What does it mean to calibrate an interest rate model to no-arbitrage conditions?

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AcadiFi · Accepted Answer

No-arbitrage calibration means the model's theoretical prices for benchmark instruments equal their market prices to within numerical tolerance. The three tiers of calibration instruments, in order of stringency: (1) discount bonds (zero curve), (2) at-the-money caplets and swaptions (volatility surface), (3) exotic or forward-start structures (smile and correlation). Consider calibrating a two-factor Hull-White model on April 2, 2026. First, time-dependent drifts theta1(t) and theta2(t) are chosen so the model reproduces the SOFR curve at 60 grid points with zero pricing error. Second, the two volatility parameters sigma1, sigma2 and the correlation rho are optimized to minimize squared errors across a grid of caplets (three-month rate on maturities from one to ten years) and European swaptions (one-by-ten, five-by-five, ten-by-ten). A typical calibration achieves average volatility errors under 30 basis points. ```mermaid flowchart LR A[Zero Curve] --> B[Theta t Solve] B --> C[Model Repricing
Check] D[Cap/Swaption Vols] --> E[Sigma Rho
Optimize] E --> C C --> F[Exotic Pricing] ``` Why it matters: an unhedged model that cannot reprice benchmarks will introduce systematic P&L bleeds. Traders value their book using the same curve they hedge against, so inconsistent models create ghost P&L. For the FRM, emphasize that calibration is about matching prices today; model risk arises because multiple calibrated models can produce very different prices for exotics.

What does it mean to calibrate an interest rate model to no-arbitrage conditions?

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