CFA L2: Valuation and Analysis of Bonds with Embedded Options

=== CLUSTER 1: BOND DECOMPOSITION & OPTION VALUE RELATIONSHIPS === CORE CONCEPT KEY FORMULAS A bond with an embedded option is decomposed into two parts: an option-free straight bond plus (or minus) an embedded option. For a callable bond, the issuer holds the call, so the investor is long the bond but short the call — the call option's value is subtracted from the straight bond's value. For a putable bond, the investor holds the put, so the investor is long both the bond and the put — the put option's value is added to the straight bond's value. The option value is never directly observed; it is computed as the residual between the arbitrage-free value of the straight bond and the observed market price of the optioned bond. Both embedded options gain value as interest rate volatility rises, since higher volatility creates more opportunities for the option to be exercised. Callable Bond Decomposition (LOS 3a) The core exam question this cluster answers: Given market prices for a callable/putable bond and the equivalent straight bond, what is the implied option value, and in which direction does it move the bond's price? Use when: pricing a callable bond or backing out call option value from price. Not when: bond is callable but deep out of the money (option value ≈ 0 → callable ≈ straight). Putable Bond Decomposition (LOS 3a) Use when: pricing a putable bond or backing out put option value from price. Not when: bond is putable but deep out of the money (option value ≈ 0 → putable ≈ straight). Implied Call Option Value (LOS 3a) ​ Use when: given both prices, need to isolate option value. Not when: you need a market-implied OAS — use OAS framework instead. Implied Put Option Value (LOS 3a) Use when: given both prices, need to isolate put option value. Not when: put is a soft put (issuer may pay in stock/notes — cash value is ambiguous). ⚠️ ANOMALIES & SPECIAL CASES ⚠​ Rising volatility + callable bond → Call option value rises, so callable bond value falls even if the underlying straight bond value is unchanged. Candidates expect higher volatility to benefit the bondholder; it only benefits putable bondholders here. ⚠​ Deeply in-the-money call → The callable bond price is effectively capped near the call price, causing it to diverge dramatically from the straight bond price, not just be slightly lower. ⚠​ Extendible bond → Economically equivalent to a putable bond (same decomposition logic applies), but the "straight bond" in the decomposition is the shorter-maturity version, not the longer one. This trips candidates who try to apply the putable formula using the extended maturity bond. ⚠​ Sinking fund bond with delivery option → Issuer holds a long straddle (call + put-like delivery option), so value increases with both rising and falling rates — not captured by simple call decomposition. ⚠​ Zero-volatility environment → Option value is not zero; it is positive as long as current forward rates make exercise probable. Candidates confuse zero-volatility assumption with zero option value. 🪤 EXAM TRAPS & YOUR ERRORS 🪤​ Candidates assume a callable bond always trades below par → Correct answer: callable bonds can trade above par if the coupon is high enough; the call merely caps further appreciation beyond the call price. ​ Candidates assume a higher coupon always increases callable bond value → Correct answer: a higher coupon increases both the straight bond value and the call option value; the net effect on the callable bond is ambiguous and depends on how deep in-the-money the call is. ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here 🪤 🪤 🪤 🪤 🪤 🪤 🪤 === CLUSTER 2: BINOMIAL TREE VALUATION WITH EMBEDDED OPTIONS === CORE CONCEPT KEY FORMULAS Valuing a bond with embedded options in the presence of interest rate volatility requires a binomial interest rate tree calibrated to the current yield curve and a volatility assumption. At each node, the one-period forward rates follow the relationship Up-Node Rate Constraint (LOS 3c) ,​ ensuring the up-node rate is always higher than the down-node rate by a volatility-scaled factor. Backward induction begins at maturity: discount each year's cash flow back one period using the node's one-year forward