3.9 Interest Rate & Term Structure

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Multivariable Calculus. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

In equity options the short rate $r$ is a scalar input to a formula. In fixed income the rate is the underlying. The whole object of study is the term structure - the continuum of yields / forward rates across maturities - and the derivative products written on it: bonds, interest-rate swaps (IRS), caps/floors, and swaptions. The pricing question is different from Black–Scholes in one structural way: you cannot hedge a rate derivative with a single tradable whose drift is observable, because the drift (the rate itself) is what moves. This is why interest-rate modelling is built on two distinctive pillars: the market price of risk (short-rate models) and the change of numeraire (forward-measure pricing), which are absent from a constant- $r$ BSM world.

This folder is the model topic-folder for the Kwant-Atlas build. It is a hub: it (a) gives the fast formula lookup below, and (b) routes you to six sub-pages that walk from raw intuition through bonds/yield curves/forward rates, short-rate models, the numeraire/HJM/market-model machinery, failure modes, and advanced extensions (multi-curve, the smile in rates, calibration).

The one-sentence essence. “Price a rate claim as the discounted expectation of its payoff under the right numeraire - the bank account for generic claims, the $T$ -bond (forward measure) for forward-LIBOR products, the annuity for swaptions - where short-rate models fix $r$ ‘s dynamics via the market price of risk, and HJM/market models build the whole forward curve and let each forward rate be a (near-)martingale under its own measure.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). All formulas below are transcribed from Brigo–Mercurio (2006, 2nd ed.) and Björk (2009, 3rd ed.), cross-checked against Shreve and Hull; the numbers in the check column were re-executed and reproduced exactly from the verified corpus (see §3).

Notation: $P (t, T)$ zero-coupon bond (ZCB) paying 1 at $T$ , $r_{t}$ short rate, $B (t) = e^{\int_{0}^{t} r_{s} d s}$ bank account, $f (t, T) = - \partial_{T} ln P (t, T)$ instantaneous forward, $L (t; T, S) = \frac{1}{τ} (\frac{P ( t , T )}{P ( t , S )} - 1)$ simple (LIBOR) forward, $F_{k} (t)$ forward-LIBOR over $[T_{k - 1}, T_{k}]$ , $R_{α, β} (t)$ forward swap rate, $C_{α, β} (t)$ annuity/PVBP, $τ$ accrual (e.g. 0.5 semi, 0.25 quarterly).

The building blocks:

Quantity	Formula	Verified check
Stochastic discount	$D (t, T) = B (t) / B (T) = e^{- \int_{t}^{T} r_{s} d s}$	random when $r$ stochastic; $\neq = P (t, T)$ in general
ZCB price (bank-account RN)	$P (t, T) = E^{Q} [e^{- \int_{t}^{T} r_{s} d s} ∣ F_{t}]$	-
Instantaneous forward	$f (t, T) = - \partial_{T} ln P (t, T)$ ; $P (t, T) = e^{- \int_{t}^{T} f (t, s) d s}$	-
Simple forward (LIBOR)	$L (t; T, S) = \frac{1}{τ} (\frac{P ( t , T )}{P ( t , S )} - 1)$	flat 4%: $L (0; 0, 0.5) = 4.0403%$
Vasicek $d r = a (b - r) d t + σ d W$	$P (t, T) = A (t, T) e^{- B (t, T) r}$ , $B = \frac{1 - e ^{- a (T - t)}}{a}$	$a = .15, b = .05, σ = .01, r = .04 : P (0, 5) = 0.807678$
CIR $d r = a (b - r) d t + σ r d W$	$P = A e^{- B r}$ , $h = a^{2} + 2 σ^{2}$ , $B = \frac{2 ( e ^{h τ} - 1 )}{( h + a ) ( e ^{h τ} - 1 ) + 2 h}$	$a = .2, b = .05, σ = .05, r = .04 : P (0, 5) = 0.804696$
Hull–White $d r = [θ (t) - a r] d t + σ d W$	$θ (t) = \partial_{T} f^{M} (0, t) + a f^{M} (0, t) + \frac{σ ^{2}}{2 a} (1 - e^{- 2 a t})$	flat 4%: $θ (0) = .0040, θ (5) = .0047$
HJM drift (Q)	$α (t, T) = σ (t, T) \int_{t}^{T} σ (t, s) d s$	$σ$ free, $α$ fixed
Caplet = Black	$Cpl = P (0, T_{i}) τ [FN (d_{1}) - K N (d_{2})]$ , $d_{1} = \frac{l n ( F / K ) + \frac{1}{2} v ^{2} T}{v T}$	$F = 4.0403%, K = 4%, v = .2, T = 1, τ = .5 : 0.001637$
Black swaption	$PS = C_{α, β} (0) [R (0) N (d_{1}) - K N (d_{2})]$	5y-into-5y ATM $v = .15$ : 0.023902
Swap rate	$R_{α, β} (t) = \frac{P ( t , T _{α} ) - P ( t , T _{β} )}{C _{α, β} ( t )}$ , $C_{α, β} (t) = \sum τ_{i} P (t, T_{i})$	flat 4% 5y semi: 4.0403%

