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CFA Level II · CFA Level II Exam

The Arbitrage-Free Valuation Framework for CFA Level II

The arbitrage-free valuation framework prices a bond by discounting each cash flow at the right rate so that no risk-free profit exists. For bonds with options, you build a binomial interest rate tree, work backward through it, and adjust values at nodes where the issuer or holder exercises the option.

What this chapter covers

This chapter shows how to value fixed-income securities so that the price is consistent with the market and leaves no riskless profit. It starts with the law of one price and spot and forward rates. It then builds a binomial interest rate tree calibrated to the current yield curve and uses it to value option-free, callable and putable bonds.

The later topics extend the same idea. Pathwise valuation and Monte Carlo simulation value securities whose cash flows depend on the path rates take, such as mortgage-backed securities. Term structure models (Vasicek, CIR and Ho-Lee) describe how interest rates evolve and relate to the rate dynamics used in trees and simulations.

The chapter links to the rest of the exam in several places. Embedded options and option-adjusted spreads carry into credit analysis and structured products in Fixed Income. Arbitrage logic returns in Derivatives, and rate-sensitive valuation shows up in Portfolio Construction. On exam day, this material appears inside an item set, so you must pull the right rates, tree values and option terms from the vignette and apply the method.

Fixed Income carries a 10-15% topic weight, and this chapter supplies the valuation engine for much of it. Item-set questions here are computational but short once you know the routine: read the tree, discount, apply the exercise rule, compare values. The same skills make later Fixed Income chapters easier, so time spent here pays off twice. Because there is no penalty for wrong answers, you should attempt every question, but accurate tree work will win you the marks that guessing cannot.

The Arbitrage-Free Valuation Framework: topics in the order to study them

  1. 1Arbitrage-Free Valuation and Law of One PriceIt sets the principle that every later method must satisfy: identical cash flows have the same price.
  2. 2Spot Rates, Forward Rates and the Forward Rate ModelYou need spot and forward rates before you can build or read a tree, and the forward rate model is used to price and compare bonds.
  3. 3Binomial Interest Rate Tree ConstructionThe tree uses forward rates and a volatility assumption, so it comes after them and before any valuation on it.
  4. 4Valuing Option-Free, Callable and Putable BondsThis is the main application of the tree. Check the option-free value against the spot-rate price first, then add the option rule.
  5. 5Pathwise Valuation and Monte Carlo SimulationIt reuses the tree's discounting logic but follows whole paths, so it is easiest once the tree is familiar.
  6. 6Term Structure Models: Vasicek, CIR and Ho-LeeThese are conceptual. They compare well once you know how trees and simulations treat rate movements, and they differ on mean reversion, volatility and fit to the current curve.

How to prepare The Arbitrage-Free Valuation Framework

Treat this chapter as one method applied several ways. Practise the calculations by hand and keep the concepts short and clear.

  1. Read the law of one price and write in your own words why an arbitrage opportunity would be exploited and removed.
  2. Practise converting between spot rates, forward rates and discount factors until it is automatic. Check each answer by rebuilding the bond price both ways.
  3. Build a small two-period binomial tree from given forward rates, and confirm that its option-free bond value matches the value from spot rates.
  4. Value a callable and a putable bond on the same tree. Apply the exercise rule only at nodes on call or put dates: at a call date, the node value is the lower of the computed value and the call price; at a put date, it is the higher of the computed value and the put price.
  5. Link values together: option-free value minus callable value gives the call option value, and putable value minus option-free value gives the put option value.
  6. Learn pathwise valuation and Monte Carlo as a sequence: generate rate paths, discount cash flows along each path, then average. Know what each step assumes.
  7. Summarise Vasicek, CIR and Ho-Lee in one table of your own: mean reversion, volatility treatment, whether rates can go negative, and fit to the current curve. Then do timed item sets.

Common mistakes in The Arbitrage-Free Valuation Framework

  • Applying the call or put rule at the wrong node or in the wrong direction.

