3.9.5 Failure Modes & Real-World Practice

Basic Prerequisites: 04 · Numeraire, HJM & Market Models.

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1. Intuition & Practical Objective

The interest-rate toolkit is mathematically elegant and fails in a small set of well-understood ways. This page names them precisely so a practitioner knows which assumption to distrust and how the failure shows up in money terms. The objective is discipline, not cynicism: know exactly where each model is an approximation so the residual risk can be measured and managed.

The failures, in one line each:

1. One factor ⇒ perfect correlation - a single short-rate Brownian motion moves the whole curve in lockstep, so curve-steepener and yield-spread products are mispriced.
2. Gaussian short rates go negative - Vasicek/Hull–White have $\mathbb{P}(r<0)>0$; and in real low/negative-rate regimes the lognormal Black machinery breaks because $\ln (F)$ is undefined for $F\le 0$.
3. Lognormal instantaneous forwards explode - HJM with $\sigma \propto f$ has drift $\sim f^2$ and blows up before $T$; only simple (LIBOR) rates are safe to model lognormally.
4. Calibration instability - HW’s exact-fit $\theta (t)$ and market-model parameter recovery need numerically unstable derivatives / inversions.

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2. Mathematical Ground Truth & Derivations

Where the assumptions live.

The one-factor correlation trap (BM Ch4). In any one-factor short-rate model $dr=a(b-r)dt+\sigma dW$, every forward rate of every maturity is a deterministic function of the same $r$. At any instant all forward rates are driven by the same $dW$, so

$$\text{corr}(f(t,T_1),f(t,T_2))=1\forall T_1,T_2.$$

Real curves move by level/slope/curvature (~3 PCA factors). The structural fix is a two-factor model: G2++ (BM Ch4) writes $r=x+y+\varphi$ with two mean-reverting Gaussians of different speeds, giving non-perfect maturity correlation and the ability to fit correlation-sensitive products like European swaptions.

The negative-rate / lognormal collapse. Black’s caplet is $P(0,T_i)\tau [FN(d_1)-KN(d_2)]$ with $d_1\propto \ln (F/K)$. If $F\le 0$ or $K\le 0$, $\ln$ is undefined. In the EUR/JPY 2015–2021 regime, forwards and strikes crossed zero; the market moved to the Bachelier (normal) model, where a caplet is

$$Cpl=P\tau [(F-K)N(d)+{\sigma }_N\sqrt{T}\varphi (d)],d=\frac{F-K}{{\sigma }_N\sqrt{T}},$$

(here $\tau \equiv 1$ is the year fraction, so the demo block drops it; the P·τ multiplier is the accrual factor for general tenors.)

well-defined for any $F,K$. The shifted lognormal model (BM Ch10.1) is the intermediate: $F_j=X_j+\alpha$, so $\ln (F_j-\alpha )$ stays finite by shifting the origin.

The HJM explosion (Shreve Ch34). With $\sigma (t,T)=\sigma f(t,T)$, the accumulated vol ${\sigma }^{\ast }=\sigma {\int }_t^{T}fds$ makes the HJM drift $\alpha =\sigma f{\sigma }^{\ast }\sim f^2$. The deterministic toy $f^{\prime }=f^2$, $f(0)=c$, blows up at $t=1/c$; HJM prove the stochastic analogue does too. Market models avoid this by modelling simple rates, whose drift is bounded by ${\gamma }^2{F}^2$ (Shreve Ch34 Remark 34.4).

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3. Computational Implementation - the failures in numbers

Stdlib only: (1) the forward-rate explosion, (2) negative/zero rates breaking Black but not Bachelier, (3) the perfect-correlation fact, (4) the HW differentiation instability.

The explosion is not a modelling nicety - $f$ goes from 5% to 1000% in 20 years, all from a “small” lognormal choice. And Black literally cannot price a strike at 0, while Bachelier does.

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4. Failure Modes & First-Principles Breakdowns (numbered)

1. The one-factor perfect-correlation trap. One Brownian motion ⇒ all forwards move together; steepeners, butterflies, and correlation-hedges are systematically mispriced. Fix: two-factor (G2++, CIR2, HW two-factor) or market models with a correlation matrix.
2. Negative rates break lognormal Black-76. $\ln (F/K)$ is undefined for $F\le 0$ or $K\le 0$. Fix: Bachelier (normal) for the full-range regime, shifted-lognormal for the intermediate, or displaced diffusion. Black vols must be converted to normal vols (and vice versa) consistently.
3. The HJM lognormal explosion. Do not model instantaneous forwards as lognormal; the drift grows like $f^2$. Model lognormal simple LIBOR rates (LFM/BGM), whose drift is bounded. This is the origin of the whole market-model framework.
4. Calibration instability. HW’s $\theta (t)$ needs numerical derivatives of the market forward curve (up to 3rd order); market-model cascade calibration inverts Black on noisy swaption vols (BM Ch7, RCCAEI exists precisely to remove negative/complex artifacts). “Exact fit” is only as good as the smoothed input curve.
5. Multi-curve / basis risk. Post-2008, the discounting curve (OIS) and the forward curve (LIBOR) diverged; pricing with a single curve misvalues swaps, and the LFM must be built off forward curves that are consistent with the discount curve. (See 06 · Advanced Extensions.)

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5. References

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

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6. Connected Graph Bridges

• Back: 04 · Numeraire, HJM & Market Models · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Implied Volatility Surfaces · Heston & SABR