes the stated rate convention and does not account for credit risk or other contract features.
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# Pricing bonds of float coupon rate by stochastic interest rate
# Pricing bonds of float coupon rate by stochastic interest rate
So I am not sure whether the following pricing of the bond is possible. Given the stochastic interest rate, one wants to price the bond with the floating coupon rate or the coupon rate being unknown. How should one price this bond if both forward interest rate and coupon rate are not known in the sense that both are random.
## Answer by Gordon (score 2)
https://quant.stackexchange.com/a/22816
Consider the calculation period $[T_1, T_2]$ and the floating coupon rate \begin{align*} L(T_1; T_1, T_2) = \frac{1}{T_2-T_1}\left(\frac{1}{P(T_1, T_2)} -1 \right) \end{align*} set at $T_1$ and paid at $T_2$, where $P(t, u)$ is the price at time $t$ of a zero-coupon bond with maturity $u$ and unit face amount.
Let $B_t= \exp\left(\int_0^t r_s ds \right)$ the money market account value to time $t$. Moreover, let $Q$ be the risk-neutral measure and $Q_{T_2}$ be the $T_2$-forward measure. Then \begin{align*} \frac{dQ}{dQ_{T_2}}\big|_{\mathcal{F}_t} = \frac{P(0, T_2)B_{t}}{P(t, T_2)}. \end{align*} Moreover, the value of the floating coupon payment is given by \begin{align*} E_Q\left(\frac{L(T_1; T_1, T_2) \times (T_2-T_1)}{B_{T_2}} \right)&=E_{Q_{T_2}}\left( \frac{dQ}{dQ_{T_2}}\big|_{\mathcal{F}_{T_2}}\frac{L(T_1; T_1, T_2) \times (T_2-T_1)}{B_{T_2}}\right)\\ &=P(0, T_2) E_{Q_{T_2}}\left( L(T_1; T_1, T_2) \times (T_2-T_1)\right)\\ &=P(0, T_2) E_{Q_{T_2}}\left( \frac{1}{P(T_1, T_2)} -1 \right)\\ &=P(0, T_2) E_{Q_{T_2}}\left( \frac{P(T_1, T_1)}{P(T_1, T_2)} -1 \right)\\ &=P(0, T_2)\left( \frac{P(0, T_1)}{P(0, T_2)} -1 \right)\\ &=P(0, T_1) - P(0, T_2). \end{align*} For a floating rate note, or bond, the value is the sum of the values of all coupon payments and the value of the notional payment at maturity. Here the valuation is model-free, no matter the interest rate is deterministic or stochastic.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.