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

Basic Prerequisites: 03 · Short-Rate Models and Stochastic Calculus & Itô’s Lemma.

---

1. Intuition & Practical Objective

This page is the mathematical engine room of interest-rate modelling: the two tools that make rate derivatives tractable. The practical objective: understand why forward-LIBOR products are priced by Black’s formula under the forward measure, and why swaptions need the annuity measure, all via a single device - the change of numeraire.

Two structural ideas, both with no constant-$r$ analogue:

1. A numeraire is just a unit of account you choose to make the payoff simple. Under the risk-neutral measure $\mathbb{Q}$ (numeraire = bank account $B$), a $T$-payoff needs the messy $\mathbb{E}[e^{-{\int }_0^{T}r}\cdot ]$ integral. Switch to the $T$-forward measure ${\mathbb{Q}}^T$ (numeraire = the $T$-bond $P(t,T)$) and the same claim prices as $P(t,T){\mathbb{E}}^{{\mathbb{Q}}^T}[\text{payoff}]$ - no discount integral, and any forward rate spanning to $T$ is a martingale under ${\mathbb{Q}}^T$. That martingale property is what makes Black’s lognormal formula exact.

2. Model the object you price, not a hidden state. Short-rate models specify $r$ and derive the curve. HJM instead specifies the whole forward curve $df(t,T)=\alpha (t,T)dt+\sigma (t,T)dW$ - which fits today’s curve by construction - and the no-arbitrage drift condition then fixes $\alpha$ from $\sigma$ alone. But lognormal instantaneous forwards explode (see §2.3). The fix is the LIBOR market model (LFM/LMM = BGM): model each simple forward LIBOR $F_k$ as lognormal under its own measure, giving exact Black caplets and avoiding the explosion.

---

2. Mathematical Ground Truth & Derivations

2.1 Change of numeraire & the forward measure (BM Ch2; Björk Ch26; Shreve Ch33)

A numeraire is any strictly positive asset. Fact One (BM Ch2): price/numeraire is a martingale under that numeraire’s measure; Fact Two: the risk-neutral price is invariant under change of numeraire. The Radon–Nikodym derivative between bank-account ($\mathbb{Q}$) and forward (${\mathbb{Q}}^T$) measures is

$$\frac{d{\mathbb{Q}}^T}{d\mathbb{Q}}=\frac{P(T,T)/B(T)}{P(0,T)/B(0)}=\frac{1}{B(T)P(0,T)}.$$

Pricing a $T$-payoff $X$:

$${\Pi }_t(X)=B(t){\mathbb{E}}^{\mathbb{Q}}\left[\frac{X}{B(T)}\right]=P(t,T){\mathbb{E}}^{{\mathbb{Q}}^T}\left[X\right].$$

The Girsanov kernel between two numeraires is their volatility difference: for numeraires $S_0,S_1$ the kernel is ${\phi }_0^1(t)={\sigma }_1(t)-{\sigma }_0(t)$ (Björk eq. 26.20). Crucially, the instantaneous forward $f(t,T)$ is a ${\mathbb{Q}}^T$-martingale, and the forward LIBOR $L(t;T,S)$ is a ${\mathbb{Q}}^S$-martingale (numeraire $P(t,S)$) - different maturity indices (Björk Lemma 26.10; BM Prop 2.5.1).

Forward vs futures (Björk Ch29 correction): forward price $f(t;T,Y)={\mathbb{E}}^{{\mathbb{Q}}^T}[Y]$; futures price $F(t;T,Y)={\mathbb{E}}^{\mathbb{Q}}[Y]$; they are equal iff the short rate is deterministic.

2.2 HJM framework (BM Ch5; Björk Ch25; Shreve Ch28/34; Hull Ch33)

Model the forward curve directly:

$$df(t,T)=\alpha (t,T)dt+\sigma (t,T)dW(t),f(0,T)=f^M(0,T).$$

No-arbitrage forces the drift condition (Björk Prop 25.2):

$$\alpha (t,T)=\sigma (t,T){\int }_t^T\sigma (t,s)ds.$$

The drift is completely determined by volatility - no freedom remains. With this, the bond satisfies $dP=rPdt-{\sigma }^{\ast }(t,T)PdW$ with accumulated vol ${\sigma }^{\ast }(t,T)={\int }_t^T\sigma (t,s)ds$. Musiela parametrization ($x=T-t$, $r(t,x)=f(t,t+x)$) turns this into an infinite-dimensional SDE (Björk §25.3). One-factor HJM with mean-reverting vol $\sigma e^{-a(T-t)}$ is the Hull–White model (BM Ch5 headline).

