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Eduzan / 06 Fixed Income

FI 3 – Valuation and Analysis of Bonds with Embedded Options

Published: 10 August 2026
Updated: 3 November 2026

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

Valuing a plain fixed-rate bond is a two-part exercise: work out the future cash flows, then discount each one at the rate appropriate to its date. Both parts are mechanical because the cash flows are known in advance. An embedded option breaks the first part. The cash flows now depend on decisions that will be taken later, and those decisions depend on where interest rates go. Valuation therefore has to model the path of interest rates before it can model the cash flows.

An embedded option is a contingency provision written into the indenture or the offering circular. It is a right, not an obligation, and it belongs either to the issuer, to the bondholder, or to neither party in the sense that it operates automatically once a threshold is crossed. Because the provision is written into the bond itself, it cannot be detached and traded on its own. That is the whole force of the word embedded.

Every embedded option sits on top of an underlying option-free bond with a defined issuer, issue date, maturity, principal and repayment pattern, coupon rate and payment pattern, and currency. That underlying instrument is called the straight bond, and it is the reference point for everything that follows. The coupon of the underlying bond may be fixed at a single rate, may step up or step down on a published schedule, or may float against a reference rate plus a credit spread, for example a market reference rate plus 100 basis points. Apart from the section on capped and floored floaters, the discussion assumes a fixed-rate, single-coupon underlying bond.

Call options

A callable bond carries an issuer option to redeem the issue before maturity. The issuer exercises when it can replace an expensive liability with a cheaper one, which happens when interest rates have fallen or when its own credit standing has improved. Until the 1990s most long-dated US corporate bonds were callable after five or ten years, the first call price stood at a premium to par that depended on the coupon, and that premium stepped down to par in the final years. Modern investment-grade corporate issues are for practical purposes non-refundable. Many carry a make-whole call whose exercise price is computed at a tight spread over a benchmark instrument, normally an on-the-run government bond such as a US Treasury or a UK gilt. The call price is then set so high that refunding for economic gain is out of the question, and a bondholder faces no risk of receiving less than the bond is worth.

Most callable bonds also grant a call protection period during which the issuer may not call at all. A ten-year bond with three years of call protection has its first possible call date three years after issue. Protection periods run from a single month to several years. High-yield issuers commonly become callable a few years after issuance, and holders of those bonds usually worry more about default than about early redemption, although that balance can shift over the life of the bond if the credit improves.

The exercise style matters for valuation:

European-style. The issuer may call on one single date and no other.
American-style. The bond is callable continuously from the first call date onward.
Bermudan-style. The bond is callable only on a set of specified dates after the protection period ends.

Callable paper is concentrated in a few places. Bonds of US government-sponsored enterprises such as Fannie Mae, Freddie Mac, the Federal Home Loan Banks and the Federal Farm Credit Banks are, with few exceptions, callable; they tend to run five to ten years, carry very short protection of three months to a year, are almost always callable at 100% of par, and often use a Bermudan schedule. Tax-exempt US municipal bonds are nearly always callable at 100% of par at any time after the end of the tenth year. Beyond those two blocks, call provisions appear across Asia Pacific, Europe, Canada and Central and South America. Most callable bonds are denominated in US dollars or euros, reflecting where investor demand sits, although Australia, the United Kingdom, Japan and Norway all support callable issuance in local currency.

Put options and extension options

A putable bond carries an investor option to sell the bond back to the issuer before maturity, normally at par. The holder exercises when rates have risen and better-yielding alternatives exist. Putable bonds also usually carry protection periods. They are European style or, rarely, Bermudan style. American-style put provisions do not exist.

An extension option achieves something very similar by a different route. The holder of an extendible bond may elect at maturity to keep the bond alive for a further period, possibly on a different coupon. The indenture terms are modified but the bond stays outstanding. A corporate example is an issue from Heathrow Funding Ltd. paying a 0.50% coupon and maturing on 17 May 2024, extendible to 7 May 2026 as a floating-rate note paying 12-month MRR plus 4.00%.

Why a putable bond and an extendible bond are the same instrument

Take a three-year 3.30% bond putable in Year 2 and a two-year 3.30% bond extendible by one year. Their cash flows are identical through Year 2. Whatever happens in Year 3 is driven by the one-year forward rate two years from now, and both instruments respond to it in the same way. If that forward rate is above 3.30%, the holder of the putable bond puts it, because the proceeds can be reinvested at a better yield, and the holder of the extendible bond declines to extend for exactly the same reason. Both bonds then pay 3.30% for two years and redeem. If that forward rate is below 3.30%, the putable bond is not put and the extendible bond is extended, because 3.30% now beats the market. Both bonds then pay 3.30% for three years and redeem. Identical cash flows in every state means identical value; any difference would be an arbitrage. The two structures are equivalent, but their underlying option-free bonds differ: two years for one, three years for the other.

More complex structures

Some bonds are callable and putable at once. A convertible bond carries a conversion option that turns debt into the issuer’s common stock, and convertibles are usually callable as well, which lets the issuer chase cheaper funding or force conversion. An estate put, sometimes called a survivor option, is available on some retail issues and allows the heirs to put the bond at par on the death of the holder; its value therefore depends on the life expectancy of the owner, which is not observable.

A sinking fund bond, or sinker, is the clearest example of a bond stacked with issuer options and no investor option at all. The underlying instrument amortises, for instance a 30-year bond that repays level annual principal instalments from the end of Year 11, so each instalment is 5% of the original principal. On top of that structure a typical sinker adds three provisions.

A standard call above par with declining premiums, available from the end of Year 10, so the whole issue can be retired from Year 10 onward.
An acceleration provision, for example a triple up, which lets the issuer repurchase at par three times the mandatory amount, here 15% of original principal, on any scheduled sinking fund date. If the issuer wants out at the end of Year 11, sinking 15% at par and calling the remainder at a premium costs less than calling everything at a premium. The provision is therefore worth more to the issuer as rates fall.
A delivery option, which lets the issuer meet a sinking fund payment by handing bonds to the trustee instead of cash. If the bonds trade at, say, 90% of par, buying them in the market is cheaper than paying par. This option is worth more to the issuer as rates rise. Its value depends on a liquid market in the bonds, and investors can defend themselves by accumulating the issue and refusing to sell at a discount.

Seen from the issuer side, holding the call and the delivery option together behaves like a long straddle: the purchase of a put and a call on the same underlying at the same strike and expiry. At expiry one leg is worthless and the other is in the money, whichever way the underlying has moved, so the position pays off on a large move in either direction and pays more the larger the move. A sinking fund bond therefore rewards the issuer both when rates fall and when they rise. Valuing the package of underlying bond plus three options is genuinely hard.

Example 1 · Worked

Two short questions on who holds which right and why.

1. What are investors in putable bonds most likely trying to take advantage of: higher interest rates, an improvement in the issuer credit rating, or movements in the price of the issuer common stock?

Solution. Higher interest rates. The put provision lets the holder hand the bond back and redeploy the proceeds into a higher-yielding instrument once rates have risen. An improving credit rating benefits the issuer, because it makes refinancing cheaper and therefore favours a call. Movements in the common stock matter to a conversion option, not to a put.

2. Who holds the conversion option in a convertible bond?

Solution. The bondholders. The conversion option is a call option on the issuer common stock, and the right to exercise it rests with the holders of the bond, not with the issuer and not with the two parties jointly.

The arbitrage-free framework says that a bond with an embedded option is worth the sum of the arbitrage-free values of its parts: the arbitrage-free value of the straight bond, plus the arbitrage-free value of each embedded option, signed according to who holds it. That single sentence generates every relationship in this reading.

Start with the callable bond. The right to call belongs to the issuer, so the investor is long the bond and short the call. A short position reduces value, so the callable bond must be worth less than the straight bond.

Rearranging gives the identity that is actually used in practice, because the straight bond and the callable bond are both easier to value than the option itself:

Equation 1. The call option is priced as a residual.

Now the putable bond. The right to put belongs to the investor, who is therefore long the bond and long the option. Two long positions add, so the putable bond must be worth more than the straight bond.

Equation 2. The put option is also priced as a residual.

The reason both options are computed as residuals is practical. Discounting a known stream of cash flows at known rates is straightforward. Pricing an option whose payoff depends on the level of interest rates on a future date is not, because the exercise decision has to be modelled at every point where exercise is possible. It is far easier to value the bond twice, once with the provision and once without, and take the difference.

Refresher: valuing a default-free option-free bond

An asset is worth the present value of the cash flows it will produce. For a default-free, option-free bond those cash flows are certain, so the only question is which discount rate applies to each one. The answer is the spot rate for that payment date. Spot rates are not usually quoted directly, but they can be extracted from the prices of actively traded on-the-run sovereign bonds. Those prices can equally be expressed as par rates, which are the coupon rates of hypothetical bonds priced at par, or as forward rates. Par rates, spot rates and forward rates carry the same information in three different presentations, and any one of them determines the other two.

The whole of this reading works from one yield curve, so it is worth building it once, carefully.

Equivalent forms of the benchmark yield curve

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 1
Maturity (year)	Par rate (%)	Spot rate (%)	One-year forward rate (%)
1	2.500	2.500	2.500 (0 years from now)
2	3.000	3.008	3.518 (1 year from now)
3	3.500	3.524	4.564 (2 years from now)

All cash flows and values in this lesson are expressed as a percentage of par, and all coupons are annual with annual compounding.

Example 2 · Worked

Build the spot curve and the forward curve from the par rates above, then value a three-year 4.25% annual coupon default-free bond.

1. Derive the one-year, two-year and three-year spot rates.

Solution. With annual coupons and annual compounding, the one-year spot rate equals the one-year par rate. The hypothetical one-year par bond pays 102.500 once, and dividing that by 1.025 returns par, so z₁ = 2.500%.

A two-year par bond carrying the 3.000% par coupon pays 3 in Year 1 and 103 in Year 2, and by construction those two discounted cash flows sum to 100. The first is discounted at the known one-year spot rate, which leaves one unknown:
3 ÷ 1.025 + 103 ÷ (1 + z₂)² = 100, giving z₂ = 3.008%.

The three-year par bond carries the 3.500% par coupon, and the same bootstrap applies with two known discount rates:
3.500 ÷ 1.02500 + 3.500 ÷ (1.03008)² + 103.500 ÷ (1 + z₃)³ = 100, giving z₃ = 3.524%.

