3.9.2 Bonds, the Yield Curve & Forward Rates

Basic Prerequisites: 01 · From Zero and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

This page is the curve toolkit: the exact relationships between bond prices, spot rates, instantaneous forward rates, and simple (LIBOR) forward rates - the objects every model and every instrument in this folder is built from. The practical objective: be able to move fluently between $P(t,T)$, $R(t,T)$, $f(t,T)$, and $L(t;T,S)$, and to build and price a plain-vanilla book - coupon bond, swap, cap, floor - directly off a zero curve.

The core intuition: a curve and a set of forward rates are the same information. The forward rate $f(t,T)$ is the rate you can lock in today for borrowing over the instant starting at $T$; the simple forward $L(t;T,S)$ is the same idea over a discrete interval. Pricing a floating-rate instrument is literally “apply the forwards that are already locked in by the curve.” This is why a swap’s floating leg has a model-free value: it is a static portfolio of forwards.

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2. Mathematical Ground Truth & Derivations

2.1 The discount curve and its rates (Björk Ch22; BM Ch1; Shreve Ch28)

The zero-coupon bond $P(t,T)$, $P(T,T)=1$, is the primitive. Its continuously-compounded spot rate, yield, instantaneous forward, and simple forward are:

$$R(t,T)=-\frac{\ln P(t,T)}{T-t},f(t,T)=-\frac{\partial }{\partial T}\ln P(t,T),P(t,T)=e^{-{\int }_t^{T}f(t,s)ds},$$ $$L(t;T,S)=\frac{1}{S-T}\left(\frac{P(t,T)}{P(t,S)}-1\right)\text{with }1+(S-T)L(t;T,S)=\frac{P(t,T)}{P(t,S)}.$$

The LIBOR rate is the simple rate on Actual/360; forward LIBOR over $[T,S]$ is exactly $L(t;T,S)$. The short rate is the zero-horizon forward: $r(t)=f(t,t)$.

Coupon bond = static portfolio of ZCBs (Björk eq. 22.13):

$$p^{cb}(t)=KP(t,T_n)+\sum _{i=1}^n{c}_{i}P(t,T_i),$$

and a floating-rate note is worth par at issue (“always trades at par”, BM Ch1) - its floating leg is exactly the locked-in forwards.

2.2 The forward swap rate (BM eq. 1.25; Björk Prop 22.7; Shreve Ch34)

The fixed rate that makes an IRS worth zero. For a swap from $T_{\alpha }$ to $T_{\beta }$ with annuity $C_{\alpha ,\beta }(t)={\sum }_{i=\alpha +1}^{\beta }{\tau }_{i}P(t,T_i)$:

$$R_{\alpha ,\beta }(t)=\frac{P(t,T_{\alpha })-P(t,T_{\beta })}{C_{\alpha ,\beta }(t)}.$$

On a flat curve this collapses to the flat rate (see §3). The forward swap rate is the martingale under the annuity (swap) numeraire ${\mathbb{Q}}^{\alpha ,\beta }$ - the key fact that powers swaption pricing (04 · Numeraire, HJM & Market Models).

2.3 Caps/floors and Black’s market formula (BM Ch1 1.26–1.29; Hull Ch29)

A cap is a portfolio of caplets; each caplet is a European call on a forward LIBOR. Under Black’s model (forward LIBOR lognormal), the caplet value is

$$Cpl(0)=P(0,T_i){\tau }_{i}Bl(K,F_i(0),v_i),Bl=FN(d_1)-KN(d_2),d_1=\frac{\ln (F/K)+\frac{1}{2}{v}^2{T}_{i-1}}{v\sqrt{T_{i-1}}}.$$

A floor is the corresponding put portfolio; a collar is long cap + short floor. Put–call parity for caps: $\text{cap}=\text{floor}+\text{swap(receive-float, pay-fixed }K)$ - a model-free identity (Hull Ch29; verified in §3).

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3. Computational Implementation - build the whole curve toolkit

Build a discount curve $P(0,T)=e^{-R(0,T)T}$ from input spot rates, derive all the forward rates, verify the $P=e^{-\int f}$ reconstruction, then price a cap and check the cap/floor put-call parity. Stdlib only.

The reconstruction error is at the level of numerical integration ($\sim 10^{-11}$), and the cap/floor put-call parity holds to $\sim 10^{-5}$ - both are pure identities, model-free. The forward rates are the curve.

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4. Failure Modes & First-Principles Breakdowns

1. Compounding-convention mixups. Simple (LIBOR, Actual/360) vs continuously-compounded vs annual yields are different numbers for the same curve. Quoting a continuously-compounded rate where the market uses simple Actual/360 misprices the payoff by exactly the $\frac{1}{\tau }\ln (1+\tau L)$ conversion.
2. The stochastic-$D$ vs bond-$P$ distinction. $D(t,T)=B(t)/B(T)=e^{-{\int }_t^{T}r}$ is random when rates are stochastic and is not equal to the bond price $P(t,T)$ (BM Ch1 precision note). Treating the discount factor as if it were the deterministic bond price is the single most common beginner error.
3. Swap/annuity measure confusion. The swap rate is a martingale under the annuity numeraire, not the $T$-forward or bank-account measure. Pricing a swaption by treating the swap rate as a $\mathbb{Q}$-martingale (bank account) is wrong; the annuity measure is the correct one (04).
4. Black’s model needs $F,K>0$. The lognormal caplet formula is undefined for zero/negative forwards or strikes - exactly the EUR/JPY 2015–2021 problem (05 · Failure Modes).

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5. References

• Brigo–Mercurio, Interest Rate Models
• Björk, Arbitrage Theory in Continuous Time
• Hull, Options, Futures, and Other Derivatives
• Shreve, Stochastic Calculus for Finance I

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Short-Rate Models · Index Hub
• Base: Stochastic Calculus & Itô