Stochastic Calculus & Finance: The Heath-Jarrow-Morton Model

This repository accompanies my presentation for the Summer 2026 University of Texas at Austin Directed Reading Program. It contains both the talk itself and a complete, working implementation of the one-factor Heath-Jarrow-Morton (HJM) model calibrated to historical Federal Reserve data.

The reading followed Steven E. Shreve's Stochastic Calculus for Finance II: Continuous-Time Models, building up the measure-theoretic probability needed to make sense of the stochastic differential equation

$$ df(t, T) = \alpha(t, T),dt + \sigma(t, T),dW(t), $$

where $f(t, T)$ is the instantaneous forward rate for maturity $T > t$ that can be locked in at time $t$. This is the HJM model, and it describes how the entire forward-rate curve evolves over time.

Contents

File	Description
main.tex / main.pdf	The presentation. A beamer talk that develops the probability theory behind HJM, from the coin-toss space up to Brownian motion, and closes with the model itself.
main.ipynb	The main deliverable. A complete one-factor HJM simulation calibrated to historical Federal Reserve yield-curve data.
basic_ex.ipynb	A short toy example. The HJM model simulated on a synthetic initial forward curve $f(0, T) = 0.05,e^{-0.3T}$, included as the coded example shown in the talk.
feds200628.csv	The historical dataset (see below).
images/	Figures used in the presentation.

The Presentation

The talk is aimed at an audience with a background in basic calculus and probability, and introduces the remaining machinery asP it is needed. Its rough ordering of topics is:

• Measures: assigning size and likelihood to events on the coin-toss space $\Omega = {-1, 1}^{\mathbb{N}}$.
• Sigma-algebras, random variables, and information: probability spaces $(\Omega, \mathcal{F}, \mathbb{P})$, and filtrations $\mathcal{F}(t)$ as a model of "what we know so far."
• Random walks and Brownian motion: from the symmetric random walk $S_n$, to the scaled random walk $W^{(n)}(t)$, to Brownian motion $W(t)$ in the limit.
• Putting it all together: the HJM model, why the forward rate is driven by a Brownian motion, and how it is used in practice.

The Notebook

main.ipynb gives a self-contained, implementation-level treatment of the one-factor HJM model. Financial quantities, modeling assumptions, and numerical methods are introduced as they are used; the measure-theoretic construction of Brownian motion and the derivation of the risk-neutral measure are left to the talk and to Shreve.

Data

The simulation is calibrated to the Gürkaynak-Sack-Wright (GSW) U.S. Treasury yield-curve dataset (feds200628.csv), a Federal Reserve staff research product covering the Treasury curve from 1961 to the present. The dataset provides the six Svensson parameters ($\beta_0, \beta_1, \beta_2, \beta_3, \tau_1, \tau_2$) for each date, from which the instantaneous forward rate at any maturity can be reconstructed:

$$ f(t, t+\tau) = \beta_0 + \beta_1 e^{-\tau/\tau_1} + \beta_2 \frac{\tau}{\tau_1} e^{-\tau/\tau_1} + \beta_3 \frac{\tau}{\tau_2} e^{-\tau/\tau_2}. $$

Method

The notebook proceeds in the following stages:

1. Build forward-rate panels. Using the Svensson parameters, the daily forward curve is reconstructed on a grid of maturities $\tau \in {1.0, 1.5, \dots, 30.0}$ years from 1990-01-01 onward, together with a one-day-ahead panel offset by $\delta = 1/252$.

2. Estimate volatility. Normalized daily forward-rate increments form an empirical covariance matrix $C$. Since a change of measure only alters the drift, this covariance (estimated under the real-world measure) is enough to recover the volatility loading $\sigma(t, T)$ under the risk-neutral measure. A single principal component captures the dominant mode of curve movement; the notebook reports the first-factor explained variance and the rank-one approximation error.

3. Enforce no arbitrage. Under the risk-neutral measure, the one-factor HJM drift is fully determined by the volatility:

$$ \alpha(t, T) = \sigma(t, T),\sigma^{\ast}(t, T), \qquad \sigma^{\ast}(t, T) = \int_t^T \sigma(t, u),du. $$

So once $\sigma$ is estimated, no further calibration is required.

4. Simulate the curve. Starting from the most recent observed forward curve as the initial condition $f(0, \cdot)$, the curve is evolved forward with an Euler-Maruyama scheme, drawing a single Brownian increment at each step and rolling the maturity grid forward as observation time advances. The result is a family of simulated forward curves out to a chosen horizon.

Running it

The notebook depends on numpy, pandas, scipy, and matplotlib. Open main.ipynb and run the cells top to bottom; feds200628.csv must be present in the repository root.

References

Steven E. Shreve. Stochastic Calculus for Finance II: Continuous-Time Models. Springer Finance, Springer, 2004.