Bonds with Embedded Options and Options on Bonds

Transcription

1 FIXED-INCOME SECURITIES Chapter 14 Bonds with Embedded Options and Options on Bonds

2 Callable and Ptable Bonds Instittional Aspects Valation Convertible Bonds Instittional Aspects Valation Options on Bonds Instittional Aspects Valation Uses Otline

3 Callable Bonds and Ptable Bonds Bond with Embedded Options Callable bonds Isser may reprchase at a pre-specified call price Typically called if interest rates fall A 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 the npleasant property for an investor to appreciate less than a normal similar bond when interest rates fall Therefore, an investor will be willing to by sch a bond at a lower price than a comparable option-free bond Examples The UK Treasry bond with copon 5.5% and matrity date 09/10/2012 can be called in fll or part from 09/10/2008 on at a price of ponds 100 The US Treasry bond with copon 7.625% and matrity date 02/15/2007 can be called on copon dates only, at a price of $100, from 02/15/2002 on Sch a bond is said to be discretely callable

4 Callable and Ptable Bonds Instittional Aspects Ptable bond holder may retire at a pre-specified price A ptable bond allows its holder to sell the bond at par vale prior to matrity in case interest rates exceed the copon rate of the isse So, he will have the opportnity to by a new bond at a higher copon rate The isser of this bond will have to isse another bond at a higher copon rate if the pt option is exercised Hence a ptable bond trades at a higher price than a comparable option-free bond

5 Callable and Ptable Bonds Yield-to-Worst Let s consider a bond with an embedded call option trading over its par vale This bond can be redeemed by its isser prior to matrity, from its first call date on One can compte a yield-to-call on all possible call dates The yield-to-worst is the lowest of the yield-to-matrity and all yields-to-call Example 10-year bond bearing an interest copon of 5%, discretely callable after 5 years and trading at 102 There are 5 possible call dates before matrity Yield-to-worst is 4.54% Yield-to-call year % year % year % year % year % Yield-to-matrity year %

6 Callable and Ptable Bonds Valation in a Binomial Model Let s assme that a binomial tree has been already bilt and calibrated as explained in Chapter 12 Recrsive procedre Price cash-flow to be disconted on period n-1 is the minimm vale of the price compted on period n and call price on period n And so on ntil we get the price P of the callable bond Example We consider a callable bond with matrity two years, annal copon 5%, callable in one year at 100 r 0 = 4%, r = 4.66% and r l = 4.57% (cf. example in Chapter 12) We have P = 105/ = and P l = 105/ = Finally, price of the callable bond P = 1 2 ( ) + 5 min( 100,100.41) min 100, % % + 5 =

7 Callable and Ptable Bonds Monte Carlo Approach Step 1: generate a large nmber of short-term interest rate paths sing some dynamic model (see Chapter 12) Step 2: along each interest rate path, the price P of the bond with embedded option is recrsively determined The price of the bond is compted as the average of its prices along all interest rate paths Period Path1 Path2 Path3 Path4 Path5 Path6 1 4,00% 4,00% 4,00% 4,00% 4,00% 4,00% 2 4,08% 4,14% 4,29% 4,24% 4,28% 4,28% 3 3,83% 4,02% 4,35% 4,27% 4,24% 4,23% 4 4,15% 3,88% 4,25% 3,87% 4,17% 4,30% 5 4,27% 4,26% 4,68% 4,58% 4,29% 3,99% 6 4,69% 4,49% 4,33% 4,29% 4,47% 4,32% 7 4,88% 5,10% 5,24% 5,08% 5,27% 4,70% 8 5,14% 4,94% 4,75% 5,54% 5,25% 5,08% 9 5,24% 5,47% 5,15% 5,26% 5,43% 5,64% 10 5,59% 5,04% 5,29% 5,58% 5,38% 5,02%

8 Callable and Ptable Bonds Monte Carlo Approach - Example Price a callable bond with annal copon 4.57%, matrity 10 years, redemption vale 100 and callable at 100 after 5 years Prices of the bond nder each scenario Path1 Path2 Path3 Path4 Path5 Path6 Price of the callable bond Price of the bond is average over all paths P=1/6( )= The Monte Carlo pricing methodology can also be applied to the valation of all kinds of interest rates derivatives