rate, then compare the resulting bond value to the exercise price. For a callable bond, if the computed value exceeds the call price, the issuer calls and the node value is reset to the call price. For a putable bond, if the computed value is below the put price, the investor puts and the node value is reset to the put price. At each interior node, the bond value is the probability-weighted (50/50) average of the two future values plus the coupon, discounted at that node's rate. The core exam question this cluster answers: Given an interest rate tree and coupon terms, what is the arbitrage-free value of the callable or putable bond, and at which nodes is the option exercised? ​ Where: = interest rate volatility; = time step (= 1 for annual). Use when: constructing or verifying the binomial tree calibration. Not when: using a lognormal tree from a vendor — rates are pre-computed. Backward Induction Node Value (LOS 3c) ​ ​= one-year Where: = coupon; forward rate at the node. Use when: computing bond value at any interior node before option-exercise check. Not when: at maturity nodes — use face value + final coupon directly. Option-Exercise Override — Callable (LOS 3c) Use when: issuer holds call option; apply after backward induction at each node. Option-Exercise Override — Putable (LOS 3c) Use when: investor holds put option; apply after backward induction at each node. ⚠️ ANOMALIES & SPECIAL CASES ⚠​ Call not exercised at zero volatility → Even with flat/inverted rates, if the call price is above every forward-discounted value, the option has zero value — the callable bond equals the straight bond. Candidates assume the call is always exercised when rates fall. ⚠​ Bermudan vs. American exercise → In a Bermudan bond (exercisable at multiple dates but not continuously), you check for exercise at each permitted node only, not at every node in the tree. ⚠​ Higher volatility does not always trigger exercise → More nodes go in-the-money, but early exercise is not automatic — the issuer compares staying with calling; a node can go in-the-money but still not be exercised if the benefit from waiting is higher. ⚠​ Coupon paid before discounting → The coupon at each node is added to the expected future value before discounting, not added after. Getting this order wrong produces a systematic undervaluation. ⚠​ Putable bond: put exercised at high-rate nodes → At the top of the tree (highest rates), bond values are lowest and the put is most likely in-the-money. Candidates erroneously check the bottom of the tree for putable exercise. 🪤 EXAM TRAPS & YOUR ERRORS 🪤​ Candidates assume that once a node value is reset to the call price, subsequent backward induction uses the original (uncapped) value → Correct answer: the reset value (min⁡\minmin or max⁡\maxmax) is the value carried backward in all subsequent calculations. ​ Candidates apply the same tree for both callable and putable valuation without adjusting which nodes trigger exercise → Correct answer: callable resets downward at low-rate nodes (bottom of tree); putable resets upward at high-rate nodes (top of tree). ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here 🪤 🪤 🪤 🪤 🪤 🪤 🪤 === CLUSTER 3: OAS, Z-SPREAD & OPTION COST === CORE CONCEPT KEY FORMULAS The Z-spread is the constant spread added to all benchmark spot rates to discount a risky bond's cash flows to its market price; it is used for option-free bonds only, because it assumes fixed cash flows. The OAS (Option-Adjusted Spread) extends this concept to bonds with embedded options: it is the constant spread added to all one-period forward rates in the binomial tree such that the arbitrage-free model value equals the bond's observed market price. Because OAS strips out the option's effect, it represents only credit and liquidity risk premium. The relationship between the three spread measures is: Z-spread = OAS + Option Cost. For a callable bond, the option cost is positive (issuer is long the option) so Z-spread > OAS. For a putable bond, the option cost is negative (investor is long the option) so Z-spread < OAS. OAS is the correct relative-value tool: comparing the OAS of a bond to peers with similar credit quality reveals whether it is cheap (high OAS), rich (low OAS), or fairly priced. Z-spread