Fixed Income Course
Callable & Putable Bonds: Hull-White Trinomial Lattice
Price callable and putable bonds on a Hull-White trinomial tree calibrated to the market term structure.
Enrol now
- Lifetime access — all lessons & updates
- 10 lessons across 2 structured parts
- Fully worked Jupyter notebooks
- Market datasets + published benchmark prices
- Free lesson previews before you buy
- 14-day money-back guarantee
Secured by a no-questions-asked 14-day refund.
Overview
Introduction
Callable and putable bonds are among the most common embedded-option products in fixed income, yet pricing them correctly requires a no-arbitrage interest rate model that is fitted exactly to the observed yield curve. The Hull-White trinomial lattice is the workhorse approach for this problem on rates desks, combining analytical tractability with exact calibration to market discount factors.
This course builds the entire lattice from first principles. You derive the trinomial branching probabilities by matching the first two moments of the short-rate process, introduce Arrow-Debreu state prices as the calibration instrument for θ(t), and then use backward induction to price vanilla bonds, callable bonds and putable bonds on the same tree — applying the exercise condition at each node.
You finish by extracting the option-adjusted spread of a 10-year callable bond, decomposing the instrument into its straight-bond value and the embedded call option cost, and verifying the tree against closed-form Hull-White bond prices. The methodology transfers directly to bermudan swaptions, range accruals and any other product with discrete exercise rights.
Hands-on
The project you'll build
The project begins with a calibrated Hull-White trinomial tree. You construct the tree geometry — node spacing dx = σ√(3Δt), branching probabilities from moment matching, special boundary nodes — and then calibrate the additive shift α(t) at each time step using Arrow-Debreu state prices so that the tree reprices every market ZCB exactly.
With the calibrated tree in hand, you price a vanilla bond by backward induction to verify the setup, then extend to a callable bond by applying the issuer's call decision at each node — hold if the continuation value exceeds the call price, exercise otherwise. You price a putable bond the same way from the holder's perspective and compute the option value as the difference from the straight-bond price.
The final step extracts the OAS: you solve numerically for the constant spread shift added to all discount rates that makes the callable bond's model price match the observed market price. You then decompose the full instrument — market price, straight price, embedded option value — and discuss how OAS is used in relative value and hedging on a rates desk.
Outcomes
What you'll learn
Hull-White tree geometry
Derive the node spacing, branching probabilities and boundary nodes of the Hull-White trinomial tree from moment-matching conditions.
Arrow-Debreu state prices
Compute state prices on the tree and use them to calibrate θ(t) so that model ZCB prices match the market term structure exactly.
Backward induction
Price cash-flow instruments on the tree by discounting from the terminal nodes backward to time zero.
Callable bond pricing
Apply the issuer's optimal call condition at each node and compute the callable bond price and the embedded call option value.
Putable bond pricing
Apply the holder's optimal put condition and verify the option floor against the straight-bond price.
Option-adjusted spread
Solve for the OAS by shifting all discount rates on the tree to match the callable bond's market price.
Option value decomposition
Separate straight-bond price from callable price to isolate the dollar value of the embedded option across rate scenarios.
Lattice methodology
Understand why lattice methods are preferred over PDE or Monte Carlo approaches for early-exercise problems with discrete exercise dates.
Stack
Tools & technology
The course teaches a workflow, not just a toolset — here is what we use and what you could swap in.
| Tool | Used for | Alternatives |
|---|---|---|
| Python 3 | Tree construction, calibration and backward induction | Julia, MATLAB, C++ |
| NumPy | Vectorised node arrays and state price computation | pure Python, JAX |
| SciPy | Root-finding for θ(t) calibration and OAS extraction | Newton-Raphson from scratch, Brent's method |
| Hull-White Trinomial Tree | The core interest rate model for early-exercise pricing | Black-Derman-Toy, BDT binomial, G2++ tree |
| Jupyter | Interactive tree construction with node-by-node visualisation | VS Code, plain Python scripts |
Syllabus
Course curriculum
10 lessons across 2 parts. Lessons marked Free preview can be read before you enrol.
Part 1 — Hull-White Trinomial LatticeConstruct the tree and calibrate it to the market curve.
- 1.Lattice methods for interest rate optionsWhy trees are used for early exercise and what a trinomial node looks like.Free preview
- 2.Hull-White tree structure and branching probabilitiesDerive dx, branching probabilities from moment matching, and boundary nodes.Free preview
- 3.Arrow-Debreu state prices for calibrationUse state prices to calibrate the additive shift α(t) to market ZCB prices.
