Short Rate Models: Vasicek, CIR & Hull-White

Short rate models are the foundational building blocks of interest rate derivatives pricing. Before the industry moved to market models, Vasicek, CIR and Hull-White were — and in many contexts still are — the standard tools for pricing bonds, options and path-dependent rate products. Understanding them at the level of their stochastic differential equations is a prerequisite for understanding everything that came after.

This course builds all three models from scratch. You implement the Ornstein-Uhlenbeck dynamics of the Vasicek model, derive the closed-form zero-coupon bond price through the A(τ)/B(τ) affine structure, and extend to the CIR square-root diffusion, which enforces non-negative rates via the Feller condition. You then move to Hull-White: the time-dependent drift θ(t) that allows the model to be calibrated exactly to the market term structure — something the time-homogeneous Vasicek and CIR models cannot do.

Every model is implemented both analytically and by simulation. You price caplets via Jamshidian decomposition, which turns a caplet into a portfolio of zero-coupon bond options priced in closed form. You finish by comparing all three models on a calibration task, understanding their respective strengths and where each breaks down — the kind of model evaluation a quant does before putting a new model into production.

The project is a single integrated interest rate modelling library that implements all three short rate models at the same level of rigour. You begin with the Vasicek SDE: you simulate the Ornstein-Uhlenbeck process, verify the mean-reversion and variance properties analytically, and then derive the A(τ) and B(τ) functions that give the closed-form zero-coupon bond price. You generate upward-sloping, flat, inverted and humped yield curves by varying the model parameters, confirming the known shapes Vasicek can and cannot produce.

You then implement the CIR model, paying careful attention to the Feller condition that prevents rates from hitting zero, and implement the non-central chi-squared bond pricing formula. Calibrating CIR to market zero rates gives you hands-on experience with the numerical challenges that affine models present. The Hull-White chapter adds the time-dependent drift: you solve for θ(t) from the initial discount curve and verify that model bond prices match market prices exactly at every tenor — a property called exact fit to the term structure.

The caplet pricing module is the culmination of the project. You use Jamshidian's decomposition to express a caplet as a portfolio of zero-coupon bond options, price each analytically under Hull-White, and sum to get the caplet price. You then run a full Monte Carlo simulation under Hull-White — simulating the short rate, constructing bond prices from simulated paths, and pricing a caplet strip — and compare the Monte Carlo output to the analytical benchmark. The final session does a head-to-head comparison of all three models on a curve-fitting task, making the trade-offs between tractability and flexibility concrete.