    Fix: Ask who benefits. The issuer calls when value is above the call price, so at nodes on call dates a callable bond's value is capped at the call price: min(computed value, call price). The holder puts when value is below the put price, so at nodes on put dates a putable bond's value has a floor at the put price: max(computed value, put price). At other nodes, do not apply the rule.

  • Forgetting to add the coupon when rolling values back one step.

    Fix: At each node, average the two next-period values after adding that period's coupon, then discount. Write the coupon next to every column of the tree.

  • Confusing spot rates with forward rates when discounting.

    Fix: Check the exhibit label. Use spot rates for discounting a single cash flow from today, and forward rates for period-by-period discounting along a tree.

  • Mixing up how option values are derived from bond values.

    Fix: Remember the call is something the issuer holds, so it lowers bond value: call value = option-free − callable. The put helps the holder: put value = putable − option-free.

  • Mixing up Vasicek, CIR and Ho-Lee features.

    Fix: Anchor each by mean reversion, volatility treatment and curve fit. Vasicek has mean reversion, constant volatility, can give negative rates and does not fit the current curve exactly. CIR has mean reversion, ties volatility to the rate level and also does not fit the curve exactly. Ho-Lee has no mean reversion and fits the current curve.

  • Skipping the check that the tree reproduces the market price.

    Fix: Whenever time allows, value the option-free bond on the tree and confirm it matches the spot-rate value. A mismatch usually signals an arithmetic slip.

Last-day revision: The Arbitrage-Free Valuation Framework

  • Law of one price: assets with identical cash flows must have the same price, or arbitrage exists.
  • An arbitrage-free bond value equals the sum of cash flows discounted at the matching spot rates.
  • Forward rates are implied by spot rates, and the forward rate model prices bonds by discounting at forward rates.
  • A binomial tree is calibrated so that its option-free bond value equals the market price from the curve.
  • Work backward through the tree: node value = average of the two next-period values (including coupon) discounted at that node's rate.
  • Callable bond: on call dates only, node value = min(computed value, call price).
  • Putable bond: on put dates only, node value = max(computed value, put price).
  • Call option value = option-free value − callable value; put option value = putable value − option-free value.
  • Monte Carlo averages discounted cash flows over many simulated rate paths and is used for path-dependent cash flows.
  • Vasicek and CIR are equilibrium models with mean reversion that do not fit the current curve exactly. Vasicek can give negative rates; CIR has volatility that scales with the square root of the rate level, which keeps rates non-negative when the Feller condition holds.
  • Ho-Lee is an arbitrage-free model that fits the current curve exactly, with constant volatility and no mean reversion.
  • Attempt every question: there is no penalty for wrong answers.

The Arbitrage-Free Valuation Framework in other exams

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

The Arbitrage-Free Valuation Framework: frequently asked questions

What is the arbitrage-free valuation framework?

It is a way of valuing bonds so that the price leaves no riskless profit. You discount each cash flow at the appropriate rate, or use a tree calibrated to the current curve, so that the model price matches market prices of comparable securities.

How do I value a callable bond with a binomial tree?

Build the tree, then work backward from maturity. At each node on a call date, compare the computed value with the call price and use the lower figure. The value at the first node is the callable bond's value.

Do I need to memorise the Vasicek, CIR and Ho-Lee formulas?

Focus on what each model assumes and how they differ, because that is what the questions test in an item set. Know the role of mean reversion and volatility in each, and whether the model fits the current curve.

How is Monte Carlo different from a binomial tree?

A tree recombines and values securities node by node. Monte Carlo simulates many full rate paths, discounts cash flows along each, and averages them, which suits path-dependent securities. A tree or a simulation is consistent with the current yield curve only when it is calibrated to it, for example with an arbitrage-free drift. Simulations under equilibrium models such as Vasicek or CIR need not fit the curve exactly.