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{| style="float: right;" border="1" width="400"
|- valign="top"
|+Example
|
*Trade Date: 1 March 2003
*Maturity Date: 6 March 2006
*Option Buyer: Bank A
*Underlying asset: FNMA Bond
*Spot Price: $101
*Strike Price: $102
||
*On the Trade Date, Bank A enters into an option with Bank B to buy certain FNMA Bonds from Bank B for the Strike Price mentioned. Bank A pays a premium to Bank B which is the premium percentage multiplied by the face value of the bonds.
*At the maturity of the option, Bank A either exercises the option and buys the bonds from Bank B at the predetermined strike price, or chooses not to exercise the option. In either case, Bank A has lost the premium to Bank B.
|}
In [[finance]], a '''bond option''' is an [[option (finance)|option]] to buy or sell a [[bond (finance)|bond]] at a certain price on or before the option expiry date.<ref name="thefreedictionary.com">{{cite web|url=http://financial-dictionary.thefreedictionary.com/Bond+option|title=Bond option}}</ref> These instruments are typically traded [[Over-the-counter (finance)|OTC]].
*A [[option style|European]] bond option is an option to buy or sell a bond at a certain date in future for a predetermined price.
*An [[option style|American]] 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]] on the bond if one believes that [[interest rate]]s will fall, causing an increase in bond prices. Likewise, one buys the [[put option]] if one believes that interest rates will rise.<ref name="thefreedictionary.com"/> 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]] associated with fluctuations in the bond price.

==Valuation==
{{hatnote|Compare: {{sectionlink|Swaption|Valuation}}}}
[[bond (finance)|Bonds]], the [[underlying|underlyers]] in this case, exhibit what is known as [[pull to par|pull-to-par]]: as the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its [[volatility (finance)|volatility]]. On the other hand, the [[Black–Scholes]] model, which assumes constant volatility, does not reflect this [[Stochastic process|process]], and cannot therefore be applied here; [http://kalotay.com/sites/default/files/private/BlackScholes.pdf] see {{slink|Black–Scholes model#Valuing bond options}}.

Addressing this, bond options are usually valued using the [[Black model]] or with a [[Lattice model (finance)|lattice-based]] [[short-rate model]] such as [[Black–Derman–Toy model|Black-Derman-Toy]], [[Ho–Lee model|Ho-Lee]] or [[Hull–White model|Hull–White]]. [http://pages.stern.nyu.edu/~eelton/debt_inst_class/option%20valuation.pdf] The latter approach is theoretically more correct, [https://books.google.com/books?id=wF8yVzLI6EYC&q=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 option|American-]] and [[Bermudan option|Bermudan-]] [[option style|styled options]], where exercise is permitted prior to maturity, only the lattice-based approach is applicable.

*Using the Black model, the [[spot price]] in the formula is not simply the market price of the [[underlying]] bond, rather it is the [[Forward price|forward]] 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 price]], and then [[Compound interest|forward valuing]] this amount to the exercise date. (These calculations are performed using today's [[yield curve]], as opposed to the bond's [[Yield to maturity|YTM]].) The reason that the Black Model may be applied in this way is that the [[numeraire]] 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 (since using the [[forward measure]] moves the discounting outside of the expectation term [https://quant.stackexchange.com/questions/61151/martingale-pricing-with-time-dependent-risk-free-rate/61159#61159]).  Thus the valuation takes place in a [[risk neutrality|risk-neutral]] "forward world" where the expected future spot rate is the forward rate, and its [[standard deviation]] is the same as in the "physical world"; [https://web.archive.org/web/20110719135831/http://finance.eller.arizona.edu/lam/fixi/creditmod/xl/BM1S09.xls] see [[Girsanov's theorem]]. The volatility used, is typically "read-off" an [[Volatility smile#Implied volatility surface|Implied volatility surface]].

