1、FIXED-INCOME SECURITIESLecture 9Options on Bonds andBonds with EmbeddedOptionsBinomial Treeu=1.1,d=.95,So=.10S010.Sd.095Su.11Suu.121Sud.1045Sdd.09025Sddd.0857Sudd.09927Suud.11495Suuu.1331Value of Two-Period Option-Free Bond:C=8 and F=100BB00597297 8598630 811096330.Bd108109598630.Bu10811197297.Bdd10
2、8Bud108Buu108Bu u1 0 91 1 2 19 7 2 3 4 6.Bu d1 0 91 1 0 4 59 8 6 8 7 2.Bd d1 0 91 0 9 0 2 59 9 9 7 7.Bu.5972346 959868729111963612Bd.5 96 687295 99 9779109598 9334B05 96361295 9893349110969521.1 0 91 0 91 0 91 0 9Value of Three-Period Option-Free BondC=9,F=100 Callable Bonds and Putable BondsBond wi
3、th Embedded OptionsCallable bonds Issuer may repurchase at a pre-specified call price Typically called if interest rates fallA callable bond has two disadvantages for an investor If it is effectively called,the investor will have to invest in another bond yielding a lower rate A callable bond has th
4、e unpleasant property for an investor to appreciate less than a normal similar bond when interest rates fall Therefore,an investor will be willing to buy such a bond at a lower price than a comparable option-free bondExamples The UK Treasury bond with coupon 5.5%and maturity date 09/10/2012 can be c
5、alled in full or part from 09/10/2008 on at a price of pounds 100 The US Treasury bond with coupon 7.625%and maturity date 02/15/2007 can be called on coupon dates only,at a price of$100,from 02/15/2002 on Such a bond is said to be discretely callable Callable and Putable Bonds Institutional Aspects
6、 Putable bond holder may retire at a pre-specified price A putable bond allows its holder to sell the bond at par value prior to maturity in case interest rates exceed the coupon rate of the issue So,he will have the opportunity to buy a new bond at a higher coupon rate The issuer of this bond will
7、have to issue another bond at a higher coupon rate if the put option is exercised Hence a putable bond trades at a higher price than a comparable option-free bondCallable and Putable Bonds Yield-to-WorstYield-to-callYear54.54%Year64.61%Year74.66%Year84.69%Year94.72%Yield-to-worstyear104.74%Let us co
8、nsider a bond with an embedded call option trading over its par valueThis bond can be redeemed by its issuer prior to maturity,from its first call date on One can compute a yield-to-call on all possible call dates The yield-to-worst is the lowest of the yield-to-maturity and all yields-to-callExampl
9、e 10-year bond bearing an interest coupon of 5%,discretely callable after 5 years and trading at 102 There are 5 possible call dates before maturity Yield-to-worst is 4.54%Callable BondsValuation in a Binomial Model the value of the callable bond is determined by selecting the minimum of the otherwi
10、se noncallable bond or the call price,and then rolling the callable bond value to the current period.Recursive procedure Price cash-flow to be discounted on period n-1 is the minimum value of the price computed on period n and call price on period n And so on until we get the price P of the callable
11、 bond,CNCttBMin BCPBB00597297 8 598 811096044.BMin BCPMinuCuNC,.,.97297 9897297Buu108BMin BCPMindCdNC,.,98630 9898Buu 108Buu1082,9%98ValueofPeriodCallablebondwithCPBM inu dC.,9 8 6 8 7 29 89 8BBMinuuC.,.5 972346 95 98 9111960516960516 98960516BBMinddC.,.5 9895 989109597 716997 7169 9897 7169BC059605
12、16 95977169 911096258.BM ind dC.,9 9 9 7 79 89 8BM inu uC.,.9 7 2 3 4 69 89 7 2 3 4 6Value of Callable Bond:C=9,F=100Value of Putable bond,PNCttBMax BPPSBBMaxuuuunpuuP121%,972346972346 97972346.,.Value of Putable Bond:C=9,F=100,PP=97SBBMaxuuuP11%5 97234695 9868729111963612963612 9797.,SBBMaxududnpud
13、P1045%,986872986872 97986872.,.SBBMaxddddnpddP9025%,9997799977 9799977.,.SBBMaxdddP95%5 98687295 999779111989334989334 97989334.,.SBP0010%5 9795 9893349110972425.An alternative but equivalent approach is to calculate the weighted average value of each possible paths defined by the binomial process.T
14、his value is known as the theoretical value.The t-period spot rate is equal to the geometric average of the current and expect one-period spot rates.Alternative Binomial Valuation Approach consider again the three-period,9%option-free bond valued with a two-period interest rate tree 11011/21/22011/3
15、1/330.10(1)(1)1(1.10)(1.095)1.0974972(1)(1)(1)1(1.10)(1.095)(1.09025)1.0950PathPathdPathdddSSSSSSSSS 312023123(1)(1)(1)PathiPathiPathiPathiCFCFCFBSSSPath 1Period tSt0CFPV10%10.198.1818189.50%20.097497297.471989.03%30.095076110983.0029698.65676Path 2Period tSt0CFPV10%10.198.1818189.50%20.097497297.47
16、19810.45%30.099826510981.9320897.58588Path 3Period tSt0CFPV10%10.198.18181811.00%20.104988797.37100710.45%30.104825810980.8248996.37771Path 4Period tSt0CFPV10%10.198.18181811.00%20.104988797.37100712.10%30.110300210979.6352395.18805Theoretical Value=(98.65676+97.58588+96.37771+95.18805)/4 =96.9521Ca
17、llable and Putable BondsMonte Carlo ApproachStep 1:generate a large number of short-term interest rate pathsStep 2:along each interest rate path,the price P of the bond with embedded option is recursively determinedThe price of the bond is computed as the average of its prices along all interest rat