9 Convertible Bonds Definition Convertible secrities are sally either convertible bonds or convertible preferred shares which are most often exchangeable into the common stock of the company issing the convertible secrity Being debt or preferred instrments, they have an advantage to the common stock in case of distress or bankrptcy Convertible bonds offer the investor the safety of a fixed income instrment copled with participation in the pside of the eqity markets Essentially, convertible bonds are bonds that, at the holder's option, are convertible into a specified nmber of shares

10 Convertible Bonds Terminology Convertible bonds Bondholder has a right to covert bond for pre-specified nmber of share of common stock Terminology Convertible price is the price of the convertible bond Bond floor or investment vale is the price of the bond if there is no conversion option Conversion ratio is the nmber of shares that is exchanged for a bond Conversion vale = crrent share price x conversion ratio Conversion premim = (convertible price conversion vale) / conversion vale Income pickp is the amont by which the yield to matrity of the convertible bond exceeds the dividend yield of the share

11 Convertible Bonds Examples Example 1: Crrent bond price = $930 Conversion ratio: 1 bond = 30 shares common Crrent stock price = $25/share Market Conversion Vale = (30 shares)x(25) = $750 Conversion Premim = ( ) / 750 = 180 / 750 = 24% Example 2: AXA Convertible Bond AXA has issed in the zone a convertible bond paying a 2.5% copon rate and matring on 01/01/2014; the conversion ratio is 4.04 On 12/13/2001, the crrent share price was and the bid-ask convertible price was / The conversion vale was eqal to = 4.04 x The conversion premim calclated with the ask price was 61.73% = ( )/ The conversion of the bond into 4.04 shares can be exected on any date before the matrity date

12 Convertible Bonds Bloomberg Description

13 Convertible Bonds Uses For the isser Issing convertible bonds enables a firm to obtain better financial conditions Copon rate of sch a bond is always lower to that of a bllet bond with the same characteristics in terms of matrity and copon freqency This comes directly from the conversion advantage which is attached to this prodct Besides the exchange of bonds for shares diminishes the liabilities of the firm isser and increases in the same time its eqity so that its debt capacity is improved For the convertible bondholder The convertible bond is a defensive secrity, 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 price decreases, price of convertible never gets below the bond floor, i.e., the price of an otherwise identical bllet bond with no conversion option

14 Convertible Bonds Determinants of Convertible Bond Prices Convertible bond is similar to a normal copon bond pls a call option on the nderlying stock With an important difference: the effective strike price of the call option will vary with the price of the bond Convertible secrities are priced as a fnction of The price of the nderlying stock Expected ftre volatility of eqity retrns Risk free interest rates Call provisions Spply and demand for specific isses Isse-specific corporate/treasry yield spread Expected volatility of interest rates and spreads Ths, there is large room for relative mis-valations

15 Convertible Bonds Convertible Bond Price as a Fnction of Stock Price Bond Price Convertible Bond Parity Straight Bond Stock Price

16 Convertible Bonds Convertible Bond Pricing Model A poplar method for pricing convertible bonds is the component model The convertible bond is divided into a straight bond component and a call option on the conversion price, with strike price eqal to the vale of the straight bond component The fair vale of the two components can be calclated with standard formlas, sch as the famos Black-Scholes valation formla. This pricing approach, however, has several drawbacks First, separating the convertible into a bond component and an option component relies on restrictive assmptions, sch as the absence of embedded options (callability and ptability, for instance, are convertible bond featres that cannot be considered in the above separation) Second, convertible bonds contain an option component with a stochastic strike price eqal to the bond price

17 Convertible Bonds Convertible Bond Pricing Models Theoretical research on convertible bond pricing was initiated by Ingersoll (1977a) and Brennan and Schwartz (1977), who both applied the contingent claims approach to the valation of convertible bonds. In their valation models, the convertible bond price depends on the firm vale as the nderlying variable. Brennan and Schwartz (1980) extend their model by inclding stochastic interest rates. These models rely heavily on the theory of stochastic processes and reqire a relatively high level of mathematical sophistication

18 Convertible Bonds Binomial Model The price of the stock only can go p to a given vale or down to a given vale S Besides, there is a bond (bank accont) that will pay interest of r S ds

19 We assme (p) > d (down) Convertible Bonds Binomial Model For Black and Scholes we will need d = 1/ For consistency we also need > (1+r) > d Example: = 1.25; d = 0.80; r = 10% S=100 S = 125 S = 80