2.3 Why lognormal instantaneous forwards explode (Shreve Ch34; Björk Ch27)

Take $\sigma (t,T)=\sigma f(t,T)$. Then ${\sigma }^{\ast }(t,T)=\sigma {\int }_t^{T}f(t,u)du$ and the HJM drift is $\alpha ={\sigma }^{2}f\int f\sim f^2$. The deterministic toy $f^{\prime }=f^2$, $f(0)=c$, solves $f(t)=c/(1-ct)$ and blows up at $t=1/c$ - HJM show the stochastic analogue does the same. Hence: model lognormal simple (LIBOR) rates instead.

2.4 LIBOR market model / BGM (BM Ch6; Björk Ch27; Shreve Ch34; Hull Ch33)

Forward LIBOR $F_k(t)=L(t;T_{k-1},T_k)$ is a martingale under its own forward measure ${\mathbb{Q}}^{T_k}$. The LFM postulates

$$d{F}_k(t)={\sigma }_k(t)F_k(t)d{Z}_k(t)\text{under }{\mathbb{Q}}^{T_k}.$$

The caplet is then priced exactly by Black’s formula (BM Prop 6.4.1; Björk Def 27.2):

$$Cp{l}_i(t)=P(t,T_i){\tau }_i[F_i(t)N(d_1)-KN(d_2)],d_1=\frac{\ln (F_i/K)+\frac{1}{2}{v}_i^2}{v_i},v_i^2={\int }_t^{T_{i-1}}{\sigma }_i^2(s)ds\text{(vol accumulated to the reset }{T}_{i-1}\text{)}.$$

Under a single common measure (e.g. terminal/spot-LIBOR ${\mathbb{Q}}^d$), the rates carry drift terms summing over other rates (BM Prop 6.3.3):

$$d{F}_k={\sigma }_k{F}_k\left[\sum _{j=\beta }^k\frac{{\rho }_{kj}{\tau }_j{\sigma }_j{F}_j}{1+{\tau }_j{F}_j}\right]dt+{\sigma }_k{F}_{k}d{Z}_k^d.$$

The forward swap rate $R_{\alpha ,\beta }$ is a martingale under the annuity (swap) measure ${\mathbb{Q}}^{\alpha ,\beta }$ (numeraire $C_{\alpha ,\beta }(t)=\sum {\tau }_{i}P(t,T_i)$), giving the Black swaption formula (BM Prop 6.7.1):

$$P{S}^{Black}(0)=C_{\alpha ,\beta }(0)[R(0)N(d_1)-KN(d_2)].$$

---

3. Computational Implementation - Black caplets, the martingale, and swaptions

Stdlib only: price a caplet by Black, verify forward-LIBOR is driftless (a martingale) under its own measure, and price a swaption under the annuity measure.

The forward LIBOR is driftless under its own measure (MC returns $F(0)$ to $10^{-4}$) - that is why the Black caplet is exact, not an approximation. The swaption uses the annuity numeraire, not the bank account.

---

4. Failure Modes & First-Principles Breakdowns

1. Modelling lognormal instantaneous forwards - the HJM explosion of §2.3. This is the structural reason market models exist; do not “fix” it by fiat.
2. Wrong numeraire for the product. Caplets need the $T$-forward measure; swaptions need the annuity measure; spot-LIBOR products need ${\mathbb{Q}}^d$. Pricing a caplet with a constant-discount shortcut, or a swaption with the bank account, is a first-principles error (BM Ch6 §6.2 rigorously avoids any “deterministic discount” approximation).
3. LFM vs LSM distributional incompatibility. A swap rate built from lognormal forwards is not exactly lognormal under the annuity measure (BM §6.8) - “mostly theoretical” in practice (Brace-Dun-Barton), but real when vols are high.
4. Forward ≠ futures with stochastic rates. The futures price is a $\mathbb{Q}$-martingale; the forward is ${\mathbb{Q}}^T$. Conflating them (Björk Ch29 corrected error) misses the convexity/marking-to-market adjustment.

---

5. References

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

---

6. Connected Graph Bridges

• Back: 03 · Short-Rate Models
• Forward: 05 · Failure Modes · Index Hub
• Base: Stochastic Calculus & Itô · No-Arbitrage & Binomial Trees