*The lattice-based model entails a tree of short rates – a zeroeth step – consistent with today's [[yield curve]] and short rate (often [[Interest rate cap and floor#Interest rate cap|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 value]] (or $1), plus coupon (in cents) if relevant; at each earlier node, it is the [[present value|discounted]] [[expected value]] 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 [[Binomial options pricing model#STEP 3: Find Option value at earlier nodes|approach for equity options]]:  at nodes in the time-step corresponding to option maturity, value is based on [[moneyness]]; 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 style]] (and other specifications – see [[Bond option#Embedded options|below]]), of the bond value at the node. [https://books.google.com/books?id=wF8yVzLI6EYC&q=Valuation+of+fixed+income+securities+and+derivatives] [https://web.archive.org/web/20110725232352/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 [[Trinomial Tree|Trinomial]]: the logic is as described, although there are then three nodes in question at each point.) See {{slink|Lattice model (finance)#Interest rate derivatives}}.

==Embedded options==
The term "bond option" is also used for option-like features of some bonds ("[[embedded option]]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 have several options embedded. [http://financial-dictionary.thefreedictionary.com/embedded+option] Bonds of this type include:

*[[Callable bond]]: allows the issuer to buy back the bond at a predetermined price at a 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 bond]]: allows the holder to demand early redemption at a predetermined price at a certain time in future. The holder of such a bond has, in effect, purchased a put option on the bond.
*[[Convertible bond]]: allows the holder to demand conversion of bonds into the stock of the issuer at a predetermined price at a certain time period in future.
*[[Extendible bond]]: allows the holder to extend the bond maturity date by a number of years.
*[[Exchangeable 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.

Callable and putable bonds can be valued using the lattice-based approach, as above, but additionally allowing that the effect of the embedded option is incorporated at each node in the tree, impacting the bond price and / or the option price as specified. [https://books.google.com/books?id=wF8yVzLI6EYC&q=Valuation+of+fixed+income+securities+and+derivatives] These bonds are also sometimes valued using [[Black–Scholes]]. Here, the bond is [[Bond valuation|priced as a "straight bond"]] (i.e. as if it had no embedded features) and the option is valued using the [[Black–Scholes#Black.E2.80.93Scholes formula|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://ocw.mit.edu/courses/sloan-school-of-management/15-414-financial-management-summer-2003/lecture-notes/lec20_options2.pdf] [https://web.archive.org/web/20110304203828/http://www.numa.com/derivs/ref/calculat/cb/calc-cbb.htm] [http://www.stern.nyu.edu/~adamodar/pc/inv2ed/ill33p6.xls]{{dead link|date=November 2016 |bot=InternetArchiveBot |fix-attempted=yes }} For convertible and exchangeable bonds, a more sophisticated approach is to model the instrument  as a "coupled system" comprising an equity component and a debt component, each with different default risks; see {{slink|Lattice model (finance)#Hybrid securities}}.

==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 floor]]s. See for example Brigo and Mercurio (2001), who also discuss bond options valuation with different models.

==References==
{{Reflist}}
*{{cite journal|first=F.|last=[[Fischer Black|Black]]|author2=[[Emanuel Derman|Derman, E.]]|author3=[[William Toy|Toy, W.]]|title=A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options|date=January–February 1990|pages=24–32|journal=Financial Analysts Journal|url=http://savage.wharton.upenn.edu/FNCE-934/syllabus/papers/Black_Derman_Toy_FAJ_90.pdf|url-status=dead|archive-url=https://web.archive.org/web/20080910041743/http://savage.wharton.upenn.edu/FNCE-934/syllabus/papers/Black_Derman_Toy_FAJ_90.pdf|archive-date=2008-09-10}}
*{{cite book | title = Interest Rate Models - Theory and Practice with Smile, Inflation and Credit| author = [[Damiano Brigo]] and [[Fabio Mercurio]] | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | isbn = 978-3-540-22149-4}}
*{{cite book | title = Investment Valuation| author = [[Aswath Damodaran]] | publisher = [[John Wiley & Sons|John Wiley]]| year = 2002| edition = 2nd| isbn = 0-471-41488-3}}, [http://pages.stern.nyu.edu/~adamodar/New_Home_Page/Inv2ed.htm#ch33 Chapter 33: Valuing Fixed Income Securities]
*{{cite book | title = Valuation of fixed income securities and derivatives | author = [[Frank Fabozzi]] | publisher = [[John Wiley & Sons|John Wiley]]| year = 1998| edition = 3rd| isbn = 978-1-883249-25-0}}
*{{cite book | title = Bond Evaluation, Selection, and Management| author = R. Stafford Johnson| publisher = [[John Wiley & Sons|John Wiley]]| year = 2010| edition = 2nd| isbn = 978-0470478356}}
*{{cite book | title = Valuation of Interest-Sensitive Financial Instruments: SOA Monograph M-FI96-1| author =  David F. Babbel | publisher = John Wiley & Sons| year = 1996| edition = 1st| isbn = 978-1883249151}}

