An embedded option is a component of a financial bond or other security, which provides the bondholder or the issuer the right to take some action against the other party. There are several types of options that can be embedded into a bond; common types of bonds with embedded options include

callable bond A callable bond (also called redeemable bond) is a type of bond (debt security) that allows the issuer of the bond to retain the privilege of redeeming the bond at some point before the bond reaches its date of maturity. In other words, on the call ...

puttable bond Puttable bond (put bond, putable or retractable bond) is a bond with an embedded put option. The holder of the puttable bond has the right, but not the obligation, to demand early repayment of the principal. The put option is exercisable on one o ...

, convertible bond, extendible bond,

exchangeable bond Exchangeable bond (or XB) is a type of hybrid security consisting of a straight bond and an embedded option An embedded option is a component of a financial bond or other security, which provides the bondholder or the issuer the right to take so ...

, and capped

floating rate note Floating rate notes (FRNs) are bonds that have a variable coupon, equal to a money market reference rate, like LIBOR or federal funds rate, plus a quoted spread (also known as quoted margin). The spread is a rate that remains constant. Almost all ...

. A bond may have several options embedded if they are not

mutually exclusive In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...

Securities A security is a tradable financial asset. The term commonly refers to any form of financial instrument, but its legal definition varies by jurisdiction. In some countries and languages people commonly use the term "security" to refer to any for ...

other than bonds that may have embedded options include senior equity,

convertible A convertible or cabriolet () is a passenger car that can be driven with or without a roof in place. The methods of retracting and storing the roof vary among eras and manufacturers. A convertible car's design allows an open-air driving expe ...

preferred stock Preferred stock (also called preferred shares, preference shares, or simply preferreds) is a component of share capital that may have any combination of features not possessed by common stock, including properties of both an equity and a debt ins ...

and exchangeable preferred stock. See

Convertible security A convertible security is a financial instrument whose holder has the right to convert it into another security of the same issuer. Most convertible securities are convertible bonds or preferred stocks that pay regular interest and can be converted ...

. The valuation of these securities couples bond- or equity-valuation, as appropriate, with

option pricing In finance, a price (premium) is paid or received for purchasing or selling options. This article discusses the calculation of this premium in general. For further detail, see: for discussion of the mathematics; Financial engineering for the impl ...

. For bonds here, there are two main approaches, as follows.''Pricing Interest Rate-dependent Financial Claims with Option Features''
Ch11 in: Richard Rendleman (2002). Applied Derivatives: Options, Futures, and Swaps (1st ed.). Wiley-Blackwell. . (Other securities with embedded derivatives are priced similarly.) Once the price has been calculated, the various yields can then be calculated for the security. # Depending on the type of option, the option price, as calculated using the Black–Scholes ( or other) model, is either added to or subtracted from the price of the "straight" bond (i.e. as if it had no optionality) and this total is then the value of the bond. # A bespoke "

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

" (usually a lattice-based

short-rate model A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,. The short rate Under a sh ...

) may be constructed where the option's effect is incorporated at each node in the tree, impacting either the bond price or the option price as specified; see

further Further or Furthur may refer to: * ''Furthur'' (bus), the Merry Pranksters' psychedelic bus * Further (band), a 1990s American indie rock band * Furthur (band), a band formed in 2009 by Bob Weir and Phil Lesh * ''Further'' (The Chemical Brothers a ...

under

bond option In finance, a bond option is an option to buy or sell a bond at a certain price on or before the option expiry date. These instruments are typically traded OTC. *A European bond option is an option to buy or sell a bond at a certain date in futu ...

. Calculating rate-sensitivities on these instruments is complicated: the embedded features make measures such as

duration Duration may refer to: * The amount of time elapsed between two events * Duration (music) – an amount of time or a particular time interval, often cited as one of the fundamental aspects of music * Duration (philosophy) – a theory of time and ...

and

convexity Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

(and

DV01 In finance, the duration of a financial asset that consists of fixed cash flows, such as a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, du ...

) less meaningful; and analysts instead use

effective duration In finance, the duration of a financial asset that consists of fixed cash flows, such as a bond, is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield, du ...

and

effective convexity In finance, bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates ( duration is the first derivative). In general, the ...

References

Fixed income analysis Commercial bonds Bond valuation {{investment-stub