- 4.Theta calibration to the term structureEnsure model ZCB prices match market prices at every pillar.
- 5.Pricing a vanilla bond on the treeBackward induction: discount cash flows through the tree.
Part 2 — Embedded Options & OASCallable and putable bond pricing, OAS and option value decomposition.
- 1.Callable bond: issuer's call optionApply the call exercise condition at each node and subtract option value.
- 2.Putable bond: holder's put optionApply the put exercise condition and compute the option floor.
- 3.Option-Adjusted Spread (OAS)Solve for the constant spread that matches the callable bond's market price.
- 4.Decomposing option value and straight priceSeparate the straight bond price from the embedded option cost.
- 5.Extensions: bermudan swaptions & exoticsFrom callable bonds to bermudan swaptions on the same tree.
Before you start
Prerequisites
- Fixed-income fundamentals: discount factors, zero-coupon bonds, par swap rates (the Yield Curve Bootstrapping course is ideal preparation).
- Basic knowledge of the Hull-White model (the Short Rate Models course covers this; the key result is the analytical ZCB price formula).
- Working Python and NumPy — arrays, loops, numerical root-finding with scipy.optimize.
- Introductory probability: expectation, variance and the normal distribution.
Fit
Who this course is for
Rates desk quants
You price vanilla instruments and want to extend your toolkit to callable bonds, bermudan swaptions and other early-exercise products.
Fixed income portfolio managers
You hold callable and putable bonds and want to understand OAS, option value decomposition and rate-sensitivity correctly.
Quant finance students
You are studying interest rate models and want a complete, working implementation of the Hull-White trinomial tree to complement your textbook.
Developers in fixed income
You support pricing systems and want a ground-up understanding of how lattice pricers work so you can maintain and extend them.
Faculty
Your instructor
QuantIndex Faculty
QuantIndex Faculty are practising quantitative analysts and former rates-desk developers with experience pricing callable bonds, bermudan swaptions and structured notes in production pricing systems. The course content is written by quants who use Hull-White lattice methods daily and is reviewed for mathematical rigour and clean, maintainable Python before each release.
Benchmarked & verified
Every price you compute is checked against published results from leading textbooks and market data. You do not finish the course hoping your model is right — you finish it knowing.
Questions
Frequently asked questions
Why a trinomial tree rather than binomial?
The Hull-White trinomial tree is the standard industry choice for this model because the three-branch structure gives an extra degree of freedom that makes the branching probabilities easier to keep non-negative across all nodes, including the boundary nodes where mean reversion is strongest. The binomial version of Hull-White requires more care at the boundaries.
Can this tree price bermudan swaptions as well?
Yes — a bermudan swaption is conceptually identical to a callable bond, just with a swap as the underlying instead of a bond. Once you have the tree calibrated and backward induction working, extending to bermudan swaptions requires only a different payoff function. The final lesson discusses this extension.
What maths do I need beyond Hull-White basics?
The key mathematical objects are the Arrow-Debreu state prices, which are just discounted risk-neutral probabilities of reaching each node. Everything is derived carefully from the moment-matching conditions and no measure theory or Girsanov's theorem is required — the tree is entirely self-contained.
Is OAS still used on a real rates desk?
Absolutely. OAS is the standard measure for comparing callable bonds of different structures and maturities on a spread basis, stripping out the embedded option cost. It is reported daily by Bloomberg and is the primary relative-value metric for callable agency bonds, MBS pass-throughs and corporate callables.
How does the calibration to the term structure work, and does it matter in practice?
Calibrating to the market curve is essential: an uncalibrated tree prices coupon bonds incorrectly and produces OAS values that are meaningless as relative-value signals. Arrow-Debreu state prices make the calibration exact — at every time step the tree is adjusted so that model ZCB prices match market prices, not just at the short end.
Is there a refund policy?
Yes — a 14-day, no-questions-asked money-back guarantee. If the course is not right for you, email us and we will refund you in full.
Ready to start?
- Lifetime access — all lessons & updates
- 10 lessons across 2 structured parts
- Fully worked Jupyter notebooks
- Market datasets + published benchmark prices
- Free lesson previews before you buy
- 14-day money-back guarantee
Secured by a no-questions-asked 14-day refund.