earlier dates. This section, therefore, focuses on the pricing of American and Bermudan options. The following example prices an option to call 100 face of a 1.5-year, 5.25% coupon bond at par on any coupon date. Assume that the risk- neutral interest rate process over six-month periods is as in the example of Chapter 9: With this tree and the techniques of Part Three, the price tree for a 5.25% coupon bond maturing in 1.5 years may be computed to be Note that all the prices in the tree are ex-coupon prices. So, for example, on date 2, state 2, the bond is worth 100.122 after the coupon payment of 2.625 has been made. The value of the option to call this bond at par is worthless on the ma- turity date of the bond since the bond is always worth par at maturity. On any date before maturity the option has two sources of value. First, it can be exercised immediately. If the price of the bond is P and the strike price is K, then the value of immediate exercise, denoted V E , is 100 99 636 99 690 100 100 006 100 122 100 655 100 100 613 100 6489 8024 3511 1976 6489 3511 . . . . . . . . ← ← ← ← ← ← ← ← ← ← ← ← 6.00% 5.00% 5.00% 5.00% 4.00% 4.00% 6489 8024 3511 1976 6489 3511 . . . . ← ← ← ← ← ← Pricing American and Bermudan Bond Options in a Term Structure Model 401 (19.5) Second, the option can be held to the next date. The value of the option in this case is like the value of any security held over a date, namely the ex- pected discounted value in the risk-neutral tree. Denote this value by V H . The option owner maximizes the value of the option by choosing on each possible exercise date whether to exercise or to hold the option. If the value of exercising is greater the best choice is to exercise, while if the value of holding is greater the best choice is to hold. Mathematically, the value of the option, V, is given by (19.6) For more intuition about the early exercise decision, consider the fol- lowing two strategies. Strategy 1 is to exercise the option and hold the bond over the next period. Strategy 2 is not to exercise and, if conditions warrant next period, to exercise then. The advantage of strategy 1 is that purchasing the bond entitles the owner to the coupon earned over the pe- riod. The advantages of strategy 2 are that the strike price does not have to be paid for another period and that the option owner has another pe- riod in which to observe market prices and decide whether to pay the strike price for the bond. With respect to the advantage of waiting to de- cide, if prices fall precipitously over the period then strategy 2 is superior to strategy 1 since it would have been better not to exercise. And if prices rise precipitously then strategy 2 is just as good as strategy 1 since the op- tion can still be exercised and the bonds bought for the same strike price of 100. To summarize, early exercise of the call option is optimal only if the value of collecting the coupon exceeds the combined values of delay- ing payment and of delaying the decision to purchase the bond at the fixed strike price. 1 Returning to the numerical example, the value of immediately exercis- ing the option on date 2 is .613, .122, and 0 in states 0, 1, and 2, respec- tively. Furthermore, since the option is worthless on date 3, the value of the option on date 2 is just the value of immediate exercise. VVV EH = () max , VPK E =− () max ,0 402 FIXED INCOME OPTIONS 1 In the stock option context, the equivalent result is that early exercise of a call op- tion is not optimal unless the dividend is large enough. On date 1, state 0, the value of immediate exercise is .655. The value of holding the option is (19.7) Therefore, on date 1, state 0, the owner of the option will choose to exer- cise and the value of the option is .655. Here, it is worth more to exercise the option on date 1 and earn a coupon rate of 5.25% in a 4.50% short- term rate environment than to hold on to the option. On date 1, state 1, the bond sells for less than par so the value of im- mediate exercise equals zero. The value of holding the option is (19.8) Hence the owner will hold the option, and its value is .042. Finally, on date 0, the value of exercising the option immediately is .006. The value of holding the option is (19.9) The owner of the option will not exercise, and the value of the option on date 0 is .159. In this situation, earning a coupon of 5.25% in a 5% short- term rate environment is not sufficient compensation for giving up an op- tion that could be worth as much as .655 on date 1. The following tree for the value of the option collects these results. States in which the option is exercised are indicated by option values in boldface. 