==External links==
;Discussion
*[https://www.ma.utexas.edu/users/mcudina/Lecture24_3.pdf Bond Options, Caps and the Black Model], Milica Cudina, [[University of Texas at Austin]]
*[http://business.nmsu.edu/~sankaran/fin545/Chapter%209.ppt Valuing Bonds with Embedded Options]{{dead link|date=November 2016 |bot=InternetArchiveBot |fix-attempted=yes }}, [[Frank J. Fabozzi]]
*[https://web.archive.org/web/20120721210243/http://www.ederman.com/new/docs/gs-valuing_convertibles.pdf Valuing Convertible Bonds as Derivatives], [[Goldman Sachs]] (authors include [[Emanuel Derman]] and [[Piotr Karasinski]])
*[http://upetd.up.ac.za/thesis/available/etd-05052009-115008/unrestricted/dissertation.pdf The Valuation and Calibration of Convertible Bonds], Sanveer Hariparsad, [[University of Pretoria]]
*[https://web.archive.org/web/20110113060801/http://wwz.unibas.ch/fileadmin/wwz/redaktion/finance/HS09/FOPT_2_090924.pdf Martingales and Measures: Black's Model], Jacqueline Henn-Overbeck, [[University of Basel]]
*[https://web.archive.org/web/20111216035130/http://www.blackwellpublishing.com/bonds/Powerpoint%20Slides/Chapter9.ppt Binomial Interest Rate Trees and the Valuation of Bonds with Embedded Options], Stafford Johnson, [[Xavier University]]
*[http://kalotay.com/sites/default/files/private/BlackScholes.pdf The Problem with Black, Scholes et al.], [[Andrew Kalotay]]
*[https://web.archive.org/web/20150501032252/http://www.mth.uct.ac.za/academics/postgrad/graduatethesis/MSc_Ariel_Zadikov.pdf Methods of Pricing Convertible Bonds], Ariel Zadikov, [[University of Cape Town]]
'''Online tools'''
*[http://www.thomasho.com/mainpages/?analysoln Black Bond Option Model], Dr. [[Thomas Ho (finance)|Thomas Ho]], thomasho.com
*[https://web.archive.org/web/20130128103932/http://lombok.demon.co.uk/financialTap/options/bond/shortterm Bond Option Pricing using the Black Model] Dr. Shing Hing Man, Thomson-Reuters' Risk Management
*[https://web.archive.org/web/20110807174135/http://lombok.demon.co.uk/financialTap/interestrates/bdtbond Pricing A Bond Using the BDT Model] Dr. Shing Hing Man, Thomson-Reuters' Risk Management
*[https://web.archive.org/web/20110316171617/http://www.cba.ua.edu/~rpascala/greeks2/GFOPMForm.php 'Greeks' Calculator using the Black model], Dr. Razvan Pascalau, [[SUNY Plattsburgh]]
*[http://www.pricing-option.com/G2pp.aspx Pricing Bond Option using G2++ model], pricing-option.com

{{Bond market}}
{{Derivatives market}}

[[Category:Bonds (finance)]]
[[Category:Options (finance)]]

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