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

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.

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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.

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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 ($V_{\text{cont}} > \text{Call Price}$), the issuer exercises the call. Therefore: $V_{\text{node, callable}} = \min(V_{\text{cont}}, \text{Call Price})$
• Putable Bond: The bondholder acts rationally to maximize asset value. If the continuation value drops below the put price ($V_{\text{cont}} < \text{Put Price}$), the investor puts the bond back to the issuer. Therefore: $V_{\text{node, putable}} = \max(V_{\text{cont}}, \text{Put Price})$

Sensitivity to Interest Rate Volatility ($\sigma$)

Because embedded options represent financial options, their theoretical value increases with higher interest rate volatility:

$\sigma \uparrow \implies V_{\text{call}} \uparrow \implies V_{\text{callable}} = (V_{\text{straight}} - V_{\text{call}}) \downarrow$ $\sigma \uparrow \implies V_{\text{put}} \uparrow \implies V_{\text{putable}} = (V_{\text{straight}} + V_{\text{put}}) \uparrow$

Exam Key Takeaway: As interest rate volatility ($\sigma$) 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 $FV = 100$, callable at $100.00$ at $t = 1$ (European style call).

Binomial Interest Rate Tree:

• $t = 0$: $i_0 = 3.0000%$
• $t = 1$: $i_{1,U} = 6.5000%$, $i_{1,L} = 4.5000%$

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

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

Step 2: Value at Time $t = 1$

• Node 1,U ($i_{1,U} = 6.50%$): $V_{\text{cont, 1U}} = \frac{106.00}{1 + 0.0650} = \frac{106.00}{1.0650} = 99.5305$ Since $V_{\text{cont}} = 99.5305 < 100.00$ (Call Price), the issuer does not call the bond. $V_{\text{callable, 1U}} = \min(99.5305, 100.00) = \mathbf{99.5305}$
• Node 1,L ($i_{1,L} = 4.50%$): $V_{\text{cont, 1L}} = \frac{106.00}{1 + 0.0450} = \frac{106.00}{1.0450} = 101.4354$ Since $V_{\text{cont}} = 101.4354 > 100.00$ (Call Price), the issuer exercises the call at 100.00. $V_{\text{callable, 1L}} = \min(101.4354, 100.00) = \mathbf{100.0000}$

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

• Callable Bond Value ($V_{\text{callable}, 0}$): $V_{\text{callable}, 0} = \frac{0.5 \times (99.5305 + 6.00) + 0.5 \times (100.0000 + 6.00)}{1 + 0.0300} = \frac{0.5 \times 105.5305 + 0.5 \times 106.0000}{1.0300} = \frac{105.7653}{1.0300} = \mathbf{102.6848}$
• Straight Option-Free Bond Value ($V_{\text{straight}, 0}$): $V_{\text{straight}, 0} = \frac{0.5 \times (99.5305 + 6.00) + 0.5 \times (101.4354 + 6.00)}{1.0300} = \frac{0.5 \times 105.5305 + 0.5 \times 107.4354}{1.0300} = \frac{106.4830}{1.0300} = \mathbf{103.3816}$
• Embedded Call Option Value ($V_{\text{call}}$): $V_{\text{call}} = V_{\text{straight}} - V_{\text{callable}} = 103.3816 - 102.6848 = \mathbf{0.6968}$

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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.

$\text{Option Cost (in bps)} = \text{Z-spread} - \text{OAS}$

Relative Value Analysis Using OAS

• If $\text{Market OAS} > \text{Fair (Credit-Justified) OAS}$, the bond is undervalued (cheap / attractive buy).
• If $\text{Market OAS} < \text{Fair (Credit-Justified) OAS}$, the bond is overvalued (rich / sell candidate).
• If the analyst assumes a higher volatility $\sigma$ in the tree: for a callable bond, $V_{\text{call}} \uparrow \implies$ model price drops $\implies$ required OAS must decrease to match market price ($\sigma \uparrow \implies \text{OAS} \downarrow$).

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

$\text{Effective Duration (ED)} = \frac{PV_- - PV_+}{2 \times \Delta y \times PV_0}$ $\text{Effective Convexity (EC)} = \frac{PV_- + PV_+ - 2PV_0}{(\Delta y)^2 \times PV_0}$

Where:

• $PV_0$ = Initial bond price.
• $PV_-$ = Bond price resulting from an instantaneous downward parallel shift of the benchmark curve by $\Delta y$.
• $PV_+$ = Bond price resulting from an instantaneous upward parallel shift of the benchmark curve by $\Delta y$.
• $\Delta y$ = Parallel shift in the benchmark yield curve (expressed in decimal form, e.g., 25 bps = $0.0025$).

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 ($PV_+ - PV_0 > PV_0 - PV_-$), producing negative convexity ($EC < 0$). 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 ($PV_0 - PV_+ < PV_- - PV_0$), producing elevated positive convexity ($EC > 0$). Effective duration shortens as the put option moves deep into the money.

$\text{Duration Hierarchy: } ED_{\text{callable}} \le ED_{\text{straight}} \quad \text{and} \quad ED_{\text{putable}} \le ED_{\text{straight}}$

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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

$\text{Conversion Value} = \text{Current Stock Price} \times \text{Conversion Ratio}$ $\text{Market Conversion Price} = \frac{\text{Convertible Bond Market Price}}{\text{Conversion Ratio}}$ $\text{Market Conversion Premium per Share} = \text{Market Conversion Price} - \text{Current Stock Price}$ $\text{Market Conversion Premium Ratio} = \frac{\text{Market Conversion Premium per Share}}{\text{Current Stock Price}}$ $\text{Premium over Straight Value} = \frac{\text{Convertible Bond Market Price}}{\text{Straight Debt Value}} - 1$

The Three Convertible Regimes

1. 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.
2. 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.
3. 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 $\approx 1.0$), with high equity sensitivity and minimal bond-floor downside protection.