20 Basic model that describes a simple world. As the nmber of steps increases, it becomes more realistic We will price and hedge an option: it applies to any other derivative secrity Key: we have the same nmber of states and secrities (complete markets) Basis for arbitrage pricing Convertible Bonds Binomial Model

21 Introdce an Eropean call option: K = 110 It matres at the end of the period Convertible Bonds Binomial Model S S = 125 C (K=110) C = 15 S=100 ds = 80 C d = 0

22 Convertible Bonds Binomial Model We can replicate the option with the stock and the bond Constrct a portfolio that pays C in state and C d in state d The price of that portfolio has to be the same as the price of the option Otherwise there will be an arbitrage opportnity

23 We by Δ shares and invest B in the bank They can be positive (by or deposit) or negative (shortsell or borrow) We want then, With soltion, ΔS ΔdS + + Convertible Bonds Binomial Model B(1 + r) B(1 + r) = = C C d Δ = C Cd Cd d C ; B = S( d) ( d)(1 + r)

24 In or example, we get for stock: Convertible Bonds Binomial Model Δ = C S( And, for bonds: B = C ( d C d d d) d)(1 + C 15 0 = = 100 ( ) r) = ( ) (1.1) 1 3 = The cost of the portfolio is, ΔS + B = = 9.09

25 The price of the Eropean call mst be Otherwise, there is an arbitrage opportnity. If the price is lower than 9.09 we wold by the call and shortsell the portfolio If higher, the opposite We have compted the price and the hedge simltaneosly: We can constrct a call by bying the stock and borrowing Short call: the opposite Convertible Bonds Binomial Model

26 Remember that ) )(1 ( ; ) ( r d C d C B d S C C d d + = Δ = And B S C + = Δ Sbstitting, ) )(1 ( ) ( r d C d C d C C C d d + + = Convertible Bonds Binomial Model

27 After some algebra, = d C d r C d d r r C ) ( ) (1 ) ( Observe the coefficients, ) ( ) (1, ) ( 1 d r d d r + + Positive Smaller than one Add p to one Like a probability. Convertible Bonds Binomial Model

28 Rewrite Where 1 C = ) 1+ r p = 1+ ( r d d) [ p C + (1 p ],1 This wold be the pricing of: A risk netral investor With sbjective probabilities p and (1-p) Convertible Bonds Binomial Model C d p = (1 + r) ( d)

29 Convertible Bonds Binomial Model Sppose the following economy, S 2 S S ds ds d 2 S We introdce an Eropean call with strike price K that matres in the second period

30 Convertible Bonds Binomial Model The price of the option will be: C = + (1 1 (1 + r) p) p (1 [ p 2 max(0, d max(0, 2 S K) p) max(0, ds 2 S K) K)] There are two paths that lead to the intermediate state (that explains the 2 ) Sppose we know the volatility σ and the time to matrity t, we can retrieve and d (see B&S) = e σ t / n ; d = 1/

31 Convertible Bond Valation Methodology Given that a convertible bond is nothing bt an option on the nderlying stock, we expect to be able to se the binomial model to price it At each node, we test a. whether conversion is optimal b. whether the position of the isser can be improved by calling the bonds It is a dynamic procedre: max(min(q1,q2),q3)), where Q1 = vale given by the rollback (neither converted nor called back) Q2 = call price Q3 = vale of stocks if conversion takes place

32 Example Convertible Bond Example We assme that the nderlying stock price trades at $50.00 with a 30% annal volatility We consider a convertible bond with a 9 months matrity, a conversion ratio of 20 The convertible bond has a $1, face vale, a 4% annal copon We frther assme that the risk-free rate is a (continosly componded) 10%, while the yield to matrity on straight bonds issed by the same company is a (continosly componded) 15% We also assme that the call price is $1, Use a 3 periods binomial model (t/n=3 months, or ¼ year)

33 Convertible Bond Example We have d p = = = e.3 1/ 4 = = r d d Actally (continosly componded rate) p = 10% exp =.547

34 Convertible Bond Example Bond is Called $78.42 G 10.00% looks like a stock: se risk-free rate $1, conversion: 78.42>1040/20=52 D $ % calling or converting does not change the bond vale becase it is already essentially eqity $1, $58.09 $58.09 B 11.03% H 10.00% looks like a stock: se risk-free rate $1, $1, conversion: 58.09>1040/20=52 $50.00 $50.00 A 12.15% E 12.27% bond shold not be converted becase 1,073.18>50*20=1,000 $1, $1, $43.04 $43.04 C 13.51% I 15.00% looks like a risky bond: se risky rate $1, $1, no conversion: 43.04<1040/20=52 $37.04 F 15.00% bond shold not be converted becase 1,001.72>50*20=1,000 $1, $31.88 J 15.00% looks like a risky bond: se risky rate $1, conversion: 31.88<1040/20=52