0 0 000 0 042 0 0 159 0 0 . . . 0.122 0.655 0.613 ← ← ← ← ← ← ← ← ← ← ← ← . . 8024 042 1976 655 1052 159 ×+ × + = . . 6489 0 3511 122 1 055 2 042 ×+ × + = . . 6489 122 3511 613 1 045 2 288 ×+ × + = Pricing American and Bermudan Bond Options in a Term Structure Model 403 Put options are priced analogously. The only change is that the value of immediately exercising a put option struck at K when the bond price is P equals (19.10) rather than (19.5). The advantage of early exercise for a put option (i.e., the right to sell) is that the strike price is received earlier. The disadvantages of exercising a put early are giving up the coupon and not being able to wait another period before deciding whether to sell the bond at the fixed strike price. Before concluding this section it should be noted that the selection of time steps takes on added importance for the pricing of American and Bermudan options. The concern when pricing any security is that a time step larger than an instant is only an approximation to the more nearly continuous process of international markets. The additional concern when pricing Bermudan or American options is that a tree may not allow for sufficiently frequent exercise decisions. Consider, for example, using a tree with annual time steps to price a Bermudan option that permits exer- cise every six months. By omitting possible exercise dates the tree does not permit an option holder to make certain decisions to maximize the value of the option. Furthermore, since on these omitted exercise dates an option holder would never make a decision that lowers the value of the option, omitting these exercise dates necessarily undervalues the Bermu- dan option. In the case of a Bermudan option the step size problem can be fixed either by reducing the step size so that every Bermudan exercise date is on the tree or by augmenting an existing tree with the Bermudan exercise dates. In the case of an American option it is impossible to add enough dates to reflect the value of the option fully. While detailed numerical analysis is beyond the scope of this book, two responses to this problem may be mentioned. First, experiment with different step sizes to determine which are accurate enough for the purpose at hand. Second, calculate option values for smaller and smaller step sizes and then extrapolate to the option value in the case of contin- uous exercise. VKP E =− () max ,0 404 FIXED INCOME OPTIONS APPLICATION: FNMA 6.25s of July 19, 2011, and the Pricing of Callable Bonds The Federal National Mortgage Association (FNMA) recently reintroduced its Callable Benchmark Program under which it regularly sells callable bonds to the public. On July 19, 2001, for example, FNMA sold an issue with a coupon of 6.25%, a maturity date of July 19, 2011, and a call feature allowing FNMA to purchase these bonds on July 19, 2004, at par. This call feature is called an embedded call because the option is part of the bond’s struc- ture and does not trade separately from the bond. In any case, until July 19, 2004, the bond pays coupons at a rate of 6.25%. On July 19, 2004, FNMA must decide whether or not to exercise its call. If FNMA does exercise, it pays par to repurchase all of the bonds. If FNMA does not exercise, the bond continues to earn 6.25% until maturity at which time principal is returned. This structure is sometimes referred to as “10NC3,” pronounced “10-non-call- three,” because it is a 10-year bond that is not callable for three years. These three years are referred to as the period of call protection. The call feature of the FNMA 6.25s of July 19, 2011, is a particularly simple example of an embedded option. First, FNMA’s option is European; it may call the bonds only on July 19, 2004. Other callable bonds give the issuer a Bermudan or American call after the period of call protection. For