Pricing Callable Bonds

In the first step, we determine the interest rate path in which we create a risk-neutral recombining lattice with the evolution of the 6-month interest rate. Therefore, the nodes of the binomial tree are for each 6-month interval, and the probability of an upward and downward movement is equal. The analysis of the interest rate evolution has a great relevance in callable bond pricing. We assume that the interest rate follows the path shown in Figure 11.4. In this example, we assume for simplicity a 2-year interest rate. We suppose that the interest rate starts at time tg and can go up and down following the geometric random walk for each period. The interest rate rg at time tg changes due to two main variables ... 

In other words, if the callable bond price is 103.82, we set this price in the binomial tree shown in Figure 11.9 and through an iterative procedure, we find an option-adjusted spread that matches the price sought. In our case, the option-adjusted spread is around 630 bps over the risk-free yield curve (Figure 11.12). 

Determine the price of a callable bond with the current market yield curve ... 

For instance, consider a hypothetical case in which a callable bond has a price equal to 103.78. Suppose that the benchmark yield curve changes by 1 basis point. For a downward parallel shift, the price obtained is 103.83, while for an upward shift the price is 103.73. Therefore, applying Formula (11.1) the bond s effective duration is around 4.93. 

The option-adjusted spread (OAS) is the most important measure of risk for bonds with embedded options. It is the average spread required over the yield curve in order to take into account the embedded option element. This is, therefore, the difference between the yield of a bond with embedded option and a government benchmark bond. The spread incorporates the future views of interest rates and it can be determined with an iterative procedure in which the market price obtained by the pricing model is equal to expected cash flow payments (coupons and principal). Also a Monte Carlo simulation may be implemented in order to generate an interest rate path. Note that the option-adjusted spread is influenced by the parameters implemented into the valuation model as the yield curve, but above all by the volatility level assumed. This is referred to volatility dependent. The higher the volatility, the lower the option-adjusted spread for a callable bond and the higher for a putable bond. 

Consider the following example. We assume to have two hypothetical bonds, a treasury bond and a callable bond. Both bonds have the same maturity of 5 years and pay semiannual coupons, respectively, of 2.4% and 5.5%. We perform a valuation in which we assume a credit spread of 300 basis points and an OAS spread of 400 basis points above the yield curve. Table 11.1 illustrates the prices of a treasury bond, conventional bond and callable bond. In particular, considering only the credit spread we find the price of a conventional bond or option-free bond. Its price is 106.81. To pricing a callable bond, we add the OAS spread over the risk-free yield curve. The price of this last bond is 99.02. We can now see that the OAS spread underlines the embedded call option of the callable bond. It is equal to 106.81-99.02, or 7.79. In Section 11.2.3, we will explain the pricing of a callable bond with the OAS methodology adopting a binomial tree. 

In this section, we illustrate the pricing of bonds with embedded options. The price of a callable bond is essentially formed by an option-free bond and an embedded option. In fact, it is given by the difference between the value of an option-free bond and a call option as follows ... 

The value of a callable bond, and therefore of a call option, depends on the interest rate path. Thus, a callable bond has a lower price than the one of a conventional bond due to the embedded option. If the value of a call option increases, the value of a callable bond decreases and vice versa. This happens when interest rates are lower than the ones at issue. 

Several factors affect the decision if exercising the option or not. The first one is the asymmetric profit-loss profile. The potential gain of the option holder is unlimited when the price of the underlying asset rises, and losing only the initial investment if the price decreases. The second one is the time of value. In fact, in callable bonds, usually the price decreases as the bmid approaches maturity. This incentