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8.3 Valuation and Analysis of Bonds with Embedded Options: OAS & Duration

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CFA Level 2 Chartered Financial Analyst

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8 Fixed Income Analysis

8.1 The Term Structure and Interest Rate Dynamics: Spot, Forward & Swap Curves 8.2 The Arbitrage-Free Valuation Framework & Binomial Interest Rate Trees 8.3 Valuation and Analysis of Bonds with Embedded Options: OAS & Duration 8.4 Credit Analysis Models: Structural vs Reduced-Form Approaches 8.5 Credit Default Swaps (CDS): Mechanics, Pricing & Trading Strategies 8.6 Managing Yield-Curve Risk: Key Rate Durations, One-Sided Durations & Yield Volatility Structure

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8.3 Valuation and Analysis of Bonds with Embedded Options: OAS & Duration

Key Takeaways Explain This Topic AI

Embedded options alter cash flow timing: Callable bonds grant the issuer the right to retire debt early (V_callable = V_straight - V_call), whereas Putable bonds grant the investor the right to sell debt back (V_putable = V_straight + V_put).

Higher interest rate volatility ( σ \sigma σ ) expands the value of embedded options: V_call increases, causing V_callable to decrease; V_put increases, causing V_putable to increase.

The Option-Adjusted Spread (OAS) is the constant spread added to all tree nodes to match market price, stripping out option risk: Option Cost (bps) = Z-spread - OAS.

Callable bonds exhibit negative convexity and compressed effective duration at low yields because price is capped at the call price, while putable bonds exhibit positive convexity supported by the put floor.

Convertible bonds combine straight debt with an equity call option: Conversion Value = Stock Price × \times × Conversion Ratio, trading across busted debt, hybrid, and equity-like regimes.

Last updated: August 2026

8.3 Valuation and Analysis of Bonds with Embedded Options: OAS & Duration

Core Insight: Traditional yield and duration metrics fail when evaluating bonds with embedded options because future cash flows vary with interest rate movements. Valuing callable and putable bonds requires backward induction on binomial trees with decision rules , separating pure credit/liquidity risk from option risk via the Option-Adjusted Spread (OAS) , and measuring interest rate sensitivity using effective duration and convexity .

1. Taxonomy of Embedded Options in Fixed Income

An embedded option is a contractual provision within a bond indenture that grants the issuer or the bondholder specific rights to alter future cash flows.

Bond Type with Embedded Option Right Holder Economic Trigger Impact on Bond Value vs. Straight Bond

Callable Bond Issuer Rates decline; issuer refinances debt at lower coupon V callable = V straight − V call ≤ V straight V_{\text{callable}} = V_{\text{straight}} - V_{\text{call}} \le V_{\text{straight}} V callable ​ = V straight ​ − V call ​ ≤ V straight ​

Putable Bond Investor Rates rise; investor puts bond and reinvests at higher yield V putable = V straight + V put ≥ V straight V_{\text{putable}} = V_{\text{straight}} + V_{\text{put}} \ge V_{\text{straight}} V putable ​ = V straight ​ + V put ​ ≥ V straight ​

Extendible Bond Investor / Issuer Right to extend maturity at predetermined coupon Acts like a combination of shorter bond + option to extend

Sinking Fund Bond Issuer Mandatory periodic principal retirement via market purchase or call Reduces default risk; call feature lowers upside at low yields

Capped Floater Issuer Rates rise above specified ceiling rate V capped = V straight floater − V cap V_{\text{capped}} = V_{\text{straight floater}} - V_{\text{cap}} V capped ​ = V straight floater ​ − V cap ​

Floored Floater Investor Rates decline below specified floor rate V floored = V straight floater + V floor V_{\text{floored}} = V_{\text{straight floater}} + V_{\text{floor}} V floored ​ = V straight floater ​ + V floor ​

Convertible Bond Investor Underlying stock price appreciates V convertible = V straight + V equity call V_{\text{convertible}} = V_{\text{straight}} + V_{\text{equity call}} V convertible ​ = V straight ​ + V equity call ​

  • 2. Valuation of Callable & Putable Bonds on Binomial Trees
  • Valuation Decision Rules at Each Tree Node

Valuing bonds with embedded options applies backward induction with a terminal decision rule evaluated at every node: Node Evaluation at Time t:

1. Compute Continuation Value: V_cont = [0.5 * (V_u + C) + 0.5 * (V_d + C)] / (1 + i_t)

2. Apply Embedded Option Rule:

  • Callable Bond: V_node = min(V_cont, Call Price)
  • Putable Bond: V_node = max(V_cont, Put Price)

Callable Bond: The issuer acts rationally to minimize liability. If the continuation value of the bond exceeds the call price ( V cont > Call Price V_{\text{cont}} > \text{Call Price} V cont ​ > Call Price ), the issuer exercises the call. Therefore: V node, callable = min ⁡ ( V cont , Call Price ) V_{\text{node, callable}} = \min(V_{\text{cont}}, \text{Call Price}) V node, callable ​ = min ( V cont ​ , Call Price )

Putable Bond: The bondholder acts rationally to maximize asset value. If the continuation value drops below the put price ( V cont < Put Price V_{\text{cont}} < \text{Put Price} V cont ​ < Put Price ), the investor puts the bond back to the issuer. Therefore: V node, putable = max ⁡ ( V cont , Put Price ) V_{\text{node, putable}} = \max(V_{\text{cont}}, \text{Put Price}) V node, putable ​ = max ( V cont ​ , Put Price )

Sensitivity to Interest Rate Volatility ( σ \sigma σ )

Because embedded options represent financial options, their theoretical value increases with higher interest rate volatility: σ ↑ ⟹ V call ↑ ⟹ V callable = ( V straight − V call ) ↓ \sigma \uparrow \implies V_{\text{call}} \uparrow \implies V_{\text{callable}} = (V_{\text{straight}} - V_{\text{call}}) \downarrow σ ↑ ⟹ V call ​ ↑ ⟹ V callable ​ = ( V straight ​ − V call ​ ) ↓ σ ↑ ⟹ V put ↑ ⟹ V putable = ( V straight + V put ) ↑ \sigma \uparrow \implies V_{\text{put}} \uparrow \implies V_{\text{putable}} = (V_{\text{straight}} + V_{\text{put}}) \uparrow σ ↑ ⟹ V put ​ ↑ ⟹ V putable ​ = ( V straight ​ + V put ​ ) ↑

Exam Key Takeaway: As interest rate volatility ( σ \sigma σ ) increases, callable bond prices decrease , while putable bond prices increase . Straight bond prices remain unchanged.

  • 3. Worked Callable Bond Backward Induction Calculation
  • Calibration & Parameters

Consider a 2-year, 6.00% annual coupon bond with face value F V = 100 FV = 100 F V = 100 , callable at 100.00 100.00 100.00 at t = 1 t = 1 t = 1 (European style call).

Binomial Interest Rate Tree:

t = 0 t = 0 t = 0 : i 0 = 3.0000 % i_0 = 3.0000\% i 0 ​ = 3.0000%

t = 1 t = 1 t = 1 : i 1 , U = 6.5000 % i_{1,U} = 6.5000\% i 1 , U ​ = 6.5000% , i 1 , L = 4.5000 % i_{1,L} = 4.5000\% i 1 , L ​ = 4.5000%

Step 1: Terminal Values at Maturity ( t = 2 t = 2 t = 2 )

At maturity t = 2 t = 2 t = 2 , cash flow received is 100 + 6.00 = 106.00 100 + 6.00 = 106.00 100 + 6.00 = 106.00 .

