Bond option
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In finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, a bond option is an option
Option (finance)
In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...
to buy or sell a bond
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
at a certain price on or before the option expiry date. http://financial-dictionary.thefreedictionary.com/Bond%2boption These instruments are typically traded OTC.
- A EuropeanOption styleIn finance, the style or family of an option is a general term denoting the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American options. These options - as well as others where the...
bond option is an option to buy or sell a bond at a certain date in future for a predetermined price. - An AmericanOption styleIn finance, the style or family of an option is a general term denoting the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American options. These options - as well as others where the...
bond option is an option to buy or sell a bond on or before a certain date in future for a predetermined price.
Generally, one buys a call option
Call option
A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...
on the bond if one believes that interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
s will fall, causing an increase in bond prices. Likewise, one buys the put option
Put option
A put or put option is a contract between two parties to exchange an asset, the underlying, at a specified price, the strike, by a predetermined date, the expiry or maturity...
if one believes that the opposite will be the case. http://financial-dictionary.thefreedictionary.com/Bond%2boption One result of trading in a bond option, is that the price of the underlying bond is "locked in" for the term of the contract, thereby reducing the credit risk
Credit risk
Credit risk is an investor's risk of loss arising from a borrower who does not make payments as promised. Such an event is called a default. Other terms for credit risk are default risk and counterparty risk....
associated with fluctuations in the bond price.
Valuation
- Compare: Swaption: Valuation
Bonds
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
, the underlyers in this case, exhibit what is known as pull-to-par
Pull to par
Pull to Par is the effect in which the price of a bond converges to par value as time passes. At maturity the price of a debt instrument in good standing should equal its par .Another name for this effect is reduction of maturity....
: as the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility. On the other hand, the Black–Scholes model, which assumes constant volatility, does not reflect this process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
, and cannot therefore be applied here;http://kalotay.com/system/files/private/BlackScholes.pdf see Black–Scholes: Valuing bond options.
Addressing this, bond options are usually valued using the Black model
Black model
The Black model is a variant of the Black–Scholes option pricing model. Its primary applications are for pricing bond options, interest rate caps / floors, and swaptions...
or with a lattice based
Lattice model (finance)
In finance, a lattice model can be used to find the fair value of a stock option; variants also exist for interest rate derivatives.The model divides time between now and the option's expiration into N discrete periods...
short rate model
Short rate model
In the context of interest rate derivatives, a short-rate model 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:...
such as Black-Derman-Toy, Ho-Lee or Hull–White. http://pages.stern.nyu.edu/~eelton/debt_inst_class/option%20valuation.pdf The latter approach is theoretically more correct, http://books.google.com/books?id=wF8yVzLI6EYC&printsec=frontcover&dq=Valuation+of+fixed+income+securities+and+derivatives, although in practice the Black Model is more widely used for reasons of simplicity and speed. For American- and Bermudan- styled options
Option style
In finance, the style or family of an option is a general term denoting the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American options. These options - as well as others where the...
, where exercise is permitted prior to maturity, only the lattice based approach is applicable.
- Using the Black model, the spot priceSpot priceThe spot price or spot rate of a commodity, a security or a currency is the price that is quoted for immediate settlement . Spot settlement is normally one or two business days from trade date...
in the formula is not simply the market price of the underlyingUnderlyingIn finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...
bond, rather it is the forwardForward priceThe forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends etc...
bond price. This forward price is calculated by first subtracting the present value of the coupons between the valuation date (i.e. today) and the exercise date from today's dirty priceDirty priceThe dirty price of a bond represents the value of a bond, exclusive of any commissions or fees. The dirty price is also called the "full price" or the "all in price."-Bond Pricing:...
, and then forward valuingCompound interestCompound interest arises when interest is added to the principal, so that from that moment on, the interest that has been added also itself earns interest. This addition of interest to the principal is called compounding...
this amount to the exercise date. (These calculations are performed using today's yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
, as opposed to the bond's YTMYTMYTM may refer to:*Yield to maturity*The designation for "medium harbor tug" in the United States Navy ship classification system*Rivière Rouge – Mont Tremblant International Airport, Canada...
.) The reason that the Black Model may be applied in this way is that the numeraireNuméraireNuméraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
is then $1 at the time of delivery (whereas under Black–Scholes, the numeraire is $1 today). This allows us to assume that (a) the bond price is a random variable at a future date, but also (b) that the risk-free rate between now and then is constant. Thus the valuation takes place in a risk-neutral "forward world" where the expected future spot rate is the forward rate, and its standard deviationStandard deviationStandard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...
is the same as in the "physical world"; http://finance.eller.arizona.edu/lam/fixi/creditmod/xl/BM1S09.xls see Girsanov's theorem. The volatilityVolatility (finance)In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...
used, is typically "read-off" an Implied volatility surface.
- The lattice based model entails a tree of short rates consistent with today's yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
and short rate (caplet) volatility, and where the final time step of the tree corresponds to the date of the underlying bond's maturity. Using this tree (1) the bond is valued at each node by "stepping backwards" through the tree: at the final nodes, bond value is simply face valueFace valueThe Face value is the value of a coin, stamp or paper money, as printed on the coin, stamp or bill itself by the minting authority. While the face value usually refers to the true value of the coin, stamp or bill in question it can sometimes be largely symbolic, as is often the case with bullion...
(or $1), plus coupon (in cents) if relevant; at each earlier node, it is the discountedPresent valuePresent value, also known as present discounted value, is the value on a given date of a future payment or series of future payments, discounted to reflect the time value of money and other factors such as investment risk...
expected valueExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the up- and down-nodes in the later time step, plus coupon payments during the current time step. Then (2), the option is valued similar to the approach for equity options: at nodes in the time-step corresponding to option maturity, value is based on moneynessMoneynessIn finance, moneyness is a measure of the degree to which a derivative is likely to have positive monetary value at its expiration, in the risk-neutral measure. It can be measured in percentage probability, or in standard deviations....