35 Convertible Bond Example At node G, the bondholder optimally choose to convert since what is obtained nder conversion ($1,568.31), is higher than the payoff nder the assmption of no conversion ($1,040.00) The same applies to node H On the other hand, at nodes I and J, the vale nder the assmption of conversion is lower than if the bond is not converted to eqity Therefore, bondholders optimally choose not to convert, and the payoff is simply the nominal vale of the bond, pls the interest payments, that is $1,040.00

36 Convertible Bond Example Working or way backward the tree, we obtain at node D the vale of the convertible bond as the disconted expected vale, sing risk-netral probabilities of the payoffs at nodes G and H $1, = e 3-10% 12 ( p 1, ( 1 p) 1,161.83) At node F, the same principle applies, except that it can be regarded as a standard bond We therefore se the rate of retrn on a non convertible bond issed by the same company, 15% $1, = e 3-15% 12 ( p 1,040 + ( 1 p) 1,040)

37 Convertible Bond Example At node E, the sitation is more interesting becase the convertible bond will end p as a stock in case of an p move (conversion), and as a bond in case of a down move (no conversion) As an approximate rle of thmb, one may se a weighted average of the riskfree and risky interest rate in the comptation, where the weighting is performed according to the (risk-netral) probability of an p verss a down move px10% + (1-p)x15% = 12.27% Then the vale is compted as $1, = e % ( p 1, ( 1 p) 1,040 )

38 Convertible Bond Example Note that at nodes D, E and F, calling or converting is not relevant becase it does not change the bond vale since the bond is already essentially eqity At node B, it can be shown that the isser finds it optimal to call the bond If the bond is indeed called by the isser, bondholders are left with the choice between not converting and getting the call price ($1,100), or converting and getting $20x58.09=1,161.8$, which is what they optimally choose This is less than $1,191.13, the vale of the convertible bond if it were not called, and this is precisely why it is called by the isser Eventally, the vale at node A, i.e., the present fair vale of the convertible bond, is compted as $1,115.41

39 Convertible Bond Allowing for Stochastic Interest Rates Common Stock Price Tree $9 $10 $12 $8 $11 $14 Interest Rate Tree 4.5% 4.0% 3.6% 5.0% 4.0% 3.2%

40 Convertible Bonds Convertible Arbitrage Convertible arbitrage strategies attempt to exploit anomalies in prices of corporate secrities that are convertible into common stocks Roghly speaking, if the isser does well, the convertible bond behaves like a stock, if the isser does poorly, the convertible bond behaves like distressed debt Convertible bonds tends to be nder-priced becase of market segmentation: investors discont secrities that are likely to change types

41 Convertible Bonds Convertible Arbitrage Convertible arbitrage hedge fnd managers typically by (or sometimes sell) these secrities and then hedge part or all of the associated risks by shorting the stock Take for example Internet company AOL's zero copon converts de Dec. 6, 2019 These bonds are convertible into shares of AOL stock With AOL common stock trading at $34.80 on Dec. 31, 2000, the conversion vale was $203 (= x 34.80) As the conversion vale is significantly below the investment vale (calclated at $450.20), the investment vale dominated and the convertible traded at $ When, or if, the stock trades above $77.15, the conversion vale will dominate the pricing of the convertible becase it will be in excess of the investment vale

42 Convertible Bonds Mechanism In a typical convertible bond arbitrage position, the hedge fnd is not only long the convertible bond position, bt also short an appropriate amont of the nderlying common stock The nmber of shares shorted by the hedge fnd manager is designed to match or offset the sensitivity of the convertible bond to common stock price changes As the stock price decreases, the amont lost on the long convertible position is contered by the amont gained on the short stock position, theoretically creating a stable net position vale As the stock price increases, the amont gained on the long convertible position is contered by the amont lost on the short stock position, theoretically creating a stable net position vale This is known as delta hedging

43 Convertible Bonds Mechanism Convertible Bond Price Convertible Bond Delta = Parity = Change in Price of Conv Bond Change in Price of Stock Stock Price Conversion Ratio Stock Price

