Identify who owns the embedded option

Learning goals

• Calculate the callable bond value.
• Explain why 108 applies to this case.
• Identify the case-specific assumption or failure mode before taking action.

Draft · approximately 85 minutes, excluding further revision.

Shared case and decision context

A straight bond with matched credit and cash flows is worth 105. The issuer’s call option is worth 4. A separate put option held by the bondholder is worth 3. Assume internally consistent option valuations. Rates fall sharply after purchase.

Economic or professional mechanism

A callable bond gives the issuer the right to retire debt, which costs value to the investor. A putable bond gives the investor an exercise right, which adds value. Falling rates can increase the likelihood of a call and cap price appreciation, producing negative convexity over relevant ranges. Embedded options require rate-contingent valuation, not just yield-to-maturity comparison.

Apply, challenge and communicate

Callable value is 101; putable value 108. Effective duration reprices option-adjusted cash flows under curve shocks. Option-adjusted spread depends on the rate model and volatility assumptions; compare OAS only on consistent inputs and do not treat it as a pure measured default probability.

Insert exercise rights into backward induction

For a callable bond at a call date the investor receives the lesser of continuation value and the contractual call payment, including coupon conventions. A putable bond instead takes the greater of continuation and the investor’s put value. At a node with continuation108 and call price102, a rational issuer call limits the applicable value to102 under the stated convention. Exercise dates and notice periods matter; never apply the call cap at a date when the issuer cannot exercise.

Trace volatility to option ownership

Higher interest-rate volatility generally increases the embedded option’s value when other modeled inputs are fixed. Because the investor is short the issuer’s call, this tends to reduce callable-bond value. Because the investor owns the put, it tends to increase putable-bond value. The sign follows ownership and option convexity rather than a rule that all bonds benefit from volatility. Recalibration or changes in the curve can offset this isolated sensitivity in observed prices.

Apply the call cap at each eligible tree node

In an option-bond tree first value terminal payments, then roll values backward. At every eligible call node compare the discounted continuation value with the contract’s call payment. Suppose child values104 and100, equal pricing weights and one-period rate2% imply continuation100; an exercisable call98 caps the node at98, absent separate coupon. Repeating this exercise test is essential; valuing a straight bond once and subtracting an arbitrary fixed amount is not a tree valuation.

Interpret an OAS change conditional on volatility

OAS is the spread added to the model discounting structure to reconcile model price with observed price while valuing embedded exercise rights. If assumed volatility rises, the issuer’s call is worth more. Holding the callable bond’s market price fixed, the calibrated OAS may need to narrow to offset the lower option-adjusted model value. This is a model-conditional effect; it does not show the issuer’s physical credit quality improved.

Compare option-adjusted durations locally

When rates are high and a call is unlikely, a callable bond can behave more like the corresponding straight bond. As rates fall and calling becomes likely, its effective duration often shortens because upside is capped. A deeply relevant put can protect downside and shorten rate exposure for a putable bond. These comparisons depend on rate level, volatility and exercise terms; there is no universal constant duration ordering for every possible contract and state.

Use asymmetric and key-rate shocks

For option bonds the price response to an upward rate shock can differ substantially from that to an equal downward shock. One-sided durations separately summarize these local responses, while key-rate durations isolate sensitivity to selected curve maturities. A single symmetric duration may conceal an approaching call boundary. Shock and revalue the exercise decision each time; holding optional cash flows fixed defeats the purpose of effective sensitivity.

Read a floating-rate cap as a sold option

A capped floating-rate note equals an otherwise matched uncapped floater minus the embedded cap owned by the issuer, under consistent modeling. A floored note equals the matched floater plus the investor’s floor. If floater100 and cap2, capped value98; if floor3, floored value103. Rate reset lags, credit spread and coupon conventions can affect real values. A cap limits high coupons to the investor, while a floor protects low coupons.

Identify the conversion contract

A convertible bond provides a contractual right to exchange the bond for a specified number of shares under stated terms. Conversion ratio, conversion price, call/put provisions, coupon and maturity shape its payoff. A ratio20 with bond principal1000 gives a simple conversion price50 per share. Conversion is an investor option unless the contract specifies mandatory or contingent features. Do not treat convertible debt as pure common equity from issuance.

Calculate conversion value and premium

Conversion value equals the conversion ratio times current share price. For20 shares at45, value is900; a convertible market price1050 implies an absolute premium150 and a conversion premium relative to parity of16.6667%. Straight-bond value and embedded conversion value provide complementary anchors. The market price need not equal the sum of independently estimated pieces if calls, puts and contingent terms interact.

Revalue hybrid claims jointly

An arbitrage-free convertible framework jointly models the equity-linked conversion right, interest rates, issuer credit and contractual exercise features. At eligible nodes compare continuation, conversion, put and issuer-call outcomes consistently with priority rules. Default can affect debt recovery and the equity value simultaneously. A simple straight-bond-plus-call approximation is educational but may miss interactions and credit-equity dependence important in stressed issuers.

Recognize changing equity sensitivity

When the underlying stock price is well below the contractual conversion price, conversion value can be much lower than the straight-bond value, and the convertible may behave primarily as credit-sensitive debt. When conversion is very attractive, it may behave more like equity. In between it can offer asymmetric participation, but downside remains exposed to credit deterioration and liquidity. The estimated bond floor depends on recovery assumptions and required spreads; it is not a guaranteed market price. Issuer calls can force conversion decisions and limit upside.

Formula and assumptions

Callable=straight−issuer call; putable=straight+holder put

Case analysis with explicit assumptions

1. A straight bond with matched credit and cash flows is worth 105. The issuer’s call option is worth 4. A separate put option held by the bondholder is worth 3. Assume internally consistent option valuations. Rates fall sharply after purchase.
2. Method: Callable=straight−issuer call; putable=straight+holder put
3. Case result: 101. Callable value is 101; putable value 108. Effective duration reprices option-adjusted cash flows under curve shocks. Option-adjusted spread depends on the rate model and volatility assumptions; compare OAS only on consistent inputs and do not treat it as a pure measured default probability.

Errors to catch

• Do not replace the stated assumptions with an unstated shortcut. Reconcile units, timing and the claim being valued or duty being assessed.
• Ignoring the option owner is not a defensible conclusion for these case facts.

Original knowledge check

Answer every question before checking. Review explanations after submitting; retry starts a new attempt.

Sources and coverage status

These original lessons are drafts. Public outlines are used to check scope; the official learning-outcome text and proprietary question banks are not reproduced here. Follow the source for exam-year requirements and updates.

• Reference · www.cfainstitute.org
• Reference · www.bis.org