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CFA Level II · CFA Level II Exam

Valuation and Analysis of Bonds with Embedded Options

A bond with an embedded option is valued by splitting it into a straight bond and an option. For callable and putable bonds you use a binomial interest rate tree and work backward, adjusting values at each node for exercise. Then you compute OAS, effective duration and convexity to measure risk.

What this chapter covers

This chapter shows you how to value bonds whose cash flows change with interest rates or with the issuer's share price. Callable, putable, capped, floored and convertible bonds cannot be priced by discounting fixed cash flows at one yield. You need a model that lets the option be exercised at the right time.

The core tool is the binomial interest rate tree. You build it from the benchmark curve, calibrate it so it reprices benchmark bonds, then roll the bond's value back through the tree. At each node you apply the option rule: the issuer calls if the value is above the call price, and the investor puts if the value is below the put price. From the tree you get the value of the option, the option-adjusted spread (OAS), and the effective duration and convexity.

This chapter links to the rest of the Fixed Income material on the yield curve, spreads and credit analysis. It also links to Derivatives, because the option logic is the same, and to Equities, because convertible bonds mix debt and equity. In the exam, it appears as an item set with a vignette, a tree or a table of prices, and questions that test whether you can read the data and apply the right step.

Fixed Income carries a 10-15% topic weight, and this chapter is one of its most computational parts. Item sets here reward a repeatable method: read the tree, apply the exercise rule, compare values, and interpret the result. Candidates who practise that method gain marks that others lose on small errors. The ideas on duration, convexity and option value also support Portfolio Construction and Derivatives questions, so the effort pays off beyond this chapter. There is no penalty for wrong answers, so you should attempt every question, but accurate method beats guessing.

Valuation and Analysis of Bonds with Embedded Options: topics in the order to study them

  1. 1Embedded Options in Bonds: Types and FeaturesStart here to learn who owns each option and how it changes bond value, which every later model depends on.
  2. 2Interest Rate Tree and Binomial ModelYou need to understand the tree, its nodes and its calibration before you can value anything on it.
  3. 3Valuing Callable and Putable BondsThis applies backward induction and the exercise rule to the tree you just learned.
  4. 4Option-Adjusted Spread (OAS)OAS builds on the tree by finding the constant spread that makes the model value equal the market price.
  5. 5Effective Duration and ConvexityYou compute these from the bond values when the curve shifts up and down, using the valuation method already learned.
  6. 6Key Rate Duration and Price-Yield BehaviorThis extends duration to non-parallel curve shifts and explains the shape of price-yield curves for option bonds.
  7. 7Valuing Capped and Floored Floating-Rate BondsIt reuses the tree and option logic on a new cash flow pattern, so it comes after the core method is secure.
  8. 8Convertible Bonds: Valuation and AnalysisFinish with convertibles, which combine a straight bond, an equity option and several comparison measures.

How to prepare Valuation and Analysis of Bonds with Embedded Options

Treat this chapter as one method applied several times. Get the tree mechanics right first, then practise variations until each step is automatic.

  1. Learn the option ownership rules first: the issuer holds a call, the investor holds a put, and write down how each affects value relative to a straight bond.
  2. Build a small tree by hand until you can roll values back without looking at notes, adding the coupon at each node and discounting at that node's rate.
  3. At each node, apply the exercise rule in the right direction: a call caps the value at the call price, and a put sets a floor at the put price.
  4. Practise the link between values: callable = straight − call option, and putable = straight + put option. Use it to check your answers.
  5. Compute effective duration and convexity from the up and down values, writing the formula each time, then check that the signs make sense, such as negative convexity for callable bonds.
  6. Do OAS and convertible questions as full item sets. Identify the needed data in the vignette, ignore the distractors, and then solve.
  7. Finish with timed practice sets, and keep a list of the errors you make so you can review it before the exam.

Common mistakes in Valuation and Analysis of Bonds with Embedded Options

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

    Fix: Ask who owns the option. The issuer lowers the value to the call price when it is above it. The investor raises the value to the put price when it is below it.

  • Forgetting to add the coupon at each node before comparing with the exercise price

    Fix: Write the node value as the average of the two next-period values plus the coupon, discounted at the node rate. Then apply the exercise test.

  • Using the wrong rate to discount at a node

    Fix: Use the rate shown at the node you are valuing, as it applies for the period starting there. Label each step as you work.

  • Mixing up the sign of the option value in the callable and putable relationships

    Fix: Remember that the issuer's call hurts the investor, so it is subtracted. The investor's put helps the investor, so it is added.

  • Treating effective duration like modified duration

    Fix: Effective duration uses full revaluation after curve shifts, so it captures changing cash flows. Use it for bonds with embedded options, because yield-based measures assume fixed cash flows.

  • Ignoring the vignette's distractor data in convertible questions

    Fix: Decide first what is being asked, such as conversion value or the premium, then pick only the inputs that define it and ignore the rest.

Last-day revision: Valuation and Analysis of Bonds with Embedded Options

  • Callable bond value = straight bond value − value of the call option.
  • Putable bond value = straight bond value + value of the put option.
  • Backward induction: start at maturity, discount each node's value, add coupon, apply the exercise rule, and move back.
  • The issuer calls when the bond's value at a node is above the call price. The investor puts when it is below the put price.
  • OAS is the constant spread added to all tree rates that makes the model value equal the market price.
  • Zero-volatility spread minus OAS equals the option cost in spread terms, for a callable bond.
  • Effective duration = (PV₋ − PV₊) ÷ (2 × PV₀ × Δcurve), where PV₋ and PV₊ are values after a curve shift down and up.
  • Effective convexity = (PV₋ + PV₊ − 2 × PV₀) ÷ (PV₀ × Δcurve²).
  • A callable bond shows negative convexity when rates are low, as price gains are limited by the call.
  • A putable bond has positive convexity, and its price is floored near the put price.
  • A capped floater is worth less than an uncapped one, and a floored floater is worth more.
  • Convertible value is at least the greater of straight bond value and conversion value.

Valuation and Analysis of Bonds with Embedded Options in other exams

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

Valuation and Analysis of Bonds with Embedded Options: frequently asked questions

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

Work backward from maturity. At each node, take the average of the next two values, add the coupon, discount at that node's rate, and then cap the result at the call price if the call is exercisable. The value at the first node is the callable bond's price.

What does OAS tell me?

OAS is the spread over the benchmark curve after removing the effect of the embedded option. It lets you compare bonds with and without options on a like-for-like basis. A higher OAS suggests more compensation for credit and liquidity risk, given the model's assumptions.

Why do I use effective duration for bonds with embedded options?

Their cash flows change when rates change, because the option may be exercised. Effective duration revalues the bond under shifted curves, so it reflects this. Yield-based duration assumes the cash flows stay fixed, so it can mislead.

Is this chapter hard to score on in the exam?

It is computational, but the steps are repeatable. If you practise the tree method and the formulas on full item sets, you can answer quickly and accurately. The main risk is small arithmetic or direction errors, which regular practice reduces.