2. Derive the one-year forward rates.

Solution. Forward rates come from indifference conditions. An investor with a two-year horizon can lock in the two-year spot rate, or invest for one year and roll into the one-year rate prevailing a year from now. If the two routes did not produce the same terminal wealth there would be an arbitrage, so:
(1 + 0.03008)² = (1 + 0.02500) × (1 + F_1,1), giving F_1,1 = 3.518%.

The same logic one year further out:
(1 + 0.03524)³ = (1 + 0.03008)² × (1 + F_2,1), giving F_2,1 = 4.564%.

3. Value the three-year 4.25% annual coupon bond, using spot rates and then using forward rates.

Solution. Discounting each cash flow at its own spot rate:
4.25 ÷ 1.02500 + 4.25 ÷ (1.03008)² + 104.25 ÷ (1.03524)³ = 102.114.

Rolling the cash flows back one year at a time through the forward rates must give the same answer, because the two curves carry the same information:
4.25 ÷ 1.02500 + 4.25 ÷ (1.02500 × 1.03518) + 104.25 ÷ (1.02500 × 1.03518 × 1.04564) = 102.114.

This 102.114 is the straight bond value that every callable and putable version of this bond will be measured against.

The forward rate presentation is the one to carry forward. Spot rates value the bond correctly but say nothing about what the bond will be worth part way through its life. Options are exercised part way through, so the model has to produce a value at every intermediate date, and only the period-by-period forward rate approach does that naturally.

Before adding volatility it is worth walking through the exercise decision in the simplest possible setting: interest rates evolve exactly along today’s forward curve and never deviate. The tree collapses to a single path, and the only question at each date is whether the option holder exercises.

The exercise rule follows from who lent and who borrowed. The issuer borrowed, so the issuer calls when the present value of the remaining cash flows exceeds the call price, because retiring the debt at the call price is then cheaper than leaving it outstanding. The investor lent, so the investor puts when the present value of the remaining cash flows falls below the put price, because handing the bond back at the put price then beats holding it.

Example 3 · Worked

Take the same three-year 4.25% annual coupon default-free bond, now in Bermudan-style callable and putable versions, each exercisable at par on either of the next two anniversaries. Use the forward rates of 2.500%, 3.518% and 4.564% derived earlier.

Backward induction at zero volatility, callable at par

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 2
	Today	Year 1	Year 2	Year 3
Cash flow		4.250	4.250	104.250
Discount rate		2.500%	3.518%	4.564%
Value before exercise	101.707	100.417	99.700
Exercise decision		Called at 100	Not called

1. Value the callable bond and the embedded call option.

Solution. Work from maturity backwards.

Year 2. Discount the Year 3 cash flow at the forward rate two years from now:
104.250 ÷ 1.04564 = 99.700. This is below the call price of 100, so a rational issuer leaves the bond outstanding.

Year 1. Add the Year 2 coupon to that value and discount at the forward rate one year from now:
(99.700 + 4.250) ÷ 1.03518 = 100.417. This exceeds the call price of 100, so the issuer calls. Reset the value at that date to 100.

Today. Add the Year 1 coupon to the reset value and discount at 2.500%:
(100 + 4.250) ÷ 1.02500 = 101.707.

Applying Equation 1 with the straight bond value of 102.114:
Value of the issuer call option = 102.114 − 101.707 = 0.407.

Backward induction at zero volatility, putable at par

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 3
	Today	Year 1	Year 2	Year 3
Cash flow		4.250	4.250	104.250
Discount rate		2.500%	3.518%	4.564%
Value before exercise	102.397	100.707	99.700
Exercise decision		Not put	Put at 100

2. Value the putable bond and the embedded put option.

Solution. Same recursion, opposite decision rule.

Year 2. 104.250 ÷ 1.04564 = 99.700, which is below the put price of 100, so the investor puts and the value is reset to 100.

Year 1. (100 + 4.250) ÷ 1.03518 = 100.707, which is above the put price, so the investor holds.

Today. (100.707 + 4.250) ÷ 1.02500 = 102.397.

Applying Equation 2:
Value of the investor put option = 102.397 − 102.114 = 0.283.

3. Sense-check the two answers against the value relationships.

Solution. The callable bond at 101.707 sits below the straight bond at 102.114, exactly as a short option position requires. The putable bond at 102.397 sits above it, exactly as a long option position requires. Note also that the same node, Year 2 at a value of 99.700, triggers no action from the issuer and immediate action from the investor. That is the whole asymmetry of the two provisions in one number.

Is it worth exercising, and is it worth exercising now?

Two separate questions face the holder of an exercisable embedded option. The first is whether exercise pays at all, and it is usually settled by comparing the value of exercising with the value of not exercising. Suppose a bond is putable at 100. If it trades above 100, putting it destroys value, because selling in the market raises more than the put price. If it trades at 100, putting it is worth serious consideration. The bond cannot trade below 100 while the put is live, because anyone could buy it cheaply and put it at 100 for a riskless gain. The issuer faces the mirror image: calling a bond that trades far below the call price is pointless when the same bonds can be bought in the market for less, whereas calling makes sense once the price sits close to the call price.

The second question only arises for American-style and Bermudan-style options. Even when exercise is profitable today, waiting might be more profitable still, and it might also be less profitable, because circumstances can improve or deteriorate. A European-style option removes the question entirely: if exercise pays on the single available date, it should be exercised. The valuation framework used throughout this reading assumes that option holders are risk neutral and exercise if and only if the benefit of exercising now exceeds the expected benefit of waiting. Real holders may be risk averse and may exercise early even when the option is worth more alive than dead.

Example 4 · Worked

A portfolio manager is reviewing three five-year annual coupon bonds issued by a sovereign government. The bonds are identical apart from their option provisions. Bond A is option free. Bond B is callable at par two years and three years from today. Bond C is callable at par two years and three years from today and also putable at par one year from today.

1. How does the value of Bond B compare with the value of Bond A?

Solution. Lower. Bond A is the underlying straight bond for Bond B. The call is an issuer option that subtracts value from the investor. If rates fall and prices rise, the appreciation of Bond B is capped, because the issuer will redeem and refinance more cheaply.

2. How does the value of Bond C compare with the value of Bond B?

Solution. Higher. Bond C carries everything Bond B carries plus a put. The put is an investor option and adds value, so Bond C must be worth more than Bond B.

3. If interest rates are expected to rise, which option in Bond C is likely to be exercised?

Solution. The put. Rising rates depress bond prices, so the holder gains by putting the bond back at par and reinvesting at the higher prevailing yield. Rising rates push the call further out of the money, so the issuer will not call.

Zero volatility is a teaching device, not a description of markets. Once rates are allowed to move, the option holder has to consider every path the yield curve might take, and the standard way of representing those paths is an interest rate tree or lattice.

Figure 1: A recombining binomial lattice of one-period forward rates

The rates shown are the 10% volatility lattice used throughout this lesson. Each node branches to a higher and a lower state with equal probability, and the branches recombine, so a two-period tree has three states rather than four.

Volatility raises the value of every embedded option

The value of an embedded option rises with interest rate volatility, and this is true of every type of embedded option regardless of who holds it. The reason is that wider dispersion of future rates means more states of the world in which exercise is worthwhile, and an option is never worth less for having more chances to pay off.

Consider a 30-year 4.50% bond callable at par in 10 years, valued on a 4% flat yield curve. The straight bond value does not move with volatility at all, because its cash flows are fixed. The call option value does move, and it moves a long way: it is worth 4.60% of par at zero volatility and 14.78% of par at 30% volatility. Since the callable bond is the straight bond minus the call, rising volatility drives the callable bond value down.

Now take a 30-year 3.75% bond putable at par in 10 years on the same 4% flat curve. Again the straight bond value is untouched by volatility. The put is worth 2.30% of par at zero volatility and 10.54% of par at 30% volatility. Since the putable bond is the straight bond plus the put, rising volatility drives the putable bond value up.

Effect of interest rate volatility on embedded option values

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 4
Instrument (4% flat curve)	Option value at 0% volatility	Option value at 30% volatility	Effect on the bond
30-year 4.50% callable at par in 10 years	4.60% of par	14.78% of par	Callable bond value falls
30-year 3.75% putable at par in 10 years	2.30% of par	10.54% of par	Putable bond value rises

The level of the curve

Take the same 30-year 4.50% callable bond and hold volatility at 15% while sliding the flat yield curve up and down. On a 5% flat curve the straight bond is worth 92.27% of par, the call is worth 5.37% of par, and the callable bond is therefore worth 86.90% of par. Drop the curve to 3% flat and the straight bond rises to 129.54% of par, a gain of 40%, while the callable bond rises only to 110.43% of par, a gain of 27%. The straight bond captures the full benefit of falling rates; the callable bond gives a large part of it away to the growing call option. That is the practical meaning of capped upside.

The putable bond shows the mirror image, and it is why the put is described as a hedge against rising rates. Moving the curve from 3% flat to 5% flat costs the straight bond 30% of its value, but the putable bond loses only 22%, because the put gains value as the straight bond loses it.

The shape of the curve

Shape matters as well as level, and the mechanism is easiest to see through the tree. The call option becomes more valuable as the curve flattens and more valuable still if it inverts. For the 30-year 4.50% callable bond at 15% volatility, the embedded call is worth roughly 8% of par when the curve slopes upward from 2% at the short end to 4% at the long end, roughly 10% of par when the curve is flat at 4%, and more than 12% of par when the curve slopes downward from 6% short to 4% long. Inverted curves are unusual but they do occur.

The intuition is entirely about the forward rates that populate the lattice. An upward-sloping curve implies high one-period forward rates, and high forward rates mean few nodes at which calling is attractive. Flatten or invert the curve and a large number of nodes carry lower forward rates, which multiplies the occasions on which the issuer would want to redeem.

The put behaves in the opposite direction for the same reason. Its value falls as the curve moves from upward sloping to flat to downward sloping. High forward rates on an upward-sloping curve are exactly the states in which the investor wants to put the bond and reinvest, so an upward-sloping curve is the friendliest environment for a put.

One consequence is worth remembering for issuance. Under a normal upward-sloping curve, a callable bond issued at par has a call option that is out of the money; the bond would not be called if the arbitrage-free forward rates at zero volatility actually came to pass. Callable bonds issued at a large premium, which happens frequently in the US municipal sector, start life with the call in the money and would be called if those forward rates prevailed.

The procedure has three steps, and every question of this type in the curriculum is an application of them.

