FI 3 – Valuation and Analysis of Bonds with Embedded Options

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%.

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.

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:

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.

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.

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

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.

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.

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.

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.

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.

1. Generate a tree of one-period forward rates consistent with the given par yield curve and the given interest rate volatility.
2. At each node, test whether the embedded option is exercised and reset the value if it is.
3. 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.

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

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%.

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.

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.

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.

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.

1. Given the current price, solve for the implied OAS to the benchmark curve at an appropriate volatility.
2. 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.
3. 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.
4. 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.