Critical numeraire facts (BM Ch2; Björk Ch26; Shreve Ch33). A numeraire is any strictly positive asset; price/numeraire is a martingale under that numeraire’s measure, and the risk-neutral price is invariant under change of numeraire. The forward measure $Q^{T}$ (numeraire $P (t, T)$ ) makes the instantaneous forward $f (t, T)$ a martingale, while the forward LIBOR $L (t; T, S)$ is a martingale under $Q^{S}$ (numeraire $P (t, S)$ ) - the two maturity indices differ (Björk Lem. 26.10; BM Prop 2.5.1). The swap (annuity) measure $Q^{α, β}$ makes $R_{α, β}$ a martingale. Forward price $= E^{Q^{T}} [Y]$ ; futures price $= E^{Q} [Y]$ ; they coincide iff $r$ is deterministic (Björk Ch29 correction).

3. Computational Implementation - the formula engine

This runs on the standard library only (math.erf gives the exact normal CDF). It reproduces every verified number above.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

One-factor perfect correlation - a single Brownian short rate drives every maturity (corr = 1), so curve steepeners / yield-spread products are mispriced (the structural route into two-factor models).
Negative rates break lognormal Black - Black-76 needs $F, K > 0$ ; EUR/JPY 2015–2021 forced Bachelier (normal) / shifted-lognormal machinery.
Lognormal instantaneous forwards explode - HJM with $σ \propto f$ has drift $\sim f^{2}$ ; the solution blows up before $T$ , which is exactly why market models use lognormal simple (LIBOR) rates instead.
HW/calibration differentiation instability - exact-fit $θ (t)$ needs numerical derivatives of the market forward curve; noisy curves break the fit.

5. References

Brigo, Damiano & Mercurio, Fabio: *Interest Rate Models
Björk, Tomas: Arbitrage Theory in Continuous Time (3rd ed., 2009)
Shreve, Steven E.: Stochastic Calculus for Finance I
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)

6. Connected Graph Bridges

Foundational base: Stochastic Calculus & Itô · No-Arbitrage Foundations & Binomial Trees
Sibling topic: Black–Scholes–Merton (the constant- $r$ seed this folder generalises)
Sub-pages (in-folder): 01 From Zero · 02 Bonds, Yield Curve & Forward Rates · 03 Short-Rate Models · 04 Numeraire, HJM & Market Models · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

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3.9 Interest Rate & Term Structure

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the formula engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.9.1 Interest Rates & Term Structure from Zero

3.9.2 Bonds, the Yield Curve & Forward Rates

3.9.3 Short-Rate Models

3.9.4 Numeraire Change, the HJM Framework & Market Models (LFM/LMM, BGM)

3.9.5 Failure Modes & Real-World Practice

3.9.6 Advanced Extensions