Generate a tree of one-period forward rates consistent with the given par yield curve and the given interest rate volatility.
At each node, test whether the embedded option is exercised and reset the value if it is.
Apply backward induction, starting at maturity and working right to left, to reach today’s value.

The recursion at each node is a single expression. The value at a node equals the coupon plus the probability-weighted average of the two values it leads to, all discounted one period at the rate attached to that node.

Backward induction with equal branching probabilities, before any exercise test.

How the lattice is calibrated

The tree is not free. It has to reprice the benchmark curve exactly, and within each time slice the rates have to be spaced by the volatility assumption. Two conditions pin it down at every step. The rate in the up state is linked to the rate in the down state by

Here σ is the interest rate volatility and t is the length of a time slice in years, so t equals 1 for annual steps.

and the rates in the slice must discount the par bond of the matching maturity back to exactly 100. There is no closed-form solution, so the rates are found iteratively. At the first maturity the par rate, the spot rate and the forward rate applying to the first year are all the same number, 2.500%, which fixes the root of the lattice. With the 10% volatility assumption, the Year 1 rates that reprice the two-year 3.000% par bond at 100 are 3.1681% in the down state and 3.8695% in the up state. Those rates are then frozen, and the Year 2 rates are solved under the same two conditions against the three-year 3.500% par bond, producing 3.7041%, 4.5242% and 5.5258%.

Example 5 · Worked

Value the three-year 4.25% annual coupon bond on the 10% volatility lattice, given that the issuer may redeem it at par in one year and again in two. The straight bond is worth 102.114 and the callable bond at zero volatility was worth 101.707.

Callable bond at 10% volatility, node by node

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 5
Node	Rate	Value before exercise	Value after exercise
Year 2, up	5.5258%	98.791	98.791
Year 2, middle	4.5242%	99.738	99.738
Year 2, down	3.7041%	100.526	100.000 (called)
Year 1, up	3.8695%	99.658	99.658
Year 1, down	3.1681%	100.922	100.000 (called)
Year 0	2.5000%	101.540	101.540

1. Work through the backward induction.

Solution. Start at Year 2, where the only remaining cash flow is 104.250 at Year 3:
98.791 = 104.250 ÷ 1.055258
99.738 = 104.250 ÷ 1.045242
100.526 = 104.250 ÷ 1.037041

Test each against the call price of 100. Only the bottom state, where the rate is 3.7041%, produces a value above par, so that node alone is reset from 100.526 to 100.

Move back to Year 1, adding the coupon and averaging the two states ahead at equal probabilities:
99.658 = [4.250 + (0.5 × 98.791 + 0.5 × 99.738)] ÷ 1.038695
100.922 = [4.250 + (0.5 × 99.738 + 0.5 × 100)] ÷ 1.031681
Note that the reset value of 100 feeds into the second calculation, not the pre-exercise 100.526. Testing again, the lower Year 1 node exceeds the call price, so it is reset from 100.922 to 100.

Finally, discount to today:
101.540 = [4.250 + (0.5 × 99.658 + 0.5 × 100)] ÷ 1.025000.

2. What is the call option worth, and how does it compare with the zero-volatility answer?

Solution. By Equation 1, the call is worth 102.114 − 101.540 = 0.574. At zero volatility it was worth 0.407. The increase is exactly what the volatility argument predicts: more dispersion, more states in which calling pays, a more valuable option and therefore a less valuable callable bond.

3. If volatility rose from 10% to 15% with nothing else changed, what happens to the callable bond value?

Solution. It falls below 101.540. Higher volatility raises the call option value, and the call value is subtracted from the straight bond value.

4. If the bond were callable at 102 instead of 100, what would it be worth?

Solution. 102.114, the straight bond value. Reading down the value column, the highest pre-exercise value anywhere on the tree is 100.922, well below a call price of 102. The call is never exercised in any state, so it is worthless and the callable bond collapses onto the straight bond.

Example 6 · Worked

Now the putable version: the same three-year 4.25% annual coupon bond on the same 10% volatility lattice, but with the holder entitled to sell it back at par in one year and again in two.

Putable bond at 10% volatility, node by node

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 6
Node	Rate	Value before exercise	Value after exercise
Year 2, up	5.5258%	98.791	100.000 (put)
Year 2, middle	4.5242%	99.738	100.000 (put)
Year 2, down	3.7041%	100.526	100.526
Year 1, up	3.8695%	100.366	100.366
Year 1, down	3.1681%	101.304	101.304
Year 0	2.5000%	102.522	102.522

1. Value the putable bond and the embedded put.

Solution. The Year 2 pre-exercise values are the same three numbers as before, because the underlying cash flows have not changed. The test is now reversed: the investor puts wherever the value falls below 100. That happens twice, in the states where the rates are 5.5258% and 4.5242%, and both are reset to 100. Neither Year 1 node falls below the put price, so no exercise occurs there. Rolling back to today gives 102.522.

By Equation 2, the put is worth 102.522 − 102.114 = 0.408, against 0.283 at zero volatility, again consistent with the volatility rule.

2. If volatility rose from 10% to 20%, what happens to the putable bond value?

Solution. It rises above 102.522. The put option gains value and the put value is added to the straight bond value.

3. If the bond were putable at 95 instead of 100, what would it be worth?

Solution. 102.114. The lowest pre-exercise value anywhere on the tree is 98.791, comfortably above a put price of 95, so the put is never exercised and is worth nothing. The putable bond collapses onto the straight bond.

Example 7 · Worked

A fixed-income associate is analysing three default-free bonds issued by Weather Analytics, a state-owned company. All three mature three years from today. Bond X pays a 5.2% annual coupon and is callable at par one year and two years from today. Bond Y is callable at par one year and two years from today, its coupon is not disclosed, and it is priced at 101.325% of par. Bond Z pays a 4.8% annual coupon and is putable at par two years from today.

The one-year, two-year and three-year par rates are 4.400%, 4.700% and 5.000%. At an estimated 15% interest rate volatility the lattice is 4.4000% at Year 0; 5.7678% and 4.2729% at Year 1; and 7.4832%, 5.5437% and 4.1069% at Year 2.

1. What is the price of Bond X: 96.057, 99.954 or 100.547% of par?

Solution. 99.954% of par. Year 2 values before exercise are 105.200 ÷ 1.074832 = 97.876, 105.200 ÷ 1.055437 = 99.674 and 105.200 ÷ 1.041069 = 101.050. Only the bottom node exceeds the call price, so it resets to 100.
Year 1 up: [5.200 + (0.5 × 97.876 + 0.5 × 99.674)] ÷ 1.057678 = 98.305, below par, so no call.
Year 1 down: [5.200 + (0.5 × 99.674 + 0.5 × 100)] ÷ 1.042729 = 100.733, above par, so it resets to 100.
Today: [5.200 + (0.5 × 98.305 + 0.5 × 100)] ÷ 1.044000 = 99.954.

2. What is the coupon rate of Bond Y: 4.200%, 5.000% or 6.000%?

Solution. 6.000%. The tree confirms it: at a 6% coupon the Year 2 values are 98.620, 100.432 and 101.818, of which the lower two reset to 100; the Year 1 values are 99.567 and 101.656, of which the lower node resets to 100; and today’s value is 101.325, which is the quoted price.

The answer can also be reached without any arithmetic. A three-year 5% straight bond must be worth exactly par, because the three-year par rate is 5%. A call provision can only reduce that, so a three-year 5% callable bond must be worth less than par and cannot possibly be 101.325. A 4.2% coupon is worth even less. Only the 6% coupon can support a price above par.

3. What is the price of Bond Z: 99.638, 100.340 or 100.778% of par?

Solution. 100.340% of par. Bond Z is putable at par two years from today only, so the exercise test applies at Year 2 and nowhere else. Year 2 values before exercise are 104.800 ÷ 1.074832 = 97.504, 104.800 ÷ 1.055437 = 99.295 and 104.800 ÷ 1.041069 = 100.666. The first two fall below the put price and reset to 100; the third is left alone.
Year 1 up: [4.800 + (0.5 × 100 + 0.5 × 100)] ÷ 1.057678 = 99.085. Year 1 down: [4.800 + (0.5 × 100 + 0.5 × 100.666)] ÷ 1.042729 = 100.825. No test applies at Year 1.
Today: [4.800 + (0.5 × 99.085 + 0.5 × 100.825)] ÷ 1.044000 = 100.340.

4. What happens to the price of Bond X at a lower volatility, and to the price of Bond Z at a higher volatility?

Solution. Bond X rises and Bond Z rises. Lower volatility reduces the value of the call embedded in Bond X, and the call is subtracted, so Bond X gains. Higher volatility raises the value of the put embedded in Bond Z, and the put is added, so Bond Z gains as well.

Everything so far has assumed the bond cannot default. That assumption is defensible for sovereign issuance in local currency and for very little else, so the framework has to be extended.

There are two ways to bring default risk into a valuation. The industry-standard route is to raise the discount rates above the default-free rates. Higher discount rates give lower present values, so a risky bond is worth less than an otherwise identical default-free bond. The alternative route models default explicitly, attaching a probability to each future period, for instance a 1% chance of default in Year 1 and a 1.25% chance in Year 2 conditional on surviving Year 1, together with an assumed recovery value such as 40% of par. Credit default swap prices are one source of the required default probabilities and recovery assumptions. This lesson uses the first route.

From Z-spread to option-adjusted spread

Building a discount curve for a risky bond can be done two ways. The better method uses an issuer-specific curve representing that issuer’s own borrowing rates across maturities, but few practitioners have access to that level of detail. The workable method lifts every one-year forward rate derived from the default-free benchmark curve by the same fixed amount, estimated from the market prices of bonds of comparable credit quality. That constant addition is the zero-volatility spread, or Z-spread.

Applying a 100 bps Z-spread to the three-year 4.25% option-free bond means discounting at 3.500%, 4.518% and 5.564% instead of 2.500%, 3.518% and 4.564%:

4.25 ÷ 1.03500 + 4.25 ÷ (1.03500 × 1.04518) + 104.25 ÷ (1.03500 × 1.04518 × 1.05564) = 99.326, against 102.114 for the default-free version.

The same idea applied to a lattice gives the option-adjusted spread. The OAS is the constant spread which, added to every one-period forward rate on the tree, makes the model value of the bond equal its observed market price. For an option-free bond the Z-spread is simply the OAS computed at zero volatility, which is a useful way of remembering what the two measures have in common.