18、e pathsCallable and Putable BondsMonte Carlo Approach-Example Price a callable bond with annual coupon 4.57%,maturity 10 years,redemption value 100 and callable at 100 after 5 years Prices of the bond under each scenario Price of the bond is average over all paths P=1/6(100.43+100.55+99.9+99.76+99.6
19、8+100.55)=100.14 The Monte Carlo pricing methodology can also be applied to the valuation of all kinds of interest rates derivativesOptions on BondsTerminology An option is a contract in which the seller(writer)grants the buyer the right to purchase from,or sell to,the seller an underlying asset(her
20、e a bond)at a specified price within a specified period of time The seller grants this right to the buyer in exchange for a certain sum of money called the option price or option premium The price at which the instrument may be bought or sold is called the exercise or strike price The date after whi
21、ch an option is void is called the expiration date An American option may be exercised any time up to and including the expiration date A European option may be exercised only on the expiration dateOptions on BondsFactors that Influence Option Prices Current price of underlying security As the price
22、 of the underlying bond increases,the value of a call option rises and the value of a put option falls Strike price Call(put)options become more(less)valuable as the exercise price decreasesTime to expiration For American options,the longer the time to expiration,the higher the option price because
23、all exercise opportunities open to the holder of the short-life option are also open to the holder of the long-life option Short-term risk-free interest rate Price of call option on bond increases and price of put option on bond decreases as short-term interest rate rises(through impact on bond pric
24、e)Expected volatility of yields(or prices)As the expected volatility of yields over the life of the option increases,the price of the option will also increaseOptions on BondsPricing Options on long-term bonds Interest payments are similar to dividends Otherwise,long-term bonds are like options on s
25、tock:We can use Black-Scholes as in options on dividend-paying equity Options on short-term bonds Problem:they are not like a stock because they quickly converge to par We cannot directly apply Black-Scholes Other shortcomings of standard option pricing models Assumption of a constant short-term rat
26、e is inappropriate for bond options Assumption of a constant volatility is also inappropriate:as a bond moves closer to maturity,its price volatility declineOptions on BondsPricing A solution to avoid the problem is to consider an interest rate model,The following figure shows a tree for the 1-year
27、rate of interest(calibrated to the current TS)The figure also shows the values for a discount bond(par=100)at each node in the treeOptions on BondsPricingConsider a 2-year European call on this 3-year bond struck at 93.5Start by computing the value at the end of the tree If by the end of the 2nd yea
28、r the short-term rate has risen to 7%and the bond is trading at 93,the option will expire worthless If the bond is trading at 94(corresponding to a short-term rate of 6%)the call option is worth 0.5 If the bond is trading at 95(short-term rate=5%),the call is worth 1.5Working our way backward the tr
29、ee01(0.5 00.5 0.5)0.23471 6.5%1(0.5 0.50.5 1.5)0.94791 5.5%1(0.50.5)0.55731 6%uludCCCCC Options on BondsPut-Call ParityAssumption no coupon payments and no premature exerciseConsider a portfolio where we purchase one zero coupon bond,one put European option,and sell(write)one European call option(sa
30、me time to maturity T and the same strike price X)Payoff at date TOptions on BondsPut-Call Parity Cont No matter what state of the world obtains at the expiration date,the portfolio will be worth X Thus,the payoff from the portfolio is risk-free,and we can discount its value at the risk-free rate r
31、We obtain the call-put relationship000000rTrTBPCXeCBPXe For coupon bonds000()rTCBPXePV CouponsConvertible BondsDefinition Convertible securities are usually either convertible bonds or convertible preferred shares which are most often exchangeable into the common stock of the company issuing the con
32、vertible security Being debt or preferred instruments,they have an advantage to the common stock in case of distress or bankruptcy Convertible bonds offer the investor the safety of a fixed income instrument coupled with participation in the upside of the equity markets Essentially,convertible bonds
33、 are bonds that,at the holders option,are convertible into a specified number of sharesConvertible BondsTerminology Convertible bonds Bondholder has a right to convert bond for pre-specified number of share of common stock Terminology Convertible price is the price of the convertible bond Bond floor
