Valuing Bonds with Embedded Options
A call, put, cap or conversion feature makes a bond's cash flows depend on the path of future interest rates — so a single discount rate no longer works. This lesson builds the tool that does: the binomial interest-rate tree, valued by backward induction. We start with benchmark rates and the three spread measures, define the option-adjusted spread (OAS) and use it for relative value, then value callable bonds, putable bonds, step-up notes, capped floaters and convertibles, and measure their interest-rate risk with effective duration and convexity.
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An embedded option is an option that is part of a bond's structure rather than a separate instrument. The common ones are call provisions (the issuer may buy the bond back early), put provisions (the holder may sell it back early), and caps on floating-rate coupons. Because exercising these depends on where rates go, the bond's future cash flows are uncertain even when default is not — so the option-free valuation (discount fixed cash flows at spot rates) breaks down. A convertible bond is harder still: its value depends both on interest rates and on the issuer's share price.
Learning outcomes
- Use relative-value analysis to judge whether a security is undervalued or overvalued.
- Interpret the three spread measures against the relevant benchmark interest rate.
- Use backward induction in a binomial interest-rate tree to value option-free and option-embedded bonds.
- Compute the value of callable and putable bonds, and the value of the embedded option.
- Explain the effect of interest-rate volatility on an embedded option's value.
- Interpret an OAS relative to a nominal spread and a benchmark.
- Calculate effective duration and effective convexity from the binomial model.
- Describe and value a convertible bond and its component values, and contrast it with the stock.
3 · Benchmark interest rates and spread measures
The presence of an embedded option complicates valuation because future cash flows depend on the future level of interest rates. A bond valuation model therefore needs two ingredients: a set of benchmark interest rates to discount against, and an interest-rate model — a probabilistic description of how the rate can change over the bond's life. We use a one-factor binomial model: given a rate model and a volatility assumption, the rate can realise one of two values each period, generating an interest-rate tree. (We will use ready-built trees, not construct them.)
Benchmark rates can be taken from any of three markets, each available as a yield curve or a spot-rate curve — six possible benchmarks in all:
- Treasury market — Treasury yield curve / Treasury spot-rate curve.
- A bond sector with a given credit rating — sector yield curve / sector spot-rate curve.
- The issuer's own securities — issuer yield curve / issuer spot-rate curve.
The spot-rate curve plots, for each maturity, the yield-to-maturity of a zero-coupon bond from today to that maturity; it is bootstrapped from the yield curve.
Nominal spread — the bond's yield minus the benchmark yield. Zero-volatility (Z-) spread — the constant spread added to every benchmark spot rate that makes the discounted cash flows equal the price; it assumes no interest-rate volatility (no option exercise). Option-adjusted spread (OAS) — the constant spread added to every rate in the binomial tree that makes the model value equal the market price. The OAS strips out the part of the spread that is compensation for the option, leaving compensation for credit and liquidity risk.
What each spread compensates for depends on the benchmark. Against the Treasury curve, the nominal and Z-spread reflect credit + option + liquidity risk, while the OAS reflects credit + liquidity (option removed). Against a same-rating sector, the same applies but credit risk is largely netted out. Against the issuer's own curve, even credit is netted out, so the OAS reflects essentially liquidity risk alone.
3.5 · OAS, the benchmark and relative value
The OAS is the workhorse of relative-value analysis, but its sign and size must be read against the benchmark. Taking the Treasury spot-rate curve as benchmark: a zero or negative OAS means the security offers no spread (or a negative spread) over Treasuries and should be avoided. A positive OAS alone does not tell you the bond is cheap — you must compare it with the OAS the market demands on comparable issues.
Call the OAS on comparable securities the required OAS, and the OAS computed for the bond in question the security OAS. Then:
- Security OAS > required OAS → the bond is cheap (undervalued).
- Security OAS < required OAS → the bond is rich (overvalued).
- Security OAS = required OAS → the bond is fairly priced.
The same logic applies when the benchmark is a same-rating bond sector; there the required OAS is the OAS of comparable issues measured against that sector.