Figure 2: Locating the option-adjusted spread by trial and error

Model value falls steadily as the spread rises. The OAS is read off where the model value line meets the market price line, and a spread that is too large produces a value below the market price.

Example 8 · Worked

A risky bond pays a 4.25% annual coupon, runs for three years and may be redeemed by the issuer at par in one year and again in two. It is otherwise identical to the bond valued on the 10% volatility lattice. Its market price is 101.000.

1. Find the option-adjusted spread.

Solution. Add a trial spread to every rate on the tree and revalue.

Adding 30 bps produces a value of 100.973, which is below the market price. Because price and yield move in opposite directions, a model value below the market price means the discount rates are too high, so the next trial spread must be smaller.

Adding 28 bps produces a value of 101.010, now slightly above the market price.

Iterating between the two gives the spread that reproduces 101.000 exactly: 28.55 bps. That is the OAS.

The 10% volatility lattice shifted by an OAS of 28.55 bps

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 7
Node	Benchmark rate	Rate plus OAS	Value at the node
Year 2, up	5.5258%	5.8114%	98.524
Year 2, middle	4.5242%	4.8097%	99.466
Year 2, down	3.7041%	3.9896%	100.250, called at 100
Year 1, up	3.8695%	4.1550%	99.126
Year 1, down	3.1681%	3.4536%	100.512, called at 100
Year 0	2.5000%	2.7855%	101.000

2. Why is the spread described as option adjusted?

Solution. Because the exercise test is applied at every node while the spread is being solved for. Two nodes here are reset to the call price of 100, so the compensation the investor demands for the risk of being called is already reflected in the model values. Whatever spread remains is compensation for credit and liquidity, with the option risk stripped out. That is precisely what makes the number comparable across bonds with different option structures.

3. How should the resulting OAS be used?

Solution. As a relative value measure against bonds of similar characteristics and credit quality. An OAS below the comparable group suggests the bond is rich and should be avoided. An OAS above the group suggests it is cheap. An OAS close to the group suggests fair pricing. The 28.55 bps figure means nothing on its own; it means something only next to a peer group.

Volatility and the OAS

The dispersion of rates across the lattice depends on the volatility assumption, and so, therefore, does the OAS. Take a 5% annual coupon bond with 23 years to maturity, callable in three years, priced at 95% of par and valued on a 4% flat curve. Its OAS falls from 138.2 bps at 0% volatility to 1.2 bps at 30% volatility.

The direction is worth reasoning through rather than memorising. Raising volatility raises the value of the call, which lowers the model value of the callable bond at any given spread. To bring the model value back up to the fixed market price, the spread must come down. So for a callable bond, higher assumed volatility means lower OAS.

The practical warning is large. A callable bond that looks cheap at 10% volatility may look fairly priced or expensive at 20%. The volatility input is not a technical detail; it drives the conclusion. Two analysts comparing OAS figures computed on different volatility assumptions are not comparing anything at all.

Example 9 · Worked

A portfolio manager has valued a 7% annual coupon bond issued by a French company with three years remaining to maturity, callable at par one year and two years from now. The valuation used the on-the-run French government curve, with one-year, two-year and three-year par rates of 4.600%, 4.900% and 5.200%. At an estimated 15% volatility the benchmark lattice is 4.6000% at Year 0; 5.9988% and 4.4440% at Year 1; and 7.7515%, 5.7425% and 4.2541% at Year 2. On that basis the callable bond is worth 102.294% of par. A colleague objects that a corporate bond is riskier than French government debt and that the valuation should carry an OAS of 200 bps.

1. How should the 200 bps be applied?

Solution. Add 200 bps to every rate in the binomial interest rate tree. The OAS is by definition the constant spread added to all the one-period forward rates on the tree. It is not a deduction from the coupon, and it is certainly not subtracted from the rates, which would raise the value rather than lower it.

2. What is the value of the callable bond at 15% volatility with the 200 bps OAS: 99.198, 99.247 or 104.288% of par?

Solution. 99.247% of par. Every rate rises by 200 bps, so the root becomes 6.6000% and the Year 2 rates become 9.7515%, 7.7425% and 6.2541%.
Year 2 values before exercise: 107.000 ÷ 1.097515 = 97.493, 107.000 ÷ 1.077425 = 99.311 and 107.000 ÷ 1.062541 = 100.702, of which only the last exceeds the call price and resets to 100.
Year 1 up: [7.000 + (0.5 × 97.493 + 0.5 × 99.311)] ÷ 1.079988 = 97.595. The Year 1 down node comes out above the call price and resets to 100.
Today: [7.000 + (0.5 × 97.595 + 0.5 × 100)] ÷ 1.066000 = 99.247.

The drop from 102.294 to 99.247 is the price of recognising the credit risk.

3. Holding that price fixed, what happens to the OAS if volatility is raised from 15% to 20%?

Solution. The OAS falls. Higher volatility means a more valuable call and a lower model value at any spread, so a smaller spread is needed to reproduce the same price.

Scenario analysis over an investment horizon

The same machinery supports scenario analysis, and it produces one result that catches people out. Over a specified horizon, performance is a trade-off between reinvestment income and the change in the value of the principal. Take a 4.5% bond with five years to maturity and a one-year horizon. If the bond is option free, higher rates lift reinvestment income but depress the terminal price. Over a horizon this short the reinvestment income is small, so the price change dominates and lower rates deliver the better outcome.

Make the same bond callable, first callable six months from now, currently priced at 99.74, and the conclusion breaks. Sharply higher rates hurt through the price, as expected. Sharply lower rates also hurt, because the bond is called away and both the coupon and the principal have to be reinvested at the new lower rates. Falling rates do not guarantee a better return on a callable bond. Making optimal, scenario-dependent exercise decisions is computationally demanding precisely because the call or put decision must be evaluated along the whole path of rates within the holding period, and assuming that the bond survives to the horizon date would overstate performance.

Measuring and managing interest rate exposure is central to fixed-income portfolio management, from hedging a book to matching assets against liabilities to monitoring a portfolio against its benchmark. Duration and convexity are the two standard measures, and for bonds with embedded options only one version of each is usable.

Duration measures the sensitivity of the full price of a bond, accrued interest included, either to a change in its own yield to maturity, in the case of yield duration measures, or to a change in benchmark interest rates, in the case of curve duration measures. Yield duration measures such as modified duration assume the expected cash flows do not change as the yield changes. For a bond with an embedded option that assumption is simply false, because the option value, and therefore the exercise decision, is contingent on interest rates. The only appropriate measure is the curve duration measure known as effective duration, sometimes called option-adjusted duration. Since effective duration works perfectly well for option-free bonds too, many practitioners use it for everything.

Equation 3. The shift is applied to the benchmark yield curve, in particular the government par curve, and the credit spread is held constant.

The three inputs are the full price after shifting the benchmark curve down by the chosen amount, the full price after shifting it up by the same amount, and the current full price. In principle the three prices need an issuer-specific curve; in practice the analyst has the market price and works from that instead, which gives a four-step procedure.

Given the current price, solve for the implied OAS to the benchmark curve at an appropriate volatility.
Shift the benchmark curve down, rebuild the lattice, and revalue the bond at the OAS from step 1. That value is the down-shifted price.
Shift the benchmark curve up by the same amount, rebuild the lattice, and revalue at the same OAS. That value is the up-shifted price.
Apply Equation 3.

The step that people skip is holding the OAS constant. The credit spread is not supposed to change when the benchmark curve moves; only the benchmark is being shifted. Recomputing the OAS at each shifted curve would mix a credit effect into what is meant to be a pure interest rate sensitivity.

Example 10 · Worked

Go back to the three-year 4.25% callable bond, redeemable at par at the issuer option in one year and again in two, on the same par curve of 2.500%, 3.000% and 3.500%, at 10% volatility, with a current full price of 101.000. Use a 30 bps parallel shift.

Effective duration inputs for the callable bond

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 8
Scenario	Root rate on the shifted lattice	Full price
Benchmark curve down 30 bps	2.4850%	101.599
No shift	2.7855%	101.000
Benchmark curve up 30 bps	3.0855%	100.407

Each root rate is the shifted one-year par rate plus the OAS of 28.55 bps.

1. Calculate the effective duration.

Solution. Step 1 gives the OAS of 28.55 bps at a price of 101.000, established in Example 8. Steps 2 and 3 rebuild the lattice on par curves of 2.200%, 2.700% and 3.200%, and of 2.800%, 3.300% and 3.800%, and revalue at that same OAS, giving 101.599 and 100.407. Step 4:

EffDur = (101.599 − 100.407) ÷ (2 × 0.0030 × 101.000) = 1.97.

2. Interpret the result.

Solution. A 100 bps parallel rise in the benchmark curve would cut the value of this callable bond by about 1.97%. Notice how far short of the three-year maturity that is. The call provision, exercised at two nodes on the tree, shortens the expected life of the bond and with it its price sensitivity.

Why a call or a put always shortens duration

Neither a call nor a put can push the effective duration of a bond above that of its straight counterpart. The reasoning runs through moneyness.

For a callable bond, when rates sit high relative to the coupon the call is out of the money and the bond is unlikely to be redeemed early. Its price responds to rate changes almost exactly as the straight bond does, and the two effective durations are close. As rates fall the call moves into the money. The issuer holds the right to retire the bond at the call price, which truncates the price appreciation, so the callable bond becomes less sensitive to rate changes than the straight bond and its effective duration shortens.

For a putable bond the argument is symmetric. When rates sit low relative to the coupon the put is out of the money and the bond behaves like the straight bond. As rates rise the put moves into the money and limits the price decline, because the holder can hand the bond back and reinvest at the higher yield, so the effective duration shortens.

At the extreme, when the embedded option is deep in the money the bond behaves like a straight bond maturing on the first exercise date, because exercise on that date is close to certain. Its effective duration converges on that of the shorter instrument.

Comparing option-free, callable and putable versions of a 4% annual coupon ten-year bond, valued on a 4% flat curve at 10% volatility with European-like options exercisable two months from now, makes the pattern visible. The effective duration of the option-free bond barely moves as rates change. The putable bond shortens as rates rise and the put moves into the money. The callable bond shortens as rates fall and the call moves into the money. Each option bites on the side of the market where it is valuable.

Effective duration by instrument type, a practitioner summary

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 9
Type of bond	Effective duration
Cash	0
Zero-coupon bond	Approximately equal to maturity
Fixed-rate bond	Less than maturity
Callable bond	At most the duration of the straight bond
Putable bond	At most the duration of the straight bond
Floater priced at the reference rate flat	Approximately the time in years to the next reset

A bond effective duration does not generally exceed its maturity, although exceptions exist, such as tax-exempt bonds analysed on an after-tax basis.