example, a Bermudan version might allow FNMA to call the bonds on any coupon date on or after the first call date of July 19, 2004, while an American version would allow FNMA to call the bonds at any time after July 19, 2004. The second reason the call feature of the FNMA issue is particularly simple is that the strike price is par. Other callable bonds require the issuer to pay a premium above par (e.g., 102 percent of par). In the Bermudan or American cases there might be a schedule of call prices. An old rule of thumb in the corporate bond market was to set the premium on the first call date equal to half the coupon rate. After the first call date the premium was set to decline linearly to par over some number of years and then to remain at par until the bond’s maturity. The pricing technique of the previous section is easily adapted to a schedule of call prices. The rest of this section and the next discuss the price behavior of callable bonds in detail. The basic idea, however, is as follows. If interest rates rise after an issuer sells a bond, the issuer wins in the sense that it is borrowing money at a relatively low rate of in- terest. Conversely, if rates fall after the sale then bondholders win in the sense that they are investing at a relatively high rate of interest. The embedded option, by allowing the is- suer to purchase the bonds at some fixed price, caps the amount by which investors can profit from a rate decline. In fact, an embedded call at par cancels any price appreciation as of the call date although investors do collect an above-market coupon rate before the call. In exchange for giving up some or all of the price appreciation from a rate decline, bondholders receive a higher coupon rate from a callable bond than from an otherwise identical noncallable bond. APPLICATION: FNMA 6.25s of July 19, 2011, and the Pricing of Callable Bonds 405 To understand the pricing of the callable bond issue, assume that there exists an oth- erwise identical noncallable bond—a noncallable bond issued by FNMA with a coupon rate of 6.25% and a maturity date of July 19, 2011. Also assume that there exists a separately traded European call option to buy this noncallable bond at par. Finally, let P C denote the price of the callable bond, let P NC denote the price of the otherwise identical noncallable bond, and let C denote the price of the European call on the noncallable bond. Then, (19.11) Equation (19.11) may be proved by arbitrage arguments as follows. Assume that P C <P NC –C. Then an arbitrageur would execute the following trades: Buy the callable bond for P C . Buy the European call option for C. Sell the noncallable bond for P NC . The cash flow from these trades is P NC –C–P C , which, by assumption, is positive. If rates are lower on July 19, 2004, and FNMA exercises the embedded option to buy its bonds at par, then the arbitrageur can unwind the trade without additional profit or loss as follows: Sell the callable bond to FNMA for 100. Exercise the European call option to purchase the noncallable bond for 100. Deliver the purchased noncallable bond to cover the short position. Alternatively, if rates are higher on July 19, 2004, and FNMA decides not to exercise its op- tion, the arbitrageur can unwind the trade without additional profit or loss as follows: Allow the European call option to expire unexercised. Deliver the once callable bond to cover the short position in the noncallable bond. Note that the arbitrageur can deliver the callable bond to cover the short in the noncallable bond because on July 19, 2004, FNMA’s embedded option expires. That once callable bond becomes equivalent to the otherwise identical noncallable bond. The preceding argument shows that the assumption P C <P NC –C leads to an initial cash flow without any subsequent losses, that is, to an arbitrage opportunity. The same argu- ment in reverse shows that P C >P NC –C also leads to an arbitrage opportunity. Hence the equality in (19.11) must hold. The intuition behind equation (19.11) is that the callable bond is equivalent to an oth- PP C CNC =− 406 FIXED INCOME OPTIONS erwise identical noncallable bond minus the value of the embedded option. The value of the option is subtracted from the noncallable bond