5 · The binomial interest-rate tree
Review: valuing an option-free bond
Start with the issuer's on-the-run par curve and bootstrap the spot rates and one-year forward rates:
A 4-year 6.5% option-free bond can be valued either by discounting each cash flow at its spot rate, or by discounting through the chain of one-year forwards — both give the same answer:
Nodes and backward induction
The tree begins at the root node \(N\) with today's one-year rate \(r_0\). Volatility is introduced by letting the rate move to one of two values next period — a higher state (subscript H) or a lower state (L) — and so on. A node is where either a random event (the rate jump) or a decision (exercise the option) occurs. To value the bond we work backwards from maturity. At each node, let \(V_H\) and \(V_L\) be the bond values one year forward in the up and down states and \(C\) the coupon. With equal (risk-neutral) probabilities of ½, the value at the node, discounted at that node's rate \(r^{*}\), is:
The tree is calibrated so that valuing the benchmark on-the-run issue through it reproduces that issue's observed market price. A correctly built tree is therefore arbitrage-free. The two one-year rates in any period are linked by the volatility assumption, e.g. with 10% volatility the rates \(5.4289\%\) and \(4.4448\%\) reprice the 3-year on-the-run issue to par.
Option-free value is volatility-independent
Value the 4-year 6.5% option-free bond on (a) the 10% -volatility tree and (b) the 20% -volatility tree.
Show solution
Rolling the bond back through either tree returns the root value \(\$104.643\) — identical to the spot-rate valuation. Because no option is being exercised, the cash flows are fixed, so the value does not depend on the volatility assumption. Volatility only matters once an embedded option lets the cash flows change.
Both trees give $104.643 — the arbitrage-free option-free value6 · Valuing and analysing a callable bond
Valuing a callable bond uses the same backward induction with one change: at each node the issuer decides whether to call. The bond value at a node is replaced by the lesser of (i) its computed value if not called and (ii) the call price. (We assume the issuer calls whenever the computed value exceeds the call price.) Capping the value at the call price at every node feeds a lower number back through the tree.
A 6.5% bond callable at 100
The 4-year 6.5% bond, callable in one year at \$100, valued on the 10% -volatility tree.
Show solution
With the call rule applied at every node the root value falls to \(\$102.899\) (versus \(\$104.643\) option-free). The embedded call is worth
$$ \$104.643 - \$102.899 = \$1.744. $$
With a softer call-price schedule (102 in yr 1, 101 in yr 2, 100 in yr 3) the issuer can call less often, so the callable bond is worth more — \(\$103.942\) — and the call is worth less.
Callable value $102.899; embedded call worth $1.744The higher the assumed volatility, the more valuable any option — including an embedded one. Re-pricing the same callable bond at 20% volatility gives \(\$102.108\), below the \(\$102.899\) found at 10%: the lower callable value means a larger call option. The issuer is short that option, so higher volatility hurts the callable bondholder.
OAS of the callable bond. If the bond trades at \(\$102.218\), the OAS is the constant spread added to every tree rate that reprices it to \(\$102.218\); here that is 35 bp at 10% volatility. At 20% volatility the OAS is \(-6\) bp. The OAS removes the option component from the nominal spread — it is a spread (point 1) that adjusts the cash flows for the option when measured against the benchmark (point 2).
6.4 · Effective duration and effective convexity
For bonds with embedded options, modified duration is inappropriate (the cash flows move with rates). Effective duration measures the approximate percentage price change for a 100 bp parallel shift, allowing the cash flows to change; effective convexity corrects the linear estimate:
where \(V_0\) is the initial price, and \(V_+\) / \(V_-\) are the values after shifting the curve up / down by \(\Delta y\). The key procedure: (1) find the bond's OAS at the market price; (2) shift the on-the-run curve by \(\Delta y\); (3) rebuild the tree on the shifted curve; (4) add the same OAS to every rate (assume OAS is unchanged); (5) re-value to get \(V_+\) (or \(V_-\)).
Effective duration of the callable bond
Using \(\Delta y = 25\) bp, \(V_0 = 102.218\), the shifted-and-OAS-adjusted trees give \(V_+ = 101.621\) and \(V_- = 102.765\). Find effective duration and convexity.
Show solution
$$ D_{\text{eff}}=\frac{102.765-101.621}{2(102.218)(0.0025)} = 2.24 $$
$$ C_{\text{eff}}=\frac{101.621+102.765-2(102.218)}{2(102.218)(0.0025)^2} = -39.13 $$
The negative convexity is the signature of a callable bond: as rates fall, the call caps the price upside.