The table is a portfolio tool rather than a valuation tool. A manager who wants to shorten the duration of a portfolio of fixed-rate bonds can add floaters, whose duration is close to the time to the next reset. An issuer that wants to shorten the duration of its own liabilities can issue callable bonds. The mechanics of changing portfolio duration are developed at Level III.

Effective duration averages the price responses to an up-shift and a down-shift of equal size. For an option-free bond that average is informative. For a bond with an embedded option it can hide more than it reveals, because the price response is not symmetric: the upside is truncated if the bond is callable and the downside is truncated if it is putable.

Take a 4.5% bond maturing in five years, immediately callable at 100, valued on a 4% flat curve at 15% volatility. Its value is 99.75. Cut rates by 30 bps and the price rises to 100. Cut them by any amount at all and the price still cannot exceed 100, because nobody will pay more than the price at which the issuer can immediately redeem the bond. Raise rates instead and the price decline has no such limit. Averaging a bounded gain against an unbounded loss produces a number that describes neither.

The fix is to report the two sides separately. The one-sided up-duration uses only the up-shift, and the one-sided down-duration uses only the down-shift. They are most informative when the embedded option is near the money, which is exactly when the asymmetry is largest.

Figure 3: Why one-sided durations differ for a callable bond

Equal shifts in the benchmark rate produce unequal price responses. The gain from the down-shift is squeezed by the call price ceiling while the loss from the up-shift is not, so the one-sided up-duration exceeds the one-sided down-duration.

Durations for a 4.5% five-year bond immediately callable at par (4% flat curve, 15% volatility)

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 10
	4% flat curve	Curve 30 bps higher	Curve 30 bps lower
Bond value	99.75	99.17	100.00
Measure and result	Effective duration 1.39	Up-duration 1.94	Down-duration 0.84

Durations for a 4.1% five-year bond immediately putable at par (4% flat curve, 15% volatility)

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 11
	4% flat curve	Curve 30 bps higher	Curve 30 bps lower
Bond value	100.45	100.00	101.81
Measure and result	Effective duration 3.00	Up-duration 1.49	Down-duration 4.51

Read the two tables together and the story is clean. For the callable bond, 1.94 against 0.84 says the bond suffers more from a rate rise than it benefits from an equal rate fall. For the putable bond, 4.51 against 1.49 says the reverse: the put floors the loss on a rate rise while leaving the gain on a rate fall untouched. In each case the two-sided effective duration, 1.39 and 3.00, sits between the two one-sided numbers and describes neither situation accurately.

Key rate durations

Effective duration assumes the whole benchmark curve moves in parallel. Real curves steepen, flatten and twist. Key rate durations, also called partial durations, answer a narrower question: what happens to the price if only one maturity point on the benchmark curve moves, say the two-year rate by 5 bps, with everything else held still? The calculation is identical to effective duration in form; only the shift is different, applied to one key point at a time in isolation. The set of key rate durations describes the shaping risk of a bond, meaning its exposure to changes in the shape rather than the level of the curve.

Key rate durations of ten-year option-free bonds valued at a 4% flat curve

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 12
Coupon (%)	Price (% of par)	Total	2-year	3-year	5-year	10-year
0	67.30	9.81	−0.07	−0.34	−0.93	11.15
2	83.65	8.83	−0.03	−0.13	−0.37	9.37
4	100.00	8.18	0.00	0.00	0.00	8.18
6	116.35	7.71	0.02	0.10	0.27	7.32
8	132.70	7.35	0.04	0.17	0.47	6.68
10	149.05	7.07	0.05	0.22	0.62	6.18

Semi-annual coupons assumed. The key rate durations across each row add to the total, subject to rounding.

Three readings come out of this table. First, for bonds not trading at par, shifting any par rate changes the value, but the maturity-matched ten-year rate matters most, because the largest single cash flow is the final coupon plus the principal. Second, the 4% row is special: a ten-year bond trading at par on a 4% flat curve is affected only by the ten-year par rate, and its other key rate durations are exactly zero. That is a definitional property of par rates, not a coincidence. Third, some key rate durations are negative for maturity points shorter than the bond, and only for zero-coupon or very low-coupon bonds.

Where the negative key rate duration of −0.93 comes from

Raise the five-year par rate and hold every other par rate still. A ten-year bond trading at par must still be worth par, because its own maturity-matched par rate has not moved. But the five-year zero-coupon rate has risen along with the five-year par rate, so the Year 5 coupon of that ten-year par bond is now worth less than before. For the total to stay at par, the remaining cash flows, including the large one at Year 10, must be discounted at slightly lower rates. So the ten-year zero-coupon rate has fallen. A ten-year zero-coupon bond has its only cash flow at Year 10, so its value rises when the five-year par rate is raised. A price rise in response to a rate rise is a negative duration, and here it comes out at −0.93.

For bonds with embedded options, key rate durations depend on the time to exercise as well as the time to maturity, and the pattern shifts systematically with the coupon.

Key rate durations of 30-year bonds callable in 10 years (4% flat curve, 15% volatility)

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 13
Coupon (%)	Price (% of par)	Total	2-year	3-year	5-year	10-year	30-year
2	64.99	19.73	−0.02	−0.08	−0.21	−1.97	22.01
4	94.03	13.18	0.00	0.02	0.05	3.57	9.54
6	114.67	9.11	0.02	0.10	0.29	6.00	2.70
8	132.27	7.74	0.04	0.17	0.48	6.40	0.66
10	148.95	7.14	0.05	0.22	0.62	6.06	0.19

The 2% coupon bond is very unlikely to be called, so it behaves like a 30-year option-free bond and its sensitivity is concentrated in the 30-year rate, at 22.01. As the coupon rises the call becomes more likely, the total effective duration shortens from 19.73 down to 7.14, and the dominant maturity point migrates from the 30-year rate to the ten-year rate. At a 10% coupon the call is virtually certain to be exercised, the bond behaves like a ten-year option-free bond, and the 30-year key rate duration of 0.19 is negligible next to the ten-year figure of 6.06.

Key rate durations of 30-year bonds putable in 10 years (4% flat curve, 15% volatility)

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 14
Coupon (%)	Price (% of par)	Total	2-year	3-year	5-year	10-year	30-year
2	83.89	9.24	−0.03	−0.14	−0.38	8.98	0.81
4	105.97	12.44	0.00	−0.01	−0.05	4.53	7.97
6	136.44	14.75	0.01	0.03	0.08	2.27	12.37
8	169.96	14.90	0.01	0.06	0.16	2.12	12.56
10	204.38	14.65	0.02	0.07	0.21	2.39	11.96

The putable table runs the other way, and for the same reason read in reverse. A high-coupon putable bond is unlikely ever to be put, so it behaves like a 30-year option-free bond: the 10% coupon row is dominated by a 30-year key rate duration of 11.96. A low-coupon putable bond is almost certain to be put, so it behaves like an option-free bond maturing on the put date, and the 2% coupon row is dominated by a ten-year key rate duration of 8.98.

Duration is a linear approximation, and bond prices are not linear functions of interest rates. The curvature that duration misses is convexity, and for bonds with embedded options that curvature can change sign. Effective convexity measures how the duration itself responds to interest rate changes.

Equation 4. Same three prices as effective duration, combined differently.

Figure 4: Price against the benchmark rate for straight, callable and putable bonds

The straight bond curve is convex everywhere. The callable bond flattens against a ceiling on the left, where the call is in the money, and that flat region is where its convexity turns negative. The putable bond is drawn only from the point at which it starts to depart from the straight bond curve, flattening against a floor on the right where the put is in the money. At low rates the putable bond and the straight bond coincide, and at high rates the callable bond and the straight bond coincide, because in each case the relevant option is out of the money.

Example 11 · Worked

Take the same callable bond again, three years to run, a 4.25% annual coupon and redeemable at par at the issuer option on each of the next two anniversaries, on the same par curve and at 10% volatility, but now assume the current full price is 100.785 rather than 101.000. The implied OAS at that price is 40 bps. With 30 bps shifts, the down-shifted and up-shifted prices are 101.381 and 100.146.

1. Calculate the effective convexity.

Solution. Substituting into Equation 4:

EffCon = (101.381 + 100.146 − 2 × 100.785) ÷ [(0.003)² × 100.785] = −47.41.

The numerator is 201.527 − 201.570 = −0.043, a small negative number, and dividing by a very small denominator turns it into a large negative convexity. Note that two conventions exist for reporting convexity: raw figures such as this one are sometimes scaled by dividing by 100.

2. What does the negative sign mean here?

Solution. That the price response is worse on the downside than it is good on the upside. At a price of 100.785 the call is near the money, so a fall in rates lifts the price only until it meets the ceiling imposed by the call, while a rise in rates depresses it without limit. The average of the two shifted prices therefore falls short of the current price, which is exactly what a negative numerator says.

The general pattern

When rates are high and the call option is worth little, the callable bond and the straight bond respond to rate changes in much the same way and both display positive convexity. The convexity of the callable bond turns negative once the call moves near the money, because the upside is capped by the call price when the exercise date is close. Putable bonds never suffer this. Their convexity is always positive, and when the put is near the money the upside is much larger than the downside, because the price is floored by the put.

Set side by side, a putable bond has more upside than an otherwise identical callable bond when rates decline, and less downside when rates rise. Both differences come from the same source: the identity of the party holding the option.

Example 12 · Worked

A portfolio manager holds two fixed-rate bonds and wants to examine their sensitivity to a parallel shift of the benchmark curve. At an assumed 10% volatility, the valuation system returns the following prices for 30 bps shifts.

Two three-year 3.75% annual coupon bonds

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 15
	Bond X	Bond Y
Type of bond	Callable at par one year from today	Putable at par one year from today
Current price (% of par)	100.594	101.330
Price with the curve shifted down 30 bps	101.194	101.882
Price with the curve shifted up 30 bps	99.860	100.924

1. Calculate the effective duration of Bond X.

Solution. EffDur = (101.194 − 99.860) ÷ (2 × 0.003 × 100.594) = 2.21.

A figure of 0.67 would be too low even if the bond were certain to be called in one year, because a single cash flow one year away carries a duration of about one. A figure of 4.42 is impossible, because it exceeds the three-year maturity.