Valuation — Option-Free Bond The core exam question this cluster answers: Given the OAS and volatility assumption, is the bond rich or cheap, and how does a change in assumed volatility affect the OAS? (LOS 3d) Where: ​= one-period forward rates from benchmark curve; = Z-spread. Use when: risky option-free bond; find spread to benchmark. Not when: bond has embedded options — use OAS instead. OAS Definition (LOS 3d) Solved iteratively: add constant spread OAS to all tree nodes until model price = market price. Use when: any risky bond with or without embedded options. Not when: comparing bonds across different volatility assumptions — OAS is not volatility-invariant. Option Cost (Spread Decomposition) (LOS 3d) For callable: → For putable: → Use when: decomposing the total yield spread into compensation components. Not when: at zero volatility — at σ=0\sigma = 0σ=0, OAS = Z-spread (option cost = 0 only if option is out-of-the-money). ⚠️ ANOMALIES & SPECIAL CASES ⚠​ Higher assumed volatility → lower OAS for callable bond → As σ\sigmaσ rises, the model assigns more value to the call option, so the model price rises at any given OAS; to bring model price back down to market price, the OAS must fall. A bond can look cheap at 10% vol and fairly priced at 20% vol — same bond, same market price. ⚠​ OAS for putable bond rises with volatility → More volatility increases the put's value, lowering the model price at any given OAS; OAS must be raised to bring model price back to market. ⚠​ Z-spread is not additive across options → You cannot decompose the Z-spread of a bond with two embedded options (e.g., callable and putable) by simply adding two option costs. The interaction of both options changes each one's marginal contribution. ⚠​ High OAS ≠ always underpriced → A high OAS may reflect a liquidity discount, model error, or an overly low volatility assumption. Always pair OAS with a sanity check on σ\sigmaσ. ⚠​ OAS at zero volatility equals the Z-spread → This is a definitional identity, but only for option-free bonds. For bonds with embedded options, OAS at zero volatility ≠ Z-spread because the option exercise path is different. 🪤 EXAM TRAPS & YOUR ERRORS 🪤​ Candidates assume Z-spread > OAS for all bonds → Correct answer: for putable bonds, Z-spread < OAS because the investor is long the option; the option cost in the decomposition is negative. ​ Candidates assume that a higher OAS always means a bond is underpriced → Correct answer: OAS is model-dependent; if the volatility input is too low, OAS is overstated and the bond may actually be fairly priced or rich at the correct volatility. ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here 🪤 🪤 🪤 🪤 🪤 🪤 🪤 === CLUSTER 4: EFFECTIVE DURATION & EFFECTIVE CONVEXITY === CORE CONCEPT KEY FORMULAS For bonds with embedded options, modified duration is inappropriate because it assumes fixed cash flows. Effective duration (also called option-adjusted duration) measures price sensitivity to a parallel shift in the benchmark yield curve, holding the OAS constant. The procedure has four steps: (1) compute the OAS at the current price; (2) shift the curve down by Δcurve\Delta\text{curve}Δcurve, regenerate the tree, revalue at that OAS → get ; (3) shift the curve up by the same amount, regenerate, revalue at the same OAS → get ; (4) apply the effective duration formula. ​ Effective convexity captures the asymmetry in price response. A callable bond has negative effective convexity when the call is near-the-money, because the upside is capped but the downside is not. A putable bond always has positive effective convexity because the downside is floored by the put. The maximum effective duration of any bond with an embedded option cannot exceed that of an otherwise identical straight bond. Effective Duration (LOS 3f) The core exam question this cluster answers: Why is modified duration inappropriate for embedded-option bonds, and what do negative effective convexity and asymmetric one-sided durations signal about a callable bond's risk profile? ​ Where: = price after downward curve shift; = price after upward curve shift; = magnitude of shift (decimal); = current full price. Use when: any bond, especially those with embedded options. Not when: you need yield duration (e.g., modified