34、 or investment value is the price of the bond if there is no conversion option Conversion ratio is the number of shares that is exchanged for a bond Conversion value=current share price x conversion ratio Conversion premium=(convertible priceconversion value)/conversion value Convertible BondsExampl
35、es Example 1:Example 1:Current bond price=$930 Conversion ratio:1 bond=30 shares common Current stock price=$25/share Market Conversion Value=(30 shares)x(25)=$750 Conversion Premium=(930 750)/750=180/750=24%Example 2:AXA Convertible BondExample 2:AXA Convertible Bond AXA has issued in the zone a co
36、nvertible bond paying a 2.5%coupon rate and maturing on 01/01/2014;the conversion ratio is 4.04 On 12/13/2001,the current share price was 24.12 and the bid-ask convertible price was 156.5971/157.5971 The conversion value was equal to 97.44=4.04 x 24.12 The conversion premium calculated with the ask
37、price 157.5971 was 61.73%=(157.5791-97.44)/97.44 The conversion of the bond into 4.04 shares can be executed on any date before the maturity dateConvertible BondsUsesFor the issuer Issuing convertible bonds enables a firm to obtain better financial conditions Coupon rate of such a bond is always low
38、er to that of a bullet bond with the same characteristics in terms of maturity and coupon frequency This comes directly from the conversion advantage which is attached to this product Besides the exchange of bonds for shares diminishes the liabilities of the firm issuer and increases in the same tim
39、e its equity so that its debt capacity is improvedFor the convertible bondholder The convertible bond is a defensive security,very sensitive to a rise in the share price and protective when the share price decreases If the share price increases,the convertible price will also increase When share pri
40、ce decreases,price of convertible never gets below the bond floor,i.e.,the price of an otherwise identical bullet bond with no conversion optionConvertible BondsDeterminants of Convertible Bond Prices Convertible bond is similar to a normal coupon bond plus a call option on the underlying stock With
41、 an important difference:the effective strike price of the call option will vary with the price of the bond Convertible securities are priced as a function of The price of the underlying stock Expected future volatility of equity returns Risk free interest rates Call provisions Supply and demand for
42、 specific issues Issue-specific corporate/Treasury yield spread Expected volatility of interest rates and spreads Thus,there is large room for relative mis-valuationsConvertible BondsConvertible Bond Price as a Function of Stock PriceConvertible Bond Price as a Function of Stock PriceConvertible Bon
43、dsConvertible Bond Pricing Model A popular method for pricing convertible bonds is A popular method for pricing convertible bonds is the component modelthe component model The convertible bond is divided into a straight bond component and a call option on the conversion price,with strike price equal
44、 to the value of the straight bond component The fair value of the two components can be calculated with standard formulas,such as the famous Black-Scholes valuation formula.This pricing approach,however,has several This pricing approach,however,has several drawbacksdrawbacks First,separating the co
45、nvertible into a bond component and an option component relies on restrictive assumptions,such as the absence of embedded options(callability and putability,for instance,are convertible bond features that cannot be considered in the above separation)Second,convertible bonds contain an option compone
46、nt with a stochastic strike price equal to the bond priceConvertible BondsConvertible Bond Pricing ModelsConvertible Bond Pricing Models Theoretical research on convertible bond pricing was initiated by Ingersoll(1977)and Brennan and Schwartz(1977),who both applied the contingent claims approach to
47、the valuation of convertible bonds In their valuation models,the convertible bond price depends on the firm value as the underlying variable Brennan and Schwartz(1980)extend their model by including stochastic interest rates.These models rely heavily on the theory of stochastic processes and require
48、 a relatively high level of mathematical sophisticationConvertible BondsBinomial Model The price of the stock only can go up to a given value or down to a given value Besides,there is a bond(bank account)that will pay interest of rConvertible BondsBinomial Model We assume u(up)d(down)For Black and S
49、choles we will need d=1/u For consistency we also need u (1+r)d Example:u=1.25;d=0.80;r=10%Convertible BondsBinomial Model Basic model that describes a simple world.As the number of steps increases,it becomes more realistic We will price and hedge an option:it applies to any other derivative securit
50、y Key:we have the same number of states and securities(complete markets)Basis for arbitrage pricingConvertible BondsBinomial Model Introduce an European call option:K=110 It matures at the end of the periodConvertible BondsBinomial Model We can replicate the option with the stock and the bond Constr