; at earlier nodes, it is the discounted expected value of the option at the up- and down-nodes in the later time step, and, depending on option styleOption styleIn finance, the style or family of an option is a general term denoting the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American options. These options - as well as others where the...
(and other specifications - see below), of the bond value at the node. http://books.google.com/books?id=wF8yVzLI6EYC&printsec=frontcover&dq=Valuation+of+fixed+income+securities+and+derivatives http://janroman.dhis.org/stud/BDT%20Thesis.pdf For both steps, the discounting is at the short rate for the tree-node in question. (Note that the Hull-White tree is usually TrinomialTrinomial TreeThe Trinomial tree is a lattice based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the Binomial options pricing model, and is conceptually similar...
: the logic is as described, although there are then three nodes in question at each point.)
Embedded options
The term "bond option" is also used for option-like features of some bonds ("embedded optionEmbedded option
An Embedded option is a component of a financial bond or other security, and usually 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. Some common types of bonds with embedded options...
s"). These are an inherent part of the bond, rather than a separately traded product. These options are not mutually exclusive, so a bond may several options embedded. http://financial-dictionary.thefreedictionary.com/embedded+option Bonds of this type include:
- Callable bondCallable bondA callable bond is a type of bond that allows the issuer of the bond to retain the privilege of redeeming the bond at some point before the bond reaches the date of maturity. In other words, on the call date, the issuer has the right, but not the obligation, to buy back the bonds from the bond...
: allows the issuer to buy back the bond at a predetermined price at certain time in future. The holder of such a bond has, in effect, sold a call option to the issuer. Callable bonds cannot be called for the first few years of their life. This period is known as the lock out period. - Puttable bondPuttable bondPuttable 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...
: allows the holder to demand early redemption at a predetermined price at certain time in future. The holder of such a bond has, in effect, purchased a put option on the bond. - Convertible bondConvertible bondIn finance, a convertible note is a type of bond that the holder can convert into shares of common stock in the issuing company or cash of equal value, at an agreed-upon price. It is a hybrid security with debt- and equity-like features...
: allows the holder to demand conversion of bonds into the stock of the issuer at a predetermined price at certain time period in future. - Extendible bondExtendible bondExtendible bond is a complex bond with the embedded option for a holder to extend its maturity date by a number of years. Such a bond may be considered as a portfolio of a straight, shorter-term bond and a call option to buy a longer-term bond. This relatively rare type of bond works to the...
: allows the holder to extend the bond maturity date by a number of years. - Exchangeable bondExchangeable bondExchangeable bond is a type of hybrid security consisting of a straight bond and an embedded option to exchange the bond for the stock of a company other than the issuer at some future date and under prescribed conditions. An exchangeable bond is different from a convertible bond...
: allows the holder to demand conversion of bonds into the stock of a different company, usually a public subsidiary of the issuer, at a predetermined price at certain time period in future.
Bonds with embedded option can be valued using the lattice-based approach, as above, but additionally allowing that the option's effect is incorporated at each node in the tree, impacting the bond price and / or the option price as specified. http://books.google.com/books?id=wF8yVzLI6EYC&printsec=frontcover&dq=Valuation+of+fixed+income+securities+and+derivatives These bonds are also sometimes valued using Black–Scholes. Here, the bond is priced as a "straight bond"
Bond valuation
Bond valuation is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value of the stream of cash flows it is expected to generate. Hence, the value of a bond is obtained by discounting the bond's expected...
(i.e. as if it had no embedded features) and the option is valued using the Black Scholes formula. The option value is then added to the straight bond price if the optionality rests with the buyer of the bond; it is subtracted if the seller of the bond (i.e. the issuer) may choose to exercise. http://www.numa.com/derivs/ref/calculat/cb/calc-cbb.htm http://www.stern.nyu.edu/~adamodar/pc/inv2ed/ill33p6.xls More sophisticated approaches view a bond as comprising an equity component and a debt component, each with different default risks, and which must be modelled as a "coupled system"; see http://www.fincad.com/derivatives-resources/articles/pricing-convertible-bonds.aspx.
Relationship with caps and floors
European Put options on zero coupon bonds can be seen to be equivalent to suitable caplets, i.e. interest rate cap components, whereas call options can be seen to be equivalent to suitable floorlets, i.e. components of interest rate floors. See for example Brigo and Mercurio (2001), who also discuss bond options valuation with different models.External links
Discussion- Bond Options, Caps and the Black Model Milica Cudina, University of Texas at AustinUniversity of Texas at AustinThe University of Texas at Austin is a state research university located in Austin, Texas, USA, and is the flagship institution of the The University of Texas System. Founded in 1883, its campus is located approximately from the Texas State Capitol in Austin...
- Martingales and Measures: Black's Model Jacqueline Henn-Overbeck, University of BaselUniversity of BaselThe University of Basel is located in Basel, Switzerland, and is considered to be one of leading universities in the country...
- The Problem with Black, Scholes et al., Andrew KalotayAndrew KalotayAndrew Kalotay is a Hungarian-born finance professor, Wall Street quant and chess master. He is best known as an authority on fixed income valuation and institutional debt management...
Online tools
- Bond Option Pricing using the Black Model Dr. Shing Hing Man, Thomson-Reuters' Risk Management
- Pricing A Bond Using the BDT Model Dr. Shing Hing Man, Thomson-Reuters' Risk Management
- 'Greeks' Calculator using the Black model, Dr. Razvan Pascalau, SUNY Plattsburgh