3.9.1 Interest Rates & Term Structure from Zero

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

This page builds the why of interest-rate modelling with no prior fixed-income knowledge needed. The objective is one idea: in fixed income the risk-free rate is not a constant - it is the thing you are trying to price derivatives on, and it varies across time and across maturities. Every model in this folder is an answer to the question “how do I describe the whole continuum of rates, and how do I price a claim whose payoff depends on one of them?”

Start with the dumbest question: why is there a “term structure” at all? A dollar paid today is worth more than a dollar paid in a year, but the compensation per year is not flat - lending for 1 month pays a different (annualised) rate than lending for 30 years. The set of rates for all maturities is the term structure of interest rates, a.k.a. the yield curve or zero curve. Three steps:

1. The fundamental building block is a zero-coupon bond (ZCB). $P(t,T)$ is the price at $t$ of a risk-free promise to pay 1 at $T$. Everything - coupon bonds, swaps, caps, swaptions - is a static portfolio of ZCBs (a coupon bond is $\sum c_{i}P(t,T_i)$; a floating-rate note is worth par). Learn to price $P(t,T)$ and you can price all of fixed income.

2. A rate is a log-slope of the bond-price curve, and there is a whole hierarchy of rates. The continuously-compounded spot rate is $R(t,T)=-\ln P(t,T)/(T-t)$; the instantaneous forward rate is the derivative $f(t,T)=-{\partial }_T\ln P(t,T)$; the simple (LIBOR) forward rate is the discrete analogue $L(t;T,S)=\frac{1}{\tau }(\frac{P(t,T)}{P(t,S)}-1)$. These are all different views of the same curve, and knowing any one of them is knowing all of them.

3. The rate you borrow at today - the short rate $r_t$ - is the single engine. It is $r_t=f(t,t)$, the instantaneous forward rate at the shortest horizon. If you can describe how $r_t$ moves (a “short-rate model”), then every bond is just the discounted expectation of 1 under that description, and every rate derivative is a claim on $r$‘s path.

The one-paragraph essence. Fixed-income quants do not model a stock; they model the discount factor. The price of a $T$-bond is the market’s answer to “what is the value of 1 at time $T$ worth today,” and every derivative written on rates is a bet on the path of that discount factor. The models divide into short-rate models (specify $r_t$‘s SDE directly) and market models (specify the forward curve / each forward rate directly).

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2. Mathematical Ground Truth & Derivations

The three facts that make term-structure modelling non-trivial.

Fact 1 - $P(t,T)$ is an expectation under the risk-neutral measure. Define the bank account $B(t)=e^{{\int }_0^t{r}_{s}ds}$, the price of rolling $1$ at the short rate. Then (BM Ch2; Shreve Ch28) the no-arbitrage value of a $T$-bond is

$$P(t,T)={\mathbb{E}}^{\mathbb{Q}}\left[e^{-{\int }_t^T{r}_{s}ds}|{\mathcal{F}}_t\right].$$

This is the fundamental pricing relation. The whole “term-structure equation” and the market price of risk come from asking which dynamics of $r$ to use inside this expectation - the drift under $\mathbb{Q}$ is what the model specifies.

Fact 2 - forwards are the log-slope; rates are linked by identities, not by models. The relations

$$f(t,T)=-\frac{\partial }{\partial T}\ln P(t,T),P(t,T)=e^{-{\int }_t^{T}f(t,s)ds},r(t)=f(t,t)$$

are pure definitions, model-free. So is the simple-forward relation $1+\tau L(t;T,S)=P(t,T)/P(t,S)$. What is not model-free is the dynamics of any of these objects - that is what a model pins down.

Fact 3 - in a one-factor world, one Brownian motion moves the whole curve. If $r$ is driven by a single $dW$, then all forward rates of all maturities are functions of the same $r$, so at any instant they are perfectly correlated (corr = 1). Real curves move by level/slope/curvature (roughly three PCA factors). This single fact motivates two-factor models (BM Ch4) and is the first real failure mode (05 · Failure Modes).

The risk-free-rate equation for a bond portfolio. Bond prices satisfy a term-structure PDE: writing $P=F(t,r;T)$,

$$F_t+(\mu -\lambda \sigma )F_r+\frac{1}{2}{\sigma }^2{F}_{rr}-rF=0,F(T,r;T)=1,$$

where $\lambda$ is the market price of risk - the excess return per unit of volatility that the market demands for bearing short-rate risk. Different choices of $\lambda$ give different $\mathbb{Q}$ and hence different bond prices (Björk Prop 23.2). This is the exact analogue of the BSM PDE, but with the drift $\mu$ not killed - only replaced by the risk-adjusted drift $\mu -\lambda \sigma$.

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3. Computational Implementation - the discount factor in motion

Simulate the short rate $r_t$ under a model and Monte Carlo the discount factor to recover a bond price - the most direct way to see Fact 1. Stdlib only.

The closed form (03 · Short-Rate Models) gives $0.807678$; the MC converges to it. This is the whole job: pick $r$‘s dynamics, integrate $e^{-\int r}$, get the bond, then price any claim on the rate.

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4. Failure Modes & First-Principles Breakdowns

1. “The yield curve is a constant” - the beginner’s assumption from equity pricing. It ignores term-structure risk: a 30-year bond and a 3-month bill do not move together. Any short-rate model is overkill-free but the constant-$r$ BSM shortcut is simply wrong for long-dated or rate-linked products.
2. Discounting under the wrong measure. $P(t,T)={\mathbb{E}}^{\mathbb{Q}}[e^{-\int r}]$ - not ${\mathbb{E}}^{\mathbb{P}}$. The physical drift of $r$ does not enter bond prices directly; only the $\mathbb{Q}$-drift (with the market price of risk folded in) does. Confusing the two misprices everything.
3. Forgetting $r_t=f(t,t)$ and the log-slope identities. These are definitions, not model output. Building a “model” that violates $P(t,T)=e^{-{\int }_t^{T}f}$ (e.g. inconsistent interpolation) is internally inconsistent before any dynamics are chosen.
4. One factor, one shape. A single Brownian motion can only move the curve in parallel (and one “direction of tilt” is forced by the model’s mean reversion). It cannot reproduce independent level/slope/curvature moves - the seed of the two-factor extension and of the failure page.

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5. References

• Björk, Arbitrage Theory in Continuous Time
• Brigo–Mercurio, Interest Rate Models
• Shreve, Stochastic Calculus for Finance I

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô · No-Arbitrage & Binomial Trees
• Continue: 02 · Bonds, Yield Curve & Forward Rates · Index Hub
• Sibling: Black–Scholes–Merton (constant $r$ vs term structure)