duration for option-free bonds quoted to yield). Effective Convexity (LOS 3f) Same variable definitions as effective duration formula. Use when: assessing curvature of price–yield relationship, especially for callable bonds. Not when: interpreting a single yield-curve scenario — convexity is a second-order measure. One-Sided Duration (Down/Up) (LOS 3g) ​ Use when: the option is near-the-money and the price–yield relationship is asymmetric. Not when: option is deep out-of-the-money (symmetric → one-sided ≈ standard EffDur). ⚠️ ANOMALIES & SPECIAL CASES ⚠​ Callable bond: negative convexity near-the-money → When the call option is near-the-money, ​(price is capped by call price) but falls normally. This ⚠​ ⚠​ ⚠​ ⚠​ produces → negative EffCon. Deep in-the-money call → effective duration ≈ duration to first call date → The callable bond behaves like a short-maturity bond; its EffDur collapses to match the time to the first call date, not the final maturity. Deep out-of-the-money call → EffDur ≈ straight bond duration → The option barely affects pricing; callable bond behaves almost like the straight bond. Putable bond one-sided duration asymmetry → When the put is in-the-money, the bond has higher duration down (prices rise freely) but lower duration up (put floors the price). This is the opposite of callable bonds — a critical distinction on the exam. Key rate durations for callable bonds → Unlike straight bonds, key rate durations for callable bonds depend on both the maturity date and the call date, and intermediate key rates (near the call date) can dominate. 🪤 EXAM TRAPS & YOUR ERRORS 🪤​ Candidates use modified duration for callable bonds → Correct answer: modified duration assumes fixed cash flows and is only valid for option-free bonds; effective duration must be used for any bond with embedded options. ​ Candidates assume callable bonds always have negative convexity → Correct answer: callable bonds have negative convexity only when the call is near-the-money; when the call is deep out-of-the-money, the callable bond exhibits positive convexity similar to the straight bond. ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here 🪤 🪤 🪤 🪤 🪤 🪤 🪤 === CLUSTER 5: INTEREST RATE SENSITIVITY & VOLATILITY EFFECTS === TOP LEFT — CORE CONCEPT TOP RIGHT — KEY FORMULAS Interest rate volatility affects embedded option bonds through two channels: the level/shape of the yield curve and the assumed volatility σ\sigmaσ. A higher σ\sigmaσ widens the dispersion of forward rates in the binomial tree, increasing the probability that any embedded option ends up in-the-money. For a callable bond, higher σ\sigmaσ → higher call value → lower callable bond price and lower OAS. For a putable bond, higher σ\sigmaσ → higher put value → higher putable bond price and higher OAS. The shape of the yield curve also matters: an upward-sloping curve raises forward rates in the tree, reducing call exercise opportunities (call option is cheaper) but increasing put exercise opportunities (put option is more valuable). A flat or inverted curve increases the call option's value and decreases the put option's value. The interaction between curve shape, volatility, and option exercise is always tested in vignettes through scenario comparisons. Callable Bond Value vs. Volatility (LOS 3b) The core exam question this cluster answers: Given a change in volatility or yield curve shape, determine the directional impact on the option value, the bond price, and the OAS for callable vs. putable bonds. where , so . Use when: assessing the impact of a change in assumed volatility on callable bond price. Not when: volatility changes affect credit spreads simultaneously — isolate effects first. Putable Bond Value vs. Volatility (LOS 3b) where , so . Use when: assessing the impact of a change in assumed volatility on putable bond price. Not when: comparing OAS across volatility assumptions — OAS is not volatility-invariant. OAS vs. Volatility for Callable (LOS 3d) Use when: a vignette asks how OAS changes if volatility assumption rises. Not when: the bond is putable — direction reverses. ⚠️ ANOMALIES & SPECIAL CASES ⚠​ Inverted yield curve → higher call option value → When the curve inverts, short-term forward rates exceed long-term rates. More nodes