price because the issuer has the option. Equivalently, the value of the option is subtracted because the bondholder has sold the em- bedded option to the issuer. Along the lines of the previous section, a term structure model may be used to price the European option on the otherwise identical noncallable bond. After that, equation (19.11) may be used to obtain a value for the callable bond. While the discussion to this point assumes that the embedded option is European, equation (19.11) applies to other op- tion styles as well. If the option embedded in the FNMA 6.25s of July 19, 2011, were Bermudan or American, then a term structure model would be used to calculate the value of that Bermudan or American option on a hypothetical noncallable FNMA bond with a coupon of 6.25% and a maturity date of July 19, 2011. Then this Bermudan or American option value would be subtracted from the value of the noncallable bond to obtain the value of the callable bond. Combining equation (19.11) with the optimal exercise rules described in the previous section reveals the following about the price of the callable bond. First, if the issuer calls the bond then the price of the callable bond equals the strike price. Second, if the issuer chooses not to call the bond (when it may do so) then the callable bond price is less than the strike price. To prove the first of these statements, note that if it is optimal to exercise, then, by equation (19.6), the value of the call option must equal the value of immediate ex- ercise. Furthermore, by equation (19.5), the value of immediate exercise equals the price of the noncallable bond minus the strike. Putting these facts together, (19.12) But substituting (19.12) into (19.11), (19.13) To prove the second statement, note that if it is not optimal to exercise, then, by equation (19.6), the value of the option is greater than the value of immediate exercise given by equation (19.5). Hence (19.14) Then, substituting (19.14) into (19.11), (19.15) By market convention, issuers pay for embedded call options through a higher coupon rate rather than by selling callable bonds at a discount from par. On July 19, 2001, PP CP P KK CNC NC NC =−<− − ( ) = CP K NC >− PP CP P KK CNC NC NC =−=− − ( ) = CP K NC =− APPLICATION: FNMA 6.25s of July 19, 2011, and the Pricing of Callable Bonds 407 for example, when FNMA sold its 6.25s of July 19, 2011, for approximately par, the yield on 10-year FNMA bonds was approximately 5.85%. FNMA could have sold a callable bond with a coupon of 5.85%. In that case the otherwise identical noncallable bond would be worth about par, and the callable bond, by equation (19.11), would sell at a discount from par. Instead, FNMA chose to sell a callable bond with a coupon of 6.25%. The otherwise identical noncallable bond was worth more than par but the embedded call option, through equation (19.11), reduced the price of the callable bond to approximately par. GRAPHICAL ANALYSIS OF CALLABLE BOND PRICING This section graphically explores the qualitative behavior of callable bond prices using the FNMA 6.25s of July 19, 2011, for settle on July 19, 2001, as an example. Begin by defining two reference bonds. The first is the otherwise identical noncallable bond referred to in the previous section—an imaginary 10-year noncallable FNMA bond with a coupon of 6.25% and a maturity of July 19, 2011. The second reference bond is an imaginary three-year noncallable FNMA bond with a coupon of 6.25% and a maturity of July 19, 2004. 2 Assuming a flat yield curve on July 19, 2001, the dashed line and thin solid line in Figure 19.4 graph the prices of these reference bonds at different yield levels. When rates are particularly low the 10-year bond is worth more than the three-year bond because the former earns an above-market rate for a longer period of time. Conversely, when rates are particularly high the three-year bond is worth more because it earns a below-market rate for a shorter period of time. Also, the 10-year bond’s price-yield curve is the steeper of the two because its DV01 is greater. The thick solid line in the figure graphs the price of the callable bond using a particular pricing model. While the shape and placement of this curve depends on the model and its parameters, the qualitative re- sults described in the rest of this section apply to any model and any set of parameters. 