FRM Part II · FRM Exam Part II

Arbitrage Pricing with Term Structure Models for FRM Part II

Arbitrage pricing with term structure models values bonds and interest rate derivatives so that no risk-free profit exists. You build a rate tree or a rate model, use risk-neutral probabilities, and discount expected payoffs backward node by node. Then you check the price against the market curve, or solve for OAS.

What this chapter covers

This chapter shows how to price bonds and interest rate derivatives without allowing arbitrage. It starts with the idea that two portfolios with the same payoffs must have the same price. From there you build a binomial interest rate tree, set risk-neutral probabilities, and roll values back from maturity to today.

Once the tree works, you use it for real products: callable and putable bonds, caps, floors and options on bonds. You then measure the option-adjusted spread (OAS), the constant spread added to tree rates so the model price equals the market price. Finally you meet the continuous models: Ho-Lee, Vasicek, and the Cox-Ingersoll-Ross (CIR) model, along with time-dependent volatility.

The chapter links to other parts of the paper. Market risk uses these models to value positions and compute sensitivities. Credit and liquidity topics use spreads and embedded options. Risk management in investment management uses OAS and effective duration to compare callable and mortgage-type securities. Learn the logic here once and it supports many questions elsewhere.

Questions here are applied: you are given a small tree or a set of parameters and asked for a price, a spread or a rate distribution. The calculations are short if your method is clean, so this chapter rewards practice more than memory. You also need to interpret results, such as why a callable bond has lower value than an otherwise identical straight bond, or why Vasicek can produce negative rates while CIR does not under its usual conditions. Candidates who drill the backward-induction routine and know each model's drift and volatility features can pick up reliable marks in a computer-based exam where each of the 80 questions carries equal weight.

Arbitrage Pricing with Term Structure Models: topics in the order to study them

1No-Arbitrage Principle and Replicating PortfoliosEverything else rests on this idea, so learn it first: same payoff means same price.
2Binomial Interest Rate TreesYou need to read and build a tree before you can price anything on it.
3Risk-Neutral Pricing and ProbabilitiesLinks the replicating-portfolio idea to the probabilities you use at each node.
4Pricing Interest Rate Derivatives on TreesApplies the tree and probabilities to bonds, caps, floors, swaps and options by backward induction.
5Option-Adjusted Spread (OAS)Builds on tree pricing: you now solve for the spread that matches a market price.
6Term Structure Models with Drift: Ho-Lee and VasicekMoves from trees to named models; compare a drift fitted to the curve with mean reversion.
7Time-Dependent Volatility and Cox-Ingersoll-Ross ModelsLast, because it extends the earlier models with changing volatility and rate-dependent volatility.

How to prepare Arbitrage Pricing with Term Structure Models

Build skill in layers. Get the pricing routine right on small trees first, then add models and interpretation.

Write the no-arbitrage argument in your own words, using a two-state example with two instruments and one replicating portfolio.
Practise one-step and two-step trees until backward induction is automatic: value at a node = [p × up value + (1 − p) × down value] ÷ (1 + rate at that node).
Price a bond, then a call or put on it, then a cap or floor on the same tree. Check each answer for reasonableness, such as a call never being worth less than zero.
Solve OAS problems by trial: add a spread to every node rate, reprice, and adjust until the model price matches the market price. Note the sign and meaning of the result.
Make a one-page table for Ho-Lee, Vasicek and CIR: the form of the rate process, the drift, how volatility behaves, and whether mean reversion is present.
Finish with timed mixed sets on a phone-friendly question bank. Review every miss and record whether it was a method error, an arithmetic error or a concept gap.

Common mistakes in Arbitrage Pricing with Term Structure Models

Using real-world probabilities instead of risk-neutral probabilities at the nodes.
Fix: Use the probabilities the question states for pricing, or derive them from no-arbitrage. Do not mix in an expected-return view.
Discounting with the wrong rate during backward induction.
Fix: Label each node with its own rate and discount each cash flow value one period using the rate at the node it is leaving.
Forgetting to apply the option exercise rule at each node.
Fix: For a callable bond, cap the node value at the call price where exercise is allowed. For a putable bond, floor it at the put price. Add the coupon in the right order.
Misreading what OAS means.
Fix: Remember OAS is the spread after removing the option's value. For a callable bond, OAS is usually below the Z-spread.
Mixing up the features of Ho-Lee, Vasicek and CIR.
Fix: Anchor on three questions: does it mean-revert, how does volatility behave, and can rates go negative. Check each model against your table.
Rounding too early in multi-step trees.
Fix: Keep four to six decimals until the final answer, then compare with the closest option.

Last-day revision: Arbitrage Pricing with Term Structure Models

No arbitrage: portfolios with identical payoffs in every state must have identical prices.
Risk-neutral pricing: discount expected payoffs at the risk-free rate using risk-neutral probabilities, not real-world ones.
Tree pricing works backward from maturity, one node at a time.
At each node, discount using the rate at that node, not the spot rate from time zero.
A callable bond's value equals the straight bond value minus the call option value.
A putable bond's value equals the straight bond value plus the put option value.
OAS is the constant spread added to tree rates that makes the model price equal the market price.
OAS removes the effect of the embedded option, so compare it across bonds with different options.
Ho-Lee: rates move with a time-dependent drift and constant volatility; no mean reversion.
Vasicek: mean-reverting rates with constant volatility; normally distributed rates, so negatives are possible.
CIR: mean reversion with volatility that rises with the level of the rate; keeps rates non-negative under its usual parameter condition.
Higher volatility raises option values, so callable bond prices fall as volatility rises.

Arbitrage Pricing with Term Structure Models practice questions

Arbitrage Pricing with Term Structure Models in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Arbitrage Pricing with Term Structure Models: frequently asked questions

Do I need to memorise the Vasicek and CIR equations?

You should know the structure of each process: the drift, the mean-reversion term and how volatility depends on the rate. You do not need heavy derivations. Focus on what each parameter does and what the model implies for rates.

How much calculation is in this chapter?

Mostly short numerical work on one- or two-period trees, plus conceptual questions on models. A calculator and a clean backward-induction layout are enough. Practise until a two-step tree takes only a few minutes.

Why is OAS better than a simple yield spread for callable bonds?

A simple spread mixes credit and liquidity compensation with the value of the embedded option. OAS strips out the option using a rate model. That lets you compare bonds with different option features more fairly.

Which Part II topic does this chapter belong to?

It supports Market Risk Measurement and Management, as it covers valuation models for interest rate products. The same ideas also help in credit and investment management questions involving spreads and embedded options.