2. Calculate the effective duration of Bond Y.

Solution. EffDur = (101.882 − 100.924) ÷ (2 × 0.003 × 101.330) = 1.58.

3. When interest rates rise, whose effective duration shortens?

Solution. Bond Y. Rising rates push the put into the money, the bond becomes more likely to be put, and it starts to behave like a shorter-maturity instrument. Rising rates push the call in Bond X out of the money, so Bond X behaves more like the straight bond, and the effective duration of a straight bond falls as rates rise, so it does not lengthen either.

4. When the put embedded in Bond Y is in the money, what do the one-sided durations show?

Solution. That the bond is more sensitive to a decrease in rates. If rates rise, the right to put at par truncates the price decline. If rates fall, the price appreciation has no ceiling. The one-sided down-duration therefore exceeds the one-sided up-duration.

5. Which par rate shifts affect the price of Bond X?

Solution. All of them, but the one-year and three-year par rates matter most. The call decision one year from today turns on the two-year forward rate one year from now, and that forward rate is driven principally by the one-year and three-year par rates.

6. What can be said about the effective convexity of Bond X, and about the relative upside of the two bonds?

Solution. The effective convexity of Bond X turns negative when the call is near the money, because the price response to falling rates is capped: the issuer refunds at the lower rate and the investor loses the upside. It does not turn negative when the option moves out of the money, which is when the bond behaves most like a straight bond.

On relative upside, Bond X has less upside than Bond Y for a given fall in rates. Falling rates raise the value of a call and lower the value of a put, so the call in Bond X caps its appreciation while Bond Y carries no such cap. Bond Y always has positive convexity, and the straight bond underlying Bond Y has low positive convexity rather than negative convexity.

Options embedded in floaters are not exercised by anybody. They operate automatically: once the coupon rate would breach the threshold, the cap or the floor takes effect. The valuation machinery is unchanged, because the exercise test at each node simply becomes a comparison of the reset rate with the threshold.

A cap stops the coupon rate rising above a stated maximum. It protects the issuer against rising rates, so it is an issuer option. The investor is long the bond and short the cap, so the cap subtracts value.

Equation 5.

A floor stops the coupon rate falling below a stated minimum. It protects the investor against falling rates, so it is an investor option. The investor is long the bond and long the floor, so the floor adds value.

Equation 6.

The benchmark for both is the uncapped, unfloored floater. If the coupon paid always equals the rate used to discount it, the floater is worth exactly 100 at every reset, so the straight bond value here is par. Everything the cap or floor does shows up as a deviation from par.

The examples below use a three-year floater paying the one-year reference rate annually, set in arrears, meaning the rate is fixed at the end of the coupon period so that the setting date and the payment date coincide. The issuer credit quality is assumed to match the reference rate swap curve, so there is no credit spread, and that swap curve is the same par curve used throughout: 2.500%, 3.000% and 3.500%, at 10% volatility. The lattice is therefore the familiar one.

Example 13 · Worked

Value the three-year reference rate floater capped at 4.500%, then the same floater floored at 3.500%.

Capped floater at 4.500%, 10% volatility

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 16
Node	Reference rate	Coupon after the cap	Value at the node
Year 2, up	5.5258%	4.5000 (capped)	99.028
Year 2, middle	4.5242%	4.5000 (capped)	99.977
Year 2, down	3.7041%	3.7041	100.000
Year 1, up	3.8695%	3.8695	99.521
Year 1, down	3.1681%	3.1681	99.989
Year 0	2.5000%	2.5000	99.761

1. Value the capped floater and the embedded cap.

Solution. Without the cap this floater would be worth 100 in every state, because the coupon paid always equals the discount rate. The cap at 4.500% bites wherever the reference rate exceeds it, which happens at two Year 2 nodes.

At the top Year 2 node the reference rate is 5.5258%, so the Year 3 cash flow is cut from the uncapped 105.5258 to 104.5000, and the node value is 104.5000 ÷ 1.055258 = 99.028. At the middle node the rate of 4.5242% also exceeds the cap, so the cash flow is 104.5000 and the value is 104.5000 ÷ 1.045242 = 99.977. At the bottom node the rate of 3.7041% is below the cap, the cash flow is 103.7041, and the value is exactly 100.000.

Rolling back: [3.8695 + (0.5 × 99.028 + 0.5 × 99.977)] ÷ 1.038695 = 99.521, and [3.1681 + (0.5 × 99.977 + 0.5 × 100.000)] ÷ 1.031681 = 99.989. Then [2.5000 + (0.5 × 99.521 + 0.5 × 99.989)] ÷ 1.025000 = 99.761.

By Equation 5, the embedded cap is worth 100 − 99.761 = 0.239.

Floored floater at 3.500%, 10% volatility

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 17
Node	Reference rate	Coupon after the floor	Value at the node
Year 2, up	5.5258%	5.5258	100.000
Year 2, middle	4.5242%	4.5242	100.000
Year 2, down	3.7041%	3.7041	100.000
Year 1, up	3.8695%	3.8695	100.000
Year 1, down	3.1681%	3.5000 (floored)	100.322
Year 0	2.5000%	3.5000 (floored)	101.133

2. Value the floored floater and the embedded floor.

Solution. The floor at 3.500% binds wherever the reference rate is below it. That happens at Year 0, where the rate is 2.5000%, and at the lower Year 1 node, where the rate is 3.1681%. Both of the corresponding interest payments are lifted to 3.5000.

Every Year 2 rate is above the floor, so all three Year 2 values are 100.000. At the lower Year 1 node the coupon is raised to 3.5000, giving [3.5000 + (0.5 × 100 + 0.5 × 100)] ÷ 1.031681 = 100.322. At the upper Year 1 node the coupon of 3.8695 is unaffected and the value is 100.000.

Today: [3.5000 + (0.5 × 100.000 + 0.5 × 100.322)] ÷ 1.025000 = 101.133.

By Equation 6, the embedded floor is worth 101.133 − 100 = 1.133.

3. Explain why the two answers sit on opposite sides of par.

Solution. A cap can only reduce a coupon, never raise one, so a capped floater is worth par or less, here 99.761. A floor can only raise a coupon, never cut one, so a floored floater is worth par or more, here 101.133. Whether either provision is worth anything at all depends on whether the threshold is actually crossed anywhere on the lattice.

Example 14 · Worked

Three variations on the same theme, all on the same par curve of 2.500%, 3.000% and 3.500% at 10% volatility unless stated otherwise.

1. A three-year floater pays the one-year reference rate annually, set in arrears, capped at 5.600%. What is it worth: 100.000, 105.600 or 105.921?

Solution. 100.000. The highest reset rate anywhere on the lattice is 5.5258%, which is below the 5.600% cap, so the cap never binds. The floater is worth what it would be worth with no cap at all, namely par.

2. The same floater is floored at 3.000% instead. What is it worth: 100.000, 100.488 or 103.000?

Solution. 100.488. The third choice can be eliminated immediately: a floor of 3.500% was shown above to give 101.133, and a lower floor cannot be worth more. The floor at 3.000% binds only at Year 0, where the reference rate is 2.5000%, lifting that one coupon from 2.5000 to 3.0000. Every other rate on the lattice is above 3.000%. Discounting the extra 0.5000 of coupon back one year at 2.5000% adds 0.488, so the floored floater is worth 100.488, and the floor is worth 0.488. Had the floor been set at 2.500%, it would never bind and the floater would be worth exactly par.

3. A eurozone issuer wants to place a three-year floating-rate note at par paying 12-month Euribor plus 300 bps. Because Euribor is at a historic low and the issuer wants protection against a sudden rise in its interest cost, its advisers propose lifting the credit spread to 320 bps and capping the coupon at 5.50%. At 8% volatility the all-in rate lattice is 3.7430% at Year 0; 5.2908% and 4.9817% at Year 1; and 5.8865%, 5.4893% and 5.1508% at Year 2. What is the capped floater worth: 92.929, 99.916 or 109.265?

Solution. 99.916. Note that the lattice already includes the 320 bps spread, so each node rate is both the coupon that resets there and the rate at which that node is discounted.

Year 2, top node: the reset of 5.8865% exceeds the 5.50% cap, so the Year 3 cash flow is capped at 105.5000 and the value is 105.5000 ÷ 1.058865 = 99.635. The middle and bottom Year 2 rates of 5.4893% and 5.1508% are both below the cap, so those nodes are worth 100.000 each.

Year 1, up node: [5.2908 + (0.5 × 99.635 + 0.5 × 100.000)] ÷ 1.052908 = 99.827. Year 1, down node: [4.9817 + (0.5 × 100 + 0.5 × 100)] ÷ 1.049817 = 100.000.

Today: [3.7430 + (0.5 × 99.827 + 0.5 × 100.000)] ÷ 1.037430 = 99.916.

The note prices just below par, so the extra 20 bps of spread does not quite pay for the cap the issuer is buying.

Every option seen so far changes the timing or the size of a bond cash flow. A conversion option does something different: it changes the security itself, turning a bond into common stock. That is what sets convertibles apart and what makes them harder to analyse.

A convertible bond is an option-free bond plus an embedded conversion option that lets the holder exchange debt for equity during a defined conversion period at a pre-determined conversion price. Investors accept a lower coupon than an otherwise identical non-convertible bond would pay, because the conversion mechanism lets them participate in the upside of the shares at a cost below market value. The issuer gains from the lower coupon and, if conversion happens, from no longer having to repay the converted debt.

This is not a free lunch on either side. Existing shareholders are diluted if conversion occurs. If the share price stays below the conversion price and the bond is never converted, the issuer must repay or refinance, possibly at a higher cost than it started with. And the investors, if conversion never happens, will have given up interest income relative to the higher-coupon non-convertible bond they could have bought instead.

The defining terms

The worked material below uses a real structure: a $1 billion 0.25% convertible bond issued by Twitter, Inc. in June 2018 and due 15 June 2024. Some features of the actual issue, such as a make-whole call, have been stripped out for clarity.

Twitter, Inc. $1 billion 0.25% convertible bonds due 15 June 2024

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 18
Term	Detail
Issued	11 June 2018
Ranking	Senior unsecured
Coupon	0.25% a year, accruing from 11 June 2018, settled twice yearly in arrears each 15 June and 15 December, starting 15 December 2018
Priced at issue	Par, 100%
Matures	15 June 2024
Shares per $1,000 of par (conversion rate)	17.5 common shares
Price per share on conversion	$57.14
Share price on the issue date	$40.10
Share price assumed for 15 June 2019	$35.14
Bond price assumed for the same date	95.225% of par
Premium at issue over the share price	42.5%

The assumed prices in fact pertain to 11 April 2019, which leaves exactly five full years to maturity and simplifies the straight bond calculation.