Step 2: Value at Time t = 1 t = 1 t = 1

Node 1,U ( i 1 , U = 6.50 % i_{1,U} = 6.50\% i 1 , U ​ = 6.50% ): V cont, 1U = 106.00 1 + 0.0650 = 106.00 1.0650 = 99.5305 V_{\text{cont, 1U}} = \frac{106.00}{1 + 0.0650} = \frac{106.00}{1.0650} = 99.5305 V cont, 1U ​ = 1 + 0.0650 106.00 ​ = 1.0650 106.00 ​ = 99.5305 Since V cont = 99.5305 < 100.00 V_{\text{cont}} = 99.5305 < 100.00 V cont ​ = 99.5305 < 100.00 (Call Price), the issuer does not call the bond. V callable, 1U = min ⁡ ( 99.5305 , 100.00 ) = 99.5305 V_{\text{callable, 1U}} = \min(99.5305, 100.00) = \mathbf{99.5305} V callable, 1U ​ = min ( 99.5305 , 100.00 ) = 99.5305

Node 1,L ( i 1 , L = 4.50 % i_{1,L} = 4.50\% i 1 , L ​ = 4.50% ): V cont, 1L = 106.00 1 + 0.0450 = 106.00 1.0450 = 101.4354 V_{\text{cont, 1L}} = \frac{106.00}{1 + 0.0450} = \frac{106.00}{1.0450} = 101.4354 V cont, 1L ​ = 1 + 0.0450 106.00 ​ = 1.0450 106.00 ​ = 101.4354 Since V cont = 101.4354 > 100.00 V_{\text{cont}} = 101.4354 > 100.00 V cont ​ = 101.4354 > 100.00 (Call Price), the issuer exercises the call at 100.00. V callable, 1L = min ⁡ ( 101.4354 , 100.00 ) = 100.0000 V_{\text{callable, 1L}} = \min(101.4354, 100.00) = \mathbf{100.0000} V callable, 1L ​ = min ( 101.4354 , 100.00 ) = 100.0000

Step 3: Present Value at Time t = 0 t = 0 t = 0

Callable Bond Value ( V callable , 0 V_{\text{callable}, 0} V callable , 0 ​ ): V callable , 0 = 0.5 × ( 99.5305 + 6.00 ) + 0.5 × ( 100.0000 + 6.00 ) 1 + 0.0300 = 0.5 × 105.5305 + 0.5 × 106.0000 1.0300 = 105.7653 1.0300 = 102.6848 V_{\text{callable}, 0} = \frac{0.5 \times (99.5305 + 6.00) + 0.5 \times (100.0000 + 6.00)}{1 + 0.0300} = \frac{0.5 \times 105.5305 + 0.5 \times 106.0000}{1.0300} = \frac{105.7653}{1.0300} = \mathbf{102.6848} V callable , 0 ​ = 1 + 0.0300 0.5 × ( 99.5305 + 6.00 ) + 0.5 × ( 100.0000 + 6.00 ) ​ = 1.0300 0.5 × 105.5305 + 0.5 × 106.0000 ​ = 1.0300 105.7653 ​ = 102.6848

Straight Option-Free Bond Value ( V straight , 0 V_{\text{straight}, 0} V straight , 0 ​ ): V straight , 0 = 0.5 × ( 99.5305 + 6.00 ) + 0.5 × ( 101.4354 + 6.00 ) 1.0300 = 0.5 × 105.5305 + 0.5 × 107.4354 1.0300 = 106.4830 1.0300 = 103.3816 V_{\text{straight}, 0} = \frac{0.5 \times (99.5305 + 6.00) + 0.5 \times (101.4354 + 6.00)}{1.0300} = \frac{0.5 \times 105.5305 + 0.5 \times 107.4354}{1.0300} = \frac{106.4830}{1.0300} = \mathbf{103.3816} V straight , 0 ​ = 1.0300 0.5 × ( 99.5305 + 6.00 ) + 0.5 × ( 101.4354 + 6.00 ) ​ = 1.0300 0.5 × 105.5305 + 0.5 × 107.4354 ​ = 1.0300 106.4830 ​ = 103.3816

Embedded Call Option Value ( V call V_{\text{call}} V call ​ ): V call = V straight − V callable = 103.3816 − 102.6848 = 0.6968 V_{\text{call}} = V_{\text{straight}} - V_{\text{callable}} = 103.3816 - 102.6848 = \mathbf{0.6968} V call ​ = V straight ​ − V callable ​ = 103.3816 − 102.6848 = 0.6968

  • 4. Option-Adjusted Spread (OAS) Mechanics
  • Z-Spread vs. OAS Market Price of Bond ---> Shift Benchmark Tree Nodes by OAS ---> Model Value = Market Price

Zero-Volatility Spread (Z-spread): The uniform spread added to the spot rate curve that sets the discounted bond cash flows equal to market price. It includes compensation for credit risk, liquidity risk, AND option risk .