408 FIXED INCOME OPTIONS 2 If the call price of the FNMA 6.25s of July 19, 2011, were 102 instead of 100, the three-year reference bond would change. For the analysis of this section to apply, this reference bond would pay 3.125 every six months, like the callable bond, but would pay 102 instead of 100 at maturity. While admittedly an odd structure, this reference bond can be priced easily. The four qualitative features of Figure 19.4 may be summarized as follows. 1. The price of the callable bond is always below the price of the three- year bond. 2. The price of the callable bond is always below the price of the 10-year bond. 3. As rates increase, the price of the callable bond approaches the price of the 10-year bond. 4. As rates decrease, the price of the callable bond approaches the price of the three-year bond. The intuition behind statement 1 is as follows. From July 19, 2001, to July 19, 2004, the callable bond and the three-year bond make exactly the same coupon payments. However, on July 19, 2004, the three-year bond will be worth par while the callable bond will be worth par or less: By the results at the end of the previous section, the callable bond will be worth par if FNMA calls the bond but less than par otherwise. But if the cash flows from the bonds are the same until July 19, 2004, and then the three- year bond is worth as much as or more than the callable bond, then, by ar- bitrage, the three-year bond must be worth more as of July 19, 2001. Statement 2 follows immediately from the fact that the price of an Graphical Analysis of Callable Bond Pricing 409 FIGURE 19.4 Price-Rate Curves for the Callable Bond and the Two Noncallable Reference Bonds 80 85 90 95 100 105 110 115 120 125 130 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% Rate Price 10-year Noncallable 3-year Noncallable Callable option is always positive. Since C>0, by (19.11) P C <P NC . In fact, rear- ranging (19.11), C=P NC –P C . Hence the value of the call option is given graphically by the distance between the price of the 10-year bond and the price of the callable bond in Figure 19.4. Statement 3 is explained by noting that, when rates are high and bond prices low, the option to call the bond at par is worth very little. More loosely, when rates are high the likelihood of the bond being called on July 19, 2004, is quite low. But, this being the case, the prices of the callable bond and the 10-year bond will be close. Finally, statement 4 follows from the observation that, when rates are low and bond prices high, the option to call the bond at par is very valu- able. The probability that the bond will be called on July 19, 2004, is high. This being the case, the prices of the callable bond and the three-year bond will be close. Figure 19.4 also shows that an embedded call option induces negative convexity. For the callable bond price curve to resemble the three-year curve at low rates and the 10-year curve at high rates, the callable bond curve must be negatively convex. Figure 19.5 illustrates the negative convexity of callable bonds more dramatically by graphing the duration of the two reference bonds and that of the callable FNMA bonds. The duration of the 10-year bond is, as ex- pected, greater than that of the three-year bond. Furthermore, the 10-year 410 FIXED INCOME OPTIONS FIGURE 19.5 Duration-Rate Curves for the Callable Bond and the Two Noncallable Reference Bonds 2.5 3.5 4.5 5.5 6.5 7.5 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% Rate Duration 10-year Noncallable 3-year Noncallable Callable [...]... of Delivery Price/ Factor 4.75% 5.50% 6.00% 6.50% 5.75% 5.00% 5.00% 11/15/08 05/15/ 09 08/15/ 09 02/15/10 08/15/10 02/15/11 08/15/11 98 . 598 102 .96 2 106.107 1 09. 671 105.081 99 .99 5 99 .99 5 0 .93 35 0 .97 18 0 .99 99 1.0305 0 .98 38 0 .93 26 0 .92 97 0.000 0.3 19 0. 496 0.828 1.170 1. 492 1. 