Effective duration ≈ 2.24; effective convexity ≈ −39.17 · Valuing a putable bond
A putable bond lets the holder force early repayment. Backward induction is the same, but at each node the value is the greater of the computed value and the put price (the holder puts when the bond is worth less than the put price). Because the holder owns the put, it adds value:
A 6.5% bond putable at 100
The 4-year 6.5% bond, putable in one year at \$100, on the 10% -volatility tree.
Show solution
Applying the "max with the put price" rule gives \(\$105.327\), above the \(\$104.643\) option-free value. The embedded put is worth \(\$105.327 - \$104.643 = \$0.684\). At 20% volatility the putable value rises to \(\$106.010\) — again, more volatility means a more valuable option. A bond that is both callable and putable is valued by applying both adjustments (call cap and put floor) at each node.
Putable value $105.327; embedded put worth $0.684A step-up callable note has a coupon that rises on a set schedule (a single step-up rises once; a multiple step-up rises several times) and is callable. The rising coupon makes the issuer more likely to call once the step-up kicks in. It is valued exactly like any callable bond — apply the call rule at each node — but using the scheduled coupon at each year.
9 · Valuing a capped floater
For a floating-rate note the coupon is reset to the reference rate at the start of each period and paid in arrears at the end. In the tree, the coupon at a node is set by the one-year rate at the previous node. An uncapped floater always values to par at every node (the coupon exactly compensates for the rate). A cap limits the coupon: whenever the reference rate exceeds the cap, the coupon is held at the cap, so the holder receives less than market — and the floater is worth less than par.
A 7.25% capped floater
Value a floater on the 10% -volatility tree (a) with no cap and (b) with a 7.25% cap.
Show solution
(a) With no cap every node values to 100.000 — par. (b) With a 7.25% cap, at nodes where the reference rate would exceed 7.25% (e.g. the 9.1987% and 7.5312% nodes) the coupon is capped, pushing those node values below par; rolled back, the floater is worth \(\$99.724\). The cap is a short call on rates owned by the issuer, so it reduces the floater's value.
Uncapped = par (100.000); 7.25% cap → 99.72410 · Analysis of convertible bonds
A convertible bond can be converted into the issuer's common stock at the holder's option; an exchangeable can be exchanged into the stock of a different firm. Its value depends on both interest rates and the share price, so it bundles bond and equity exposure.
Conversion value = stock price × conversion ratio — what the bond is worth if converted now. Straight value = value as an ordinary (option-free) bond — the assumed price floor. Market conversion price = convertible price ÷ conversion ratio — the effective price paid per share. Market premium per share = market conversion price − stock price. Premium payback period = market premium per share ÷ favourable income differential per share — how long the bond's extra income takes to recoup the premium.
ADC 5.75% convertible
Conversion ratio 25.320 shares per \$1,000 par; stock at \$33; convertible at \$106.50 (i.e. \$1,065 per \$1,000). Find the conversion value and market conversion price.
Show solution
$$ \text{Conversion value}=\$33\times25.320=\$835.6\ \text{per \$1,000 par.} $$
$$ \text{Market conversion price}=\frac{\$1{,}065}{25.320}=\$42.06\ \text{per share,} $$
a market premium of \(\$42.06-\$33=\$9.06\) per share over buying the stock directly. The premium is the price of the bond's downside protection (its straight value).
Conversion value ≈ $836; market conversion price ≈ $42.06; premium ≈ $9.06/shInvestment character depends on the share price. If the stock is low, straight value > conversion value and the bond behaves like a fixed-income equivalent (a "busted convertible"). If the stock is high, conversion value > straight value and it is a common-stock equivalent (small premium). In between it is a hybrid, sharing both characters — giving the asymmetric profile convertibles are bought for: most of the stock's upside, much less of its downside (e.g. for ADC, a +25% move in the stock gives the convertible +6.9%, while a −25% move costs it only −0.3%).
When cash flows depend on the rate path, value the bond on an arbitrage-free binomial tree by backward induction. Insert the option as a decision at each node: cap the value at the call price (callable) or floor it at the put price (putable). Then \(V_{\text{call}}=V_{\text{free}}-V_{\text{callable}}\) and \(V_{\text{put}}=V_{\text{putable}}-V_{\text{free}}\), and higher volatility raises every embedded option's value. The OAS strips the option out of the spread for relative value (security vs required OAS), and effective duration/convexity — computed by shifting the curve and holding the OAS fixed — capture the interest-rate risk, with callables showing the tell-tale negative convexity. Convertibles add an equity call on top of the straight bond floor.