on the binomial tree have low long-term rates → more call exercise opportunities → call value rises. Counterintuitive because an inverted curve is associated with high current rates and lower bond prices. ⚠​ Upward-sloping curve → higher put option value → High forward rates in the tree create more scenarios where bond values fall below the put price. Putable bond investors benefit from steep curves. This is the opposite of what happens with callable bonds. ⚠​ Callable bond return is non-monotonic in rate direction → Steeply declining rates trigger a call and force reinvestment at lower rates. Steeply rising rates depress bond price. Maximum return is achieved at a moderate rate decline — not the steepest decline. ⚠​ At very high volatility, OAS of callable bond approaches zero → Option cost consumes nearly the entire Z-spread. The bond's yield advantage over benchmarks is almost entirely explained by the embedded call, leaving no credit/liquidity premium. ⚠​ Straight bond value is unaffected by volatility → The benchmark for all sensitivity comparisons is the straight bond, which by definition has no optionality and thus no volatility sensitivity. Candidates erroneously 🪤 EXAM TRAPS & YOUR ERRORS 🪤​ Candidates assume rising interest rates always hurt callable bondholders more than straight bondholders → Correct answer: when the call is deep out-of-the-money, callable and straight bonds have nearly identical price sensitivity to rising rates; the divergence is largest when the call is near-the-money and rates fall. ​ Candidates assume that higher volatility always increases bond value → Correct answer: higher volatility increases the value of the option, which helps putable bondholders but hurts callable bondholders (the option is held by the issuer). ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here ​ ................ ← add your mock exam errors here 🪤 🪤 🪤 🪤 🪤 🪤 🪤 adjust the straight bond value when changing . TOP LEFT — CORE CONCEPT TOP RIGHT — KEY FORMULAS Interest rate volatility affects embedded option bonds through two channels: the level/shape of the yield curve and the assumed volatility σ\sigmaσ. A higher σ\sigmaσ widens the dispersion of forward rates in the binomial tree, increasing the probability that any embedded option ends up in-the-money. For a callable bond, higher σ\sigmaσ → higher call value → lower callable bond price and lower OAS. For a putable bond, higher σ\sigmaσ → higher put value → higher putable bond price and higher OAS. The shape of the yield curve also matters: an upward-slopin g curve raises Callable Bond Value vs. Volatility (LOS 3b) Vcallable=Vstraight−Vcall(σ)V_{\text{callable}} = V_{\text{straight}} V_{\text{call}}(\sigma)Vcallable​=Vstraight​−Vcall​(σ) where ∂Vcall∂σ>0\frac{\partial V_{\text{call}}}{\partial \sigma} > 0∂σ∂Vcall​​>0, so ∂Vcallable∂σ<0\frac{\partial V_{\text{callable}}}{\partial \sigma} < 0∂σ∂Vcallable​​<0. Use when: assessing the impact of a change in assumed volatility on callable bond price. Not when: volatility changes affect credit spreads simultaneously — isolate effects first. Putable Bond Value vs. Volatility (LOS 3b) Vputable=Vstraight+Vput(σ)V_{\text{putable}} = V_{\text{straight}} + V_{\text{put}}(\sigma)Vputable​=Vstraight​+Vput​(σ) where ∂Vput∂σ>0\frac{\partial V_{\text{put}}}{\partial \sigma} > 0∂σ∂Vput​​>0, so ∂Vputable∂σ>0\frac{\partial V_{\text{putable}}}{\partial \sigma} > 0∂σ∂Vputable​​>0. Use when: assessing the impact of a change in assumed volatility on putable bond price. Not when: comparing OAS across volatility assumptions — OAS is not volatility-invariant. OAS vs. Volatility for Callable (LOS 3d) ↑σ⇒↑Vcall⇒↑Vmodel at fixed OAS⇒OAS must ↓ to match market price↑σ⇒↑Vcall​⇒↑Vmodel at fixed OAS​⇒OAS must ↓ to match market price Use when: a vignette asks how OAS changes if volatility assumption rises. Not when: the bond is putable — direction reverses. forward rates in the tree, reducing call exercise opportunities (call option is cheaper) but increasing put exercise opportunities (put option is more valuable). A flat or inverted curve increases the call option's value and decreases the put option's value. The interaction between curve shape, volatility, and option exercise is always tested in vignettes through scenario