798 105.6215 105 .95 01 106.1174 106.42 49 106.81 09 107.2213 107.5558 ( ) FT = ( ) P CTD T cf CTD (20.3) where T denotes the last delivery... the Notional Coupon Coupon Maturity Conversion Factor Approximation Error 4.75% 5.50% 6.00% 6.50% 5.75% 5.00% 5.00% 11/15/08 05/15/ 09 08/15/ 09 02/15/10 08/15/10 02/15/11 08/15/11 0 .93 35 0 .97 18 0 .99 99 1.0305 0 .98 38 0 .93 26 0 .92 97 0 .93 24 0 .97 13 0 .99 99 1.0310 0 .98 36 0 .93 18 0 .92 90 –0.0011 –0.0005 0.0000 0.0005 –0.0002 –0.0008 –0.0007 Imperfection of Conversion Factors and the Delivery Option at Expiration... example of Table 20.3, the CTD would still be the 4.75s of November 15, 2008, but the futures price would be 98 . 598 The 5s of February 15, 2011, and the 5s of August 15, 2011, would tie for the next to CTD at a price of 99 .99 5 The cost of delivering either of these would be 99 .99 5 98 . 598 or 1. 397 , much more than the 32-cent cost of delivering the next to CTD when the actual conversion factors are used... FUTURES TABLE 20.1 The Deliverable Basket into TYH2 Coupon Maturity Conversion Factor 4.75% 5.50% 6.00% 6.50% 5.75% 5.00% 5.00% 11/15/08 05/15/ 09 08/15/ 09 02/15/10 08/15/10 02/15/11 08/15/11 0 .93 35 0 .97 18 0 .99 99 1.0305 0 .98 38 0 .93 26 0 .92 97 the 9. 125s of May 15, 20 09: While this bond matures in a little less than 7.25 years from March 1, 2002, it was issued as a U.S Treasury bond rather than a U.S Treasury... option is 99 -181/4 corresponding to a yield of 5.063% At the bottom right of the screen, the repo rate is 1.58% which, given the bond price, gives a forward price of 99 -181/4 The option is, FIGURE 19. 7 Bloomberg’s Option Valuation Screen for Options on the 5s of February 15, 2001 Source: Copyright 2002 Bloomberg L.P 5 For more details, see Hull (2000), pp 533–537 418 FIXED INCOME OPTIONS therefore, an... strikes is not far from 9. 087% 11 10.5 Price Volatility (%) 10 9. 5 9 Normal Model 8.5 8 Lognormal Model 7.5 7 6.5 6 85 90 95 100 Strike 105 110 FIGURE 19. 8 Black’s Model Implied Volatility as a Function of Strike for a Normal and a Lognormal Short-Rate Model 115 422 FIXED INCOME OPTIONS Cumulative Probability 1.00 0.80 0.60 0.40 0.20 –5% 0.00 0% 5% 10% 15% Rate Normal Lognormal FIGURE 19. 9 Cumulative Normal... yield volatility, it follows from equation ( 19. 20) that σP = y fwd 10, 000 × DV 01 fwd σ y Pfwd ( 19. 21) In the example of Figure 19. 7, 9. 087% = 5.063% 10, 000 × 06875σ y 99 − 18 14 ( 19. 22) Note that all of these inputs are on the Bloomberg screen Since the strike is equal to the forward price, the yield corresponding to the strike is the forward yield Also, the forward DV01 is computed next to the symbol... calculations for TYH2 as of March 28, 2002, assuming that all bonds yield 5% and that the final settlement price is 105.6215 For example, the cost of delivering the 6s of August 15, 20 09, is 106.107 − 99 99 × 105.6215 = 496 (20.2) In the example of Table 20.3, the 4.75s of November 15, 2008, are the bonds with the lowest cost of delivery, which in this case is zero The next to CTD are the 5.5s of May 15, 20 09, ... expirations Table 19. 1 illustrates a subset of this range of offerings as of January, 2002 The rows 414 FIXED INCOME OPTIONS TABLE 19. 1 Swaption Volatility Grid, January 2002 Underlying Swap Maturity 1 year Swaption Maturity 1 month 3 months 6 months 1 year 2 years 5 years 10 years 2 Years 5 Years 10 Years 30 Years 47.6% 43.5% 42 .9% 34.0% 26.0% 21.4% 17.1% 40.3% 37.8% 35.0% 29. 0% 24.4% 20 .9% 16.8% 29. 1% 28.7%... percentage of the bond’s forward price The main part of the option valuation screen shows that at a percentage price volatility of 9. 087% put and call prices equal 2.521.6 This means, for example, that an option on $100,000,000 of the 5s of February 15, 2011, on July 15, 2002, at 99 -181/4 costs $100, 000, 000 × 2.521 = $2, 521, 000 100 ( 19. 19) The price volatility is labeled “Price I Vol” for “Price Implied . is 100 99 636 99 690 100 100 006 100 122 100 655 100 100 613 100 64 89 8024 3511 197 6 64 89 3511 . . . . . . . . ← ← ← ← ← ← ← ← ← ← ← ← 6.00% 5.00% 5.00% 5.00% 4.00% 4.00% 64 89 8024 3511 197 6. value of immediate exercise given by equation ( 19. 5). Hence ( 19. 14) Then, substituting ( 19. 14) into ( 19. 11), ( 19. 15) By market convention, issuers pay for embedded call options through a higher coupon. Pricing 4 09 FIGURE 19. 4 Price-Rate Curves for the Callable Bond and the Two Noncallable Reference Bonds 80 85 90 95 100 105 110 115 120 125 130 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9. 00% Rate Price 10-year