Option-Adjusted Spread (OAS): The constant spread added to each forward rate node in the binomial interest rate tree that equates the tree-derived valuation to the market price of the bond. The OAS isolates credit and liquidity risk by stripping out the embedded option's value . Option Cost (in bps) = Z-spread − OAS \text{Option Cost (in bps)} = \text{Z-spread} - \text{OAS} Option Cost (in bps) = Z-spread − OAS

Security Type Option Owner Z-spread vs. OAS Option Cost Sign Economic Rationale

Callable Bond Issuer Z-spread > OAS \mathbf{\text{Z-spread} > \text{OAS}} Z-spread > OAS > 0 \mathbf{> 0} > 0 (Positive) Investor demands higher total yield (Z-spread) to compensate for call risk.

Putable Bond Investor Z-spread < OAS \mathbf{\text{Z-spread} < \text{OAS}} Z-spread < OAS < 0 \mathbf{< 0} < 0 (Negative) Investor accepts lower total yield (Z-spread) in exchange for the put protection.

Option-Free Bond None Z-spread = OAS \mathbf{\text{Z-spread} = \text{OAS}} Z-spread = OAS = 0 \mathbf{= 0} = 0 (Zero) No embedded option exists; credit/liquidity spread is identical.

Relative Value Analysis Using OAS

If Market OAS > Fair (Credit-Justified) OAS \text{Market OAS} > \text{Fair (Credit-Justified) OAS} Market OAS > Fair (Credit-Justified) OAS , the bond is undervalued (cheap / attractive buy).

If Market OAS < Fair (Credit-Justified) OAS \text{Market OAS} < \text{Fair (Credit-Justified) OAS} Market OAS < Fair (Credit-Justified) OAS , the bond is overvalued (rich / sell candidate).

If the analyst assumes a higher volatility σ \sigma σ in the tree: for a callable bond, V call ↑ ⟹ V_{\text{call}} \uparrow \implies V call ​ ↑ ⟹ model price drops ⟹ \implies ⟹ required OAS must decrease to match market price ( σ ↑ ⟹ OAS ↓ \sigma \uparrow \implies \text{OAS} \downarrow σ ↑ ⟹ OAS ↓ ).

  • 5. Effective Duration, Effective Convexity & Negative Convexity Dynamics
  • The Failure of Modified Duration

Modified duration assumes cash flows are fixed and invariant to interest rate shifts. Because embedded options cause cash flows to change when rates change (e.g., bonds get called), interest rate risk must be measured using Effective Duration (ED) and Effective Convexity (EC) .

Computational Formulas Effective Duration (ED) = P V − − P V + 2 × Δ y × P V 0 \text{Effective Duration (ED)} = \frac{PV_- - PV_+}{2 \times \Delta y \times PV_0} Effective Duration (ED) = 2 × Δ y × P V 0 ​ P V − ​ − P V + ​ ​ Effective Convexity (EC) = P V − + P V + − 2 P V 0 ( Δ y ) 2 × P V 0 \text{Effective Convexity (EC)} = \frac{PV_- + PV_+ - 2PV_0}{(\Delta y)^2 \times PV_0} Effective Convexity (EC) = ( Δ y ) 2 × P V 0 ​ P V − ​ + P V + ​ − 2 P V 0 ​ ​

Where:

P V 0 PV_0 P V 0 ​ = Initial bond price.

P V − PV_- P V − ​ = Bond price resulting from an instantaneous downward parallel shift of the benchmark curve by Δ y \Delta y Δ y .

P V + PV_+ P V + ​ = Bond price resulting from an instantaneous upward parallel shift of the benchmark curve by Δ y \Delta y Δ y .