44 Convertible Bonds Mechanism In the AOL example, the delta for the convertible is approximately 50% This means that for every $1 change in the conversion vale, the convertible bond price changes by 50 cents To delta hedge the eqity exposre in this bond we need to short half the nmber of shares that the bond converts into, for example 2.9 shares (5.8338\2) The combined long convertible bond/short stock position shold be relatively insensitive to small changes in the price of AOL's stock Over-hedging is sometimes appropriate when there is concern abot defalt, as the excess short position may partially hedge against a redction in credit qality

45 Convertible Bonds Risks Involved Becase a convertible bond is essentially a bond pls an option to switch so that these strategies will typically make money if expected volatility increases (long vega) make money if the stock price increases rapidly (long gamma) pay time-decay (short theta) make money if the credit qality of the isser improves (short the credit differential) The risks involved relate to changes in the price of the nderlying stock (eqity market risk) changes in the interest rate level (fixed income market risk) changes in the expected volatility of the stock (volatility risk) changes in the credit standing of the isser (credit risk) The convertible bond market as a whole is also prone to liqidity risk as demand can dry p periodically, and bid/ask spreads on bonds can widen significantly There is also the risk that the HF manager will be nable to sstain the short position in the nderlying common shares In addition, convertible arbitrage hedge fnds se varying degrees of leverage, which can magnify both risks and retrns

46 Options on Bonds Terminology An option is a contract in which the seller (writer) grants the byer the right to prchase from, or sell to, the seller an nderlying asset (here a bond) at a specified price within a specified period of time The seller grants this right to the byer in exchange for a certain sm of money called the option price or option premim The price at which the instrment may be boght or sold is called the exercise or strike price The date after which an option is void is called the expiration date An American option may be exercised any time p to and inclding the expiration date A Eropean option may be exercised only on the expiration date

47 Options on Bonds Factors that Inflence Option Prices Crrent price of nderlying secrity As the price of the nderlying bond increases, the vale of a call option rises and the vale of a pt option falls Strike price Call (pt) options become more (less) valable as the exercise price decreases Time to expiration For American options, the longer the time to expiration, the higher the option price becase all exercise opportnities 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 pt option on bond decreases as short-term interest rate rises (throgh impact on bond price) 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 increase

48 Options on long-term bonds Interest payments are similar to dividends. Otherwise, long-term bonds are like options on stock: Options on Bonds Pricing We can se Black-Scholes as in options on dividend-paying eqity Options on short-term bonds They do not pay dividends Problem: they are not like a stock becase they qickly converge to par We cannot directly apply Black-Scholes Other shortcomings of standard option pricing models Assmption of a constant short-term rate is inappropriate for bond options Assmption of a constant volatility is also inappropriate: as a bond moves closer to matrity, its price volatility decline

49 Options on Bonds Pricing A soltion to avoid the problem is to consider an interest rate model, as described in Chapter 12 The following figre shows a tree for the 1-year rate of interest (calibrated to the crrent TS) The figre also shows the vales for a discont bond (par = 100) at each node in the tree Interest rates 6% Bond prices % 5.5% % 6% 5% % 6.5% 5.5% 4.5%

50 Options on Bonds Pricing Consider a 2-year Eropean call on this 3-year bond strck at 93.5 Start by compting the vale at the end of the tree If by the end of the 2nd year 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 1.5 Working or way backward the tree C C C l 1 = % 1 = % 1 = 1+ 6% ( ) ( ) =.2347 =.9479 (.5 C +.5 C ) d =

51 Options on Bonds Pt-Call Parity Assmption no copon payments and no prematre exercise Consider a portfolio where we prchase one zero copon bond, one pt Eropean option, and sell (write) one Eropean call option (same time to matrity T and the same strike price X) Payoff at date T B T < X: B T X: Yo hold the bond: B T The call option is worthless: 0 The pt option is worth: X - B T Ths, yor net position is: X Yo hold the bond: B T The call option is worth: -(B T - X) The pt option is worthless: 0 Ths, yor net position is: X

52 Options on Bonds Pt-Call Parity Con t No matter what state of the world obtains at the expiration date, the portfolio will be worth X Ths, the payoff from the portfolio is risk-free, and we can discont its vale at the risk-free rate r We obtain the call-pt relationship B 0 rt + C0 P0 = Xe P0 = B0 + C0 For copon bonds Xe P0 = B0 + C0 Xe rt PV ( Copons) rt