comparisons. The core exam question this cluster answers: Given a change in volatility or yield curve shape, determine the directional impact on the option value, the bond price, and the OAS for callable vs. putable bonds. ⚠️ ANOMALIES & SPECIAL CASES 🪤 EXAM TRAPS & YOUR ERRORS ⚠️ Inverted yield curve → higher call option value → When the curve inverts, short-term forward rates exceed long-term rates. More nodes on the binomial tree have low long-term rates → more call exercise opportunities → call value rises. Counterintuitive because an inverted curve is associated with high current rates and lower bond prices. Upward-slopin g curve → higher put option value → High forward rates in the tree create more scenarios where bond values fall below the put price. Putable bond investors benefit from steep curves. This is the opposite of what happens with callable bonds. ⚠️ 🪤 Candidates assume rising interest rates always hurt callable bondholders more than straight bondholders → Correct answer: when the call is deep out-of-the-money, callable and straight bonds have nearly identical price sensitivity to rising rates; the divergence is largest when the call is near-the-money and rates fall. Candidates assume that higher volatility always increases bond value → Correct answer: higher volatility increases the value of the option, which helps putable bondholders but hurts callable bondholders (the option is held by the issuer). ................ ← add your mock exam errors here ................ ← add your mock exam errors here ................ ← add your mock exam errors here ................ ← add your mock exam errors here ................ ← add your mock exam errors here ................ ← add your mock exam errors here 🪤 ⚠️ Callable bond return is non-monotonic in rate direction → Steeply declining rates trigger a call and force reinvestment at lower rates. Steeply rising rates depress bond price. Maximum return is achieved at a moderate rate decline — not the steepest decline. At very high volatility, OAS of callable bond approaches zero → Option cost consumes nearly the entire Z-spread. The bond's yield advantage over benchmarks is almost entirely explained by the embedded call, leaving no credit/liquidity premium. Straight bond value is unaffected by volatility → The benchmark for ⚠️ ⚠️ all sensitivity comparisons is the straight bond, which by definition has no optionality and thus no volatility sensitivity. Candidates erroneously adjust the straight bond value when changing σ\sigmaσ. === MASTER CHEAT ROW: All 5 Clusters at a Glance === Cluster Key Concept (1-line) OAS Behavior Effective Duration Effective Convexity 1. Bond Decomposition Callable = Straight − Call; Putable = Straight + Put N/A (OAS concept in Cluster 3) Callable ≤\leq≤ Straight; Putable ≤\leq≤ Straight Callable: can be negative (near-money); Putable: always positive 2. Binomial Tree Valuation Backward induction with option-exercise override at each node OAS added to all tree rates to match market price Callable EffDur collapses when call deep in-money; Putable EffDur collapses when put Both callable and putable show asymmetric sensitivity to up vs. down shifts deep in-money 3. OAS, Z-Spread & Option Cost Z-spread = OAS + Option Cost; OAS is the credit/liquidity spread net of option value Callable: OAS held constant when computing EffDur shifts N/A 4. Effective Duration & Convexity Use EffDur (not ModDur) for all optioned bonds; EffCon turns negative for callable near-money OAS is locked in Step 1 of EffDur calculation; shifts in curve do not change OAS assumption Callable: low EffDur when call in-money; Putable: low EffDur when put in-money Callable: negative when call near-money; Putable: always positive (price is floored) 5. Volatility & Rate Sensitivity Higher σ\sigmaσ → higher option value; helps putable holders, hurts callable holders; curve shape sets base exercise probability Callable OAS falls as σ\sigmaσ rises; Putable OAS rises as σ\sigmaσ rises; both converge toward Z-spread as Callable EffDur shortens as rises (more call nodes); Putable EffDur shortens as σ\sigmaσ rises (more put nodes) Callable negative convexity intensifies with OAS; Putable: OAS and flat/inverted curve; Putable positive convexity intensifies with and steep curve