Δ y \Delta y Δ y = Parallel shift in the benchmark yield curve (expressed in decimal form, e.g., 25 bps = 0.0025 0.0025 0.0025 ).

  • Price-Yield Profiles and Negative Convexity Price ($)
  • | / Straight Bond (Positive Convexity everywhere)
  • | . - '
  • | Call Price Ceiling ----. - '
  • | . - ' \ Callable Bond (Negative Convexity at low yields)
  • | . - '
  • | . - '
  • | . - '
  • +--------------------------------------------------------> Yield (%)
  • Low Yields High Yields
  • (Negative Convexity) (Positive Convexity)

Callable Bonds & Negative Convexity: At low interest rates, the bond price is capped by the call price. As yields drop, price appreciation slows and compresses ( P V + − P V 0 > P V 0 − P V − PV_+ - PV_0 > PV_0 - PV_- P V + ​ − P V 0 ​ > P V 0 ​ − P V − ​ ), producing negative convexity ( E C < 0 EC < 0 EC < 0 ) . Effective duration shortens dramatically as the bond is virtually certain to be called.

Putable Bonds & Positive Convexity: At high interest rates, the bond price is supported by the put price floor. Price declines are cushioned ( P V 0 − P V + < P V − − P V 0 PV_0 - PV_+ < PV_- - PV_0 P V 0 ​ − P V + ​ < P V − ​ − P V 0 ​ ), producing elevated positive convexity ( E C > 0 EC > 0 EC > 0 ) . Effective duration shortens as the put option moves deep into the money. Duration Hierarchy: E D callable ≤ E D straight and E D putable ≤ E D straight \text{Duration Hierarchy: } ED_{\text{callable}} \le ED_{\text{straight}} \quad \text{and} \quad ED_{\text{putable}} \le ED_{\text{straight}} Duration Hierarchy: E D callable ​ ≤ E D straight ​ and E D putable ​ ≤ E D straight ​

6. Convertible Bonds: Mechanics and Regimes

A convertible bond grants the investor the right to convert the bond into a specified number of common shares of the issuing company.

Key Analytical Formulas Conversion Value = Current Stock Price × Conversion Ratio \text{Conversion Value} = \text{Current Stock Price} \times \text{Conversion Ratio} Conversion Value = Current Stock Price × Conversion Ratio Market Conversion Price = Convertible Bond Market Price Conversion Ratio \text{Market Conversion Price} = \frac{\text{Convertible Bond Market Price}}{\text{Conversion Ratio}} Market Conversion Price = Conversion Ratio Convertible Bond Market Price ​ Market Conversion Premium per Share = Market Conversion Price − Current Stock Price \text{Market Conversion Premium per Share} = \text{Market Conversion Price} - \text{Current Stock Price} Market Conversion Premium per Share = Market Conversion Price − Current Stock Price Market Conversion Premium Ratio = Market Conversion Premium per Share Current Stock Price \text{Market Conversion Premium Ratio} = \frac{\text{Market Conversion Premium per Share}}{\text{Current Stock Price}} Market Conversion Premium Ratio = Current Stock Price Market Conversion Premium per Share ​ Premium over Straight Value = Convertible Bond Market Price Straight Debt Value − 1 \text{Premium over Straight Value} = \frac{\text{Convertible Bond Market Price}}{\text{Straight Debt Value}} - 1 Premium over Straight Value = Straight Debt Value Convertible Bond Market Price ​ − 1

The Three Convertible Regimes

Busted Convertible (Debt-Like): Stock price is far below conversion price. Conversion value is negligible; the security trades strictly on credit yield and interest rate duration, behaving like straight debt.

Hybrid Regime (Asymmetric Risk/Reward): Stock price is close to conversion price. Offers equity upside participation via the call option while providing downside protection anchored by the straight debt floor.

Equity-Like Regime: Stock price is deep in the money far above conversion price. Convertible price moves virtually 1:1 with the underlying stock (delta ≈ 1.0 \approx 1.0 ≈ 1.0 ), with high equity sensitivity and minimal bond-floor downside protection.