Arbitrage Pricing on Rate Trees: The Science of Term Structure Models

A cap on a floating-rate loan has a value that depends on what six-month rates will be in one year, two years, and three years. A callable corporate bond can be redeemed whenever rates fall low enough to make refinancing attractive. These are not payoffs that can be replicated by static bond positions — they require a model of how short rates can evolve, and a method for pricing contingent payoffs against that model. Rate trees are that method.

This lesson establishes three results that underlie all short-rate models in this topic:

1. Arbitrage pricing by replication: given a rate tree, any derivative's value is pinned by the price of its replicating portfolio in the underlying bonds.
2. Risk-neutral pricing: there exist probabilities (the risk-neutral probabilities) under which every bond and derivative can be priced by expected discounted value — an enormous computational shortcut.
3. The continuous limit: as the time step $\Delta t \to 0$, the binomial distribution of rate changes converges to a normal distribution, bridging the discrete tree and the continuous SDEs of subsequent lessons.

• construct a one-period binomial rate tree and build the corresponding price tree for a zero-coupon bond;
• replicate a derivative security using the two underlying bonds and compute its arbitrage price;
• find the risk-neutral probability $p$ from one observable bond price and use it to price any derivative on the same tree;
• extend the single-period tree to multiple periods using backwards induction, and explain why recombining trees are computationally essential;
• state what changes as $\Delta t \to 0$ and why the short-rate approach is essential for fixed-income derivatives (unlike equity).

1. 1. Rate and price trees
2. 1. Arbitrage pricing of derivatives
3. 1. Risk-neutral pricing
4. 1. Multi-period extension
5. 1. Reducing the time step
6. 1. Fixed income versus equity derivatives