The conversion price is the share price at which the holder can convert, here $57.14. The conversion rate, also called the conversion ratio, is the number of shares received per unit of par, here 17.5 shares per $1,000. The two are reciprocal: a holder of $10,000 of par receives $10,000 divided by $57.14, which is 175 shares. Conversion may be permitted over a continuous window or only at set intervals.

The conversion price stated at issuance is the initial conversion price. Corporate actions such as stock splits, bonus issues and rights or warrant issues change the share price and would otherwise erode the benefit of conversion, so the terms of issuance specify precisely how the conversion price and conversion ratio adjust. A 2:1 stock split, for instance, would halve the conversion price to $28.57 and double the conversion rate to 35 shares per $1,000 of nominal value.

Dividends need handling too. While the bond is outstanding the holder receives interest, and common shareholders receive any dividend the issuer declares. Terms of issuance range from no compensation at all for dividends paid during the life of the bond, at one extreme, to full protection through a downward adjustment of the conversion price for every dividend, at the other. The common middle ground defines a threshold dividend: annual dividends below the threshold leave the conversion price alone, while dividends above it trigger a downward adjustment that compensates the convertible holders.

Change of control needs handling as well. If the issuer is acquired or merged, holders may no longer wish to lend to the surviving entity, so the prospectus defines change-of-control events and typically offers holders a choice between two remedies: a put option exercisable during a specified window after the event, redeeming the nominal value in full; or an adjusted conversion price below the initial conversion price, which lets them convert earlier and on better terms and take part in the transaction as shareholders.

Beyond change of control, convertibles frequently carry ordinary put options exercisable during specified periods. A hard put obliges the issuer to redeem for cash. A soft put gives the holder the right to put but leaves the issuer to choose the settlement medium, which may be cash, common stock, subordinated notes or some combination.

Call provisions and forced conversion

It is even more common for a convertible to be callable. The issuer may call for the ordinary reasons, namely falling interest rates or an improved credit rating that permits cheaper refinancing. It may also call because it expects its own share price to rise materially and wants to protect existing shareholders relative to the convertible holders. Convertible bonds therefore usually carry a protection period, after which they can be called but at a premium that declines as maturity approaches.

The distinctive case is forced conversion. When the share price has risen above the conversion price, calling the bond gives holders a stark choice: accept the redemption value, which is now worth less than the shares, or convert. Rational holders convert, and the issuer stops paying coupons. The issuer may do this even when refinancing offers no saving, simply to exploit favourable equity market conditions. Forced conversion strengthens the capital structure and removes the risk that a later fall in the share price leaves the bonds unconverted and repayable in cash at maturity.

The investment metrics

The conversion value, also called parity value, is what the bond would be worth if converted at today’s share price.

The minimum value of a convertible is the greater of its conversion value and the value of the underlying option-free bond. In theory the straight value could be read off the market price of a non-convertible bond of the same issuer with matching terms, but such a bond rarely exists, so the straight value is computed within the arbitrage-free framework by discounting the future cash flows at appropriate rates.

This minimum acts as a floor, but a moving one. The straight value is not fixed. A rise in interest rates lowers it. So does a widening of the issuer credit spread, for instance after a downgrade from investment grade to non-investment grade. The floor beneath a convertible can therefore fall away exactly when the holder most wants it.

Investors who buy in the secondary market rather than at issuance need to know what they are paying for the equity exposure. The market conversion price is the effective per-share cost of acquiring the stock through the bond, and the market conversion premium per share is the excess of that over the actual share price.

The market conversion price is a break-even price. Once the share price rises above it, any further rise in the share price is certain to lift the value of the convertible by at least the same percentage.

Why pay a premium at all? Because the straight value acts as a floor, so a falling share price does not drag the convertible below the straight value. In that respect the premium resembles an option premium: the buyer of a call limits the downside to the premium paid, and the buyer of a convertible limits the downside to the straight value. The analogy is not exact, and the difference matters. The call buyer knows the maximum loss precisely. The convertible buyer knows only that the maximum loss is the gap between the price paid and the straight value, and that gap moves as rates and credit spreads move. Market conversion discounts, where the market conversion price falls below the share price, are rare but possible, because the bond and the shares trade in different markets with different participants; very volatile share prices can produce them.

Downside is commonly summarised by one more ratio:

All else equal, a higher premium over straight value makes the convertible less attractive, because more of the price is exposed above the floor. Despite its currency in practice, this measure is flawed for exactly the reason already given: the straight value it is measured against is not a constant. The upside, by contrast, depends chiefly on the prospects for the underlying shares, which is a question of equity analysis rather than fixed-income analysis.

Example 15 · Worked

Work through the Twitter convertible using the terms in the table above.

1. Calculate the conversion value at issuance and on 15 June 2019.

Solution. At issuance: $40.10 × 17.5 = $701.75.
On 15 June 2019: $35.14 × 17.5 = $614.95.
The share price has fallen, so conversion is worth less than it was, and both figures sit far below the $1,000 par value.

2. Calculate the minimum value at issuance and on 15 June 2019, assuming a 2.5% flat yield curve on the later date.

Solution. At issuance the straight value is $1,000, because the bond was issued at 100% of par, so:
Minimum value = the greater of $701.75 and $1,000 = $1,000.

On 15 June 2019 exactly five years remain, so ten semiannual coupons of $1.25 each are outstanding on $1,000 of par, discounted at 2.5% divided by 2 per period:
$1.25 ÷ (1.0125) + $1.25 ÷ (1.0125)² + … + $1,001.25 ÷ (1.0125)¹⁰ = $894.86.
Minimum value = the greater of $614.95 and $894.86 = $894.86. The straight value is the binding floor.

3. Show why the convertible cannot trade below its minimum value.

Solution. Two cases.

Suppose the convertible traded at $850.00 on 15 June 2019, below the straight value of $894.86. The convertible would then be cheap relative to the straight bond, offering a higher yield than an otherwise identical non-convertible bond. Investors would buy it and the price would be pushed back up to the straight value.

Now suppose that on the same date the yield on otherwise identical non-convertible bonds were 12.00% rather than 2.50%. The straight value would then be $567.59. If the convertible traded at that level, it would sit below its conversion value of $614.95. An arbitrageur could buy the bond for $567.59, convert into 17.5 shares and sell them at $35.14 each for $614.95, banking $614.95 − $567.59 = $47.36. Repeated by enough arbitrageurs, that trade drives the price up to the conversion value.

4. Calculate the market conversion price, the market conversion premium per share and the market conversion premium ratio on 15 June 2019.

Solution. The convertible is priced at 95.225% of par, or $952.25.
Market conversion price = $952.25 ÷ 17.5 = $54.40.
Market conversion premium per share = $54.40 − $35.14 = $19.26.
Market conversion premium ratio = $19.26 ÷ $35.14 = 54.8%.

Buying the equity exposure through the bond costs 54.8% more per share than buying the shares outright. What that premium buys is the protection of the straight value floor.

5. Calculate the premium over straight value and comment on it.

Solution. Premium over straight value = ($952.25 ÷ $894.86) − 1 = 6.41%.

Read literally, only about 6.41% of the price is at risk above the bond floor, which sounds reassuring against a 54.8% conversion premium. The reassurance is partly illusory, because $894.86 was computed on a 2.5% flat curve with the issuer credit unchanged. A rate rise or a credit deterioration moves the floor down and the true exposure up.

Convertible bonds have always been awkward to value because they sit across three disciplines at once: fixed income, equity and derivatives. The instruments have also grown more complicated. Contingent convertible bonds, or CoCos, pay a higher coupon than otherwise identical non-convertible bonds but are usually deeply subordinated and convert into equity, or suffer a principal write-down, if regulatory capital ratios are breached. Convertible contingent convertible bonds, or CoCoCos, bolt a traditional convertible onto a CoCo: the holder may convert at will, capturing share price upside, and the instrument still converts or writes down on a regulatory capital breach. Both are issued mainly by financial institutions, particularly in Europe.

A useful signal of how hard this valuation is: prospectuses frequently appoint an independent financial valuer to determine the conversion price, and in effect the value of the bond, under different scenarios. Partly because of that complexity, convertibles in many markets carry selling restrictions, come in very large denominations and are offered only to professional or institutional investors, since regulators regard them as too risky for direct retail purchase.

Breaking the price into its components is therefore the only way to see what drives it, and analysing one properly means doing three jobs. First, the ordinary credit work: the ability of the issuer to service and repay the debt, and a review of collateral, credit enhancements, covenants and contingent provisions. Convertibles often carry lighter covenants than comparable non-convertible bonds and are frequently subordinated, so this work matters more, not less. Second, the ordinary bond work, principally interest rate exposure. Third, equity work, because the investment characteristics depend on the share price: dividend policy, corporate actions such as acquisitions, disposals and rights issues, and exogenous factors that could depress the share price even when the issuer is performing well and thereby prevent conversion.

The arbitrage-free decomposition

Advanced models exist, but the workhorse remains the arbitrage-free framework, and it simply extends the building-block logic already used for callable and putable bonds. A traditional convertible is a straight bond plus a call option on the shares of the issuer.

Add an issuer call on the bond and it is subtracted, because it belongs to the issuer:

Add an investor put on the bond and it is added, because it belongs to the investor:

However many options are stacked on, the procedure does not change: build a lattice of interest rates from the given curve and volatility assumptions, test at each node whether each embedded option is exercised, and apply backward induction to reach the present value.

Where a convertible sits between a bond and a share

The risk and return character of a convertible is determined almost entirely by where the share price sits relative to the conversion price.

Figure 5: Convertible bond value across the range of share prices

At low share prices the convertible hugs the straight bond floor and behaves like a bond. At high share prices it converges onto the conversion value line and behaves like the shares. Between the two it is a hybrid, rising with the share price but by less than the share price.

Well below the conversion price, the convertible is described as a busted convertible and behaves mostly like the straight bond. The call on the shares is out of the money, so share price moves barely register, and the price is driven instead by interest rate moves and credit spreads. As the share price approaches zero, the value of the bond falls toward the present value of the recovery rate in bankruptcy. Bond-like behaviour is even more pronounced when the option is out of the money and the conversion period is nearing its end, because the time value of the option is decaying toward zero and expiry worthless becomes very likely.

Well above the conversion price, the convertible behaves mostly like the common stock. The option is in the money, so the price is driven by the share price and is largely insensitive to the interest rate factors that move an option-free bond. Conversion is likely to be exercised, since the shares received are worth more than the redemption value, and the bond trades close to its conversion value with price movements that track the stock.

In between, the option component grows in value as the share price approaches the conversion price. The convertible rises materially, but by less in percentage terms than the shares themselves, because the conversion price has not yet been reached. Once the share price passes the conversion price and keeps going, the change in the convertible price converges on the change in the share price. That is the same point made earlier about the market conversion price acting as a break-even.

Why would a holder not convert when the share price is above the conversion price? The option may be European style and not yet exercisable. Even if it is American style, waiting may be better than exercising, for the same reason that applies to any in-the-money American option. And the holder may prefer simply to sell the convertible rather than convert it.

One qualification. Outside busted convertibles, the share price is the dominant valuation factor, but large moves in interest rates or credit spreads still matter. For a fixed-coupon convertible, all else equal, a large fall in rates raises its value and a large rise lowers it. Likewise a marked improvement in the issuer credit quality raises its value and a deterioration lowers it.

Example 16 · Worked

An analyst is preparing a note on a convertible bond issued by Heavy Element Inc., a chemicals company, using the following data from the prospectus and the market.

Heavy Element Inc. convertible bond

FI 3 – Valuation and Analysis of Bonds with Embedded Options — table 19
Term	Detail
Issued	15 September 2020
Matures	15 September 2025
Coupon	3.75%, paid once a year
Size of the issue	$100,000,000
Priced at issue	Par, $1,000
Shares per $1,000 of par	23.26
Bond price, 16 September 2022	$1,230
Share price, same date	$52

1. What is the conversion price?

Solution. Par value divided by the conversion ratio: $1,000 ÷ 23.26 = $43 per share.

2. What is the conversion value on 16 September 2022?

Solution. Share price times conversion ratio: $52 × 23.26 = $1,209.

3. What is the market conversion premium per share?

Solution. Market conversion price minus share price: ($1,230 ÷ 23.26) − $52 = $52.88 − $52 = $0.88. The premium is tiny, which is what you would expect when the shares are trading well clear of the conversion price and the bond is tracking parity.

4. What do the risk and return characteristics resemble on that date?

Solution. The common stock of Heavy Element. At $52 the share price is well above the $43 conversion price, so the conversion option is comfortably in the money. This is not a busted convertible, which requires the shares to trade at a significant discount to the conversion price, and it is not behaving like an otherwise identical non-convertible bond, which describes the same busted case.

5. Credit spreads for the chemicals sector narrow, lowering borrowing rates for companies like Heavy Element. What happens to the price of the convertible?

Solution. It does not change significantly. With the shares well above the conversion price, the bond is driven by the share price, not by the factors that move an option-free bond. Narrower credit spreads would matter a great deal to a busted convertible and matter very little here.

6. Suppose the convertible were available in the secondary market at $1,050 on that date. How would an arbitrageur profit?

Solution. Buy the convertible, exercise the conversion option and sell the resulting shares. At $1,050 the bond trades below its minimum value of $1,209. Buying at $1,050, converting into 23.26 shares and selling them at $52 each realises $1,209, a profit of $159. Shorting the convertible in either direction would be wrong, because that puts the arbitrageur short the underpriced asset and long an overpriced one.

7. Some months later, costly environmental legislation follows chemical spills at a competitor site, and share prices across the sector fall sharply. Heavy Element now trades at $28. What do the risk and return characteristics resemble?

Solution. A bond. At $28 the share price is well below the $43 conversion price, so this is now a busted convertible with mostly bond-like characteristics, sensitive to interest rates and credit spreads rather than to the shares. A hybrid character would require the share price to be close to the conversion price, and stock-like character would require it to be above the conversion price.

Check yourself

1. A three-year 5% annual coupon bond is callable at par in one and two years. On a lattice built from a par curve whose three-year par rate is 5%, before any exercise test, no node value exceeds 100.4. If the call price were raised to 101, what would the callable bond be worth relative to the straight bond?

Exactly the same as the straight bond. A call price of 101 is above every pre-exercise value on the tree, so the option is never exercised in any state of the world and is worth zero. The callable bond therefore collapses onto the straight bond. The general point is that an embedded option is worth nothing unless the exercise threshold is actually crossed somewhere on the lattice.

2. An analyst values a callable bond at 10% volatility and obtains an OAS of 60 bps, which looks generous against a peer group averaging 45 bps. A colleague argues the correct volatility assumption is 20%. Without recalculating, what happens to the conclusion?

The apparent cheapness shrinks or disappears. Raising volatility raises the value of the call, which lowers the model value of the callable bond at any given spread, so a smaller spread is needed to reproduce the same market price. The OAS falls. Since the peer comparison is only meaningful when every bond is valued on the same volatility assumption, the original 60 bps against 45 bps comparison is not evidence of value on its own.

3. Why is modified duration inappropriate for a putable bond, and what is used instead?

Modified duration is a yield duration measure and assumes the expected cash flows of the bond stay fixed when the yield changes. For a putable bond that assumption fails, because a rate rise makes exercise of the put more likely and therefore changes the cash flows themselves. The correct measure is effective duration, a curve duration measure, computed by shifting the benchmark curve up and down, rebuilding the lattice each time, and revaluing at a constant OAS.

4. A callable bond has a one-sided up-duration of 2.10 and a one-sided down-duration of 0.90. What does the pair tell you that the effective duration of 1.50 does not?

That the exposure is heavily asymmetric and the call is near the money. A 30 bps rise in rates costs roughly 2.10 times as much per unit of shift as the bond is worth, while a 30 bps fall gains only 0.90. The two-sided figure of 1.50 averages a bounded gain against an unbounded loss and describes neither case. For hedging purposes the up-duration is the number that matters, because that is the direction in which the loss is unconstrained.

5. Two 30-year bonds callable in 10 years are valued on a 4% flat curve. One carries a 2% coupon and the other a 10% coupon. Which maturity point on the benchmark curve dominates each, and why?

The 2% bond is dominated by the 30-year rate, because it is deeply out of the money for the call and will almost certainly run to maturity, so it behaves like a 30-year option-free bond. The 10% bond is dominated by the 10-year rate, because the call is virtually certain to be exercised on the first call date, so it behaves like a 10-year option-free bond. Key rate durations for bonds with embedded options depend on time to exercise as well as time to maturity.

6. A floater pays the one-year reference rate annually with no credit spread and is floored at 2.000%. Every rate on the lattice is above 2.000%. What is the floater worth, and what is the floor worth?

The floater is worth exactly par and the floor is worth zero. When the coupon paid always equals the rate used to discount it, the floater prices at par at every reset. A floor set below every rate on the lattice never binds, so it never changes a single cash flow. This is the mirror image of a cap set above every rate on the lattice.

7. A convertible bond has a conversion ratio of 20, a straight value of $900 and trades at $1,100 while the shares trade at $50. Calculate the conversion value, the market conversion premium ratio and the premium over straight value, then say which risk the last measure understates.

Conversion value is $50 × 20 = $1,000. Market conversion price is $1,100 ÷ 20 = $55, so the market conversion premium per share is $5 and the premium ratio is $5 ÷ $50 = 10%. Premium over straight value is ($1,100 ÷ $900) − 1 = 22.2%. That last measure understates downside risk because the $900 straight value is not fixed: a rise in interest rates or a widening of the issuer credit spread lowers the floor, so the true amount at risk is larger than 22.2% of the price.

8. An issuer of a callable convertible sees its share price move well above the conversion price while its own credit rating is unchanged and interest rates have not fallen. Why might it still call the bond?

To force conversion. Calling when the shares are worth more than the redemption value leaves holders no rational choice but to convert, which stops the coupon payments, converts debt into equity and strengthens the capital structure. It also removes the risk that a later correction in the share price leaves the bonds unconverted and repayable in cash at maturity. The usual refinancing motives are absent here, but the equity motive is sufficient on its own.

End of lesson.

Back to 06 Fixed Income

FI 3 – Valuation and Analysis of Bonds with Embedded Options

Embedded options and the securities that carry them

Call options

Put options and extension options

More complex structures

How the bond, the straight bond and the option values fit together

Refresher: valuing a default-free option-free bond

Valuing callable and putable bonds when interest rates do not move

Is it worth exercising, and is it worth exercising now?

Interest rate volatility and the level and shape of the yield curve

Volatility raises the value of every embedded option

The level of the curve

The shape of the curve

Valuing callable and putable bonds from a binomial interest rate tree

How the lattice is calibrated

Option-adjusted spread and the valuation of risky bonds

From Z-spread to option-adjusted spread

Volatility and the OAS

Scenario analysis over an investment horizon

Effective duration for bonds with embedded options

Why a call or a put always shortens duration

One-sided durations and key rate durations

Key rate durations

Effective convexity and the asymmetry of embedded options

The general pattern

Capped and floored floating-rate bonds

Convertible bonds: features and component values

The defining terms

Call provisions and forced conversion

The investment metrics

Valuing convertibles and comparing risk and return

The arbitrage-free decomposition

Where a convertible sits between a bond and a share

FI 3 – Valuation and Analysis of Bonds with Embedded Options

Embedded options and the securities that carry them

Call options

Put options and extension options

More complex structures

How the bond, the straight bond and the option values fit together

Refresher: valuing a default-free option-free bond

Valuing callable and putable bonds when interest rates do not move

Is it worth exercising, and is it worth exercising now?

Interest rate volatility and the level and shape of the yield curve

Volatility raises the value of every embedded option

The level of the curve

The shape of the curve

Valuing callable and putable bonds from a binomial interest rate tree

How the lattice is calibrated

Option-adjusted spread and the valuation of risky bonds

From Z-spread to option-adjusted spread

Volatility and the OAS

Scenario analysis over an investment horizon

Effective duration for bonds with embedded options

Why a call or a put always shortens duration

One-sided durations and key rate durations

Key rate durations

Effective convexity and the asymmetry of embedded options

The general pattern

Capped and floored floating-rate bonds

Convertible bonds: features and component values

The defining terms

Call provisions and forced conversion

The investment metrics

Valuing convertibles and comparing risk and return

The arbitrage-free decomposition

Where a convertible sits between a bond and a share

Related lessons