Heath-Jarrow-Morton Framework

The Heath-Jarrow-Morton framework is one of the most important results in mathematical finance. Rather than modelling a single short rate and deriving the whole curve from it, HJM takes the entire forward rate curve as the state variable. The payoff is enormous: any arbitrage-free interest rate model — Vasicek, Hull-White, the LIBOR Market Model — can be expressed as a special case of HJM. Understanding it means understanding the deep structure behind all of them.

The framework's central result is the no-arbitrage drift restriction: under the risk-neutral measure, the drift of every instantaneous forward rate is completely determined by its volatility and the integrals of volatility up to that maturity. You do not choose the drift — the market pins it for you. This is why HJM models are automatically consistent with the initial term structure, and it is the conceptual key that unlocks both the LMM and the whole family of modern interest rate models.

This course builds the framework step by step and then puts it to work in code. You implement single-factor and two-factor HJM models, show that exponential volatility recovers Hull-White prices exactly, simulate the full forward curve with Monte Carlo, and price caplets in both cases. It is a course for practitioners who want to understand not just how to use these models but why they work the way they do.

The project is a forward curve simulation library. You begin from the HJM SDE for f(t, T) and derive the no-arbitrage drift restriction step by step, showing that the drift is a deterministic function of the volatility structure. You implement a discrete tenor grid and an Euler-Maruyama simulation of the forward curve, evolving f(t, Tj) for each tenor simultaneously on each path — the direct numerical realisation of the HJM framework.

The single-factor model uses exponential volatility σ(t, T) = σ·exp(−a(T − t)), which corresponds to the Hull-White model. After building the pricer, you verify this connection explicitly: you price a set of zero-coupon bonds using HJM Monte Carlo and compare the results to Hull-White closed-form prices, confirming that the two implementations agree to simulation error. This benchmarking step validates the code and deepens the conceptual connection between the two models.

The second half of the project extends the model to two factors with a second volatility term σ2·exp(−a2(T − t)) and a correlation coefficient between the two Brownian drivers. You re-run the simulation, observe how the additional factor affects term structure dynamics — in particular the ability to generate richer curvature moves — and price caplets under both the single- and two-factor models. The final lesson introduces the Musiela parametrisation, which rewrites the HJM equation in time-to-maturity coordinates, and explains how it leads naturally to the LIBOR/SOFR Market Model.