What is the Heath-Jarrow-Morton framework for forward rates?

Q: What is the Heath-Jarrow-Morton framework for forward rates?

HJM models the entire forward rate curve directly rather than just the short rate. The fundamental object is f(t,T), the instantaneous forward rate at time t for maturity T. Under HJM, df(t,T) = alpha(t,T) dt + sigma(t,T) dW under the real measure.\n\nThe critical HJM result: under the risk-neutral measure, the drift alpha(t,T) is uniquely determined by the volatility structure sigma(t,T). Specifically, alpha(t,T) = sigma(t,T) * integral from t to T of sigma(t,s) ds. This is the HJM no-arbitrage condition — the drift is not free; it's locked to the vol.\n\nConsequences:\n- You choose the forward-rate volatility function freely\n- Drift follows automatically to prevent arbitrage\n- Most short-rate models (Ho-Lee, Hull-White) are special cases of HJM\n\n```mermaid\ngraph TD\n A[Choose sigma(t,T)] --> B[HJM Condition]\n B --> C[alpha(t,T) Determined]\n C --> D[Forward Curve Evolution]\n```\n\nA drawback is that instantaneous forwards aren't directly observable or tradeable. The Libor Market Model (BGM) fixes this by modeling discrete forward LIBOR rates, which ARE observable in swap and cap markets. At a shop like Kestrel Rates Trading, HJM is the theoretical backbone; BGM is the practical implementation.\n\nFor CFA Level II: know that HJM is a framework, works with forward rates, and the drift is pinned by volatility under no-arbitrage.

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AcadiFi · Accepted Answer

HJM models the entire forward rate curve directly. The drift alpha(t,T) is uniquely determined by the volatility structure sigma(t,T).

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The critical HJM result: under the risk-neutral measure, the drift alpha(t,T) is uniquely determined by the volatility structure sigma(t,T). Specifically, alpha(t,T) = sigma(t,T) * integral from t to T of sigma(t,s) ds. This is the HJM no-arbitrage condition — the drift is not free; it's locked to the vol.

Consequences:
- You choose the forward-rate volatility function freely
- Drift follows automatically to prevent arbitrage
- Most short-rate models (Ho-Lee, Hull-White) are special cases of HJM

```mermaid
graph TD
    A[Choose sigma(t,T)] --> B[HJM Condition]
    B --> C[alpha(t,T) Determined]
    C --> D[Forward Curve Evolution]
```

A drawback is that instantaneous forwards aren't directly observable or tradeable. The Libor Market Model (BGM) fixes this by modeling discrete forward LIBOR rates, which ARE observable in swap and cap markets. At a shop like Kestrel Rates Trading, HJM is the theoretical backbone; BGM is the practical implementation.

For CFA Level II: know that HJM is a framework, works with forward rates, and the drift is pinned by volatility under no-arbitrage.

What is the Heath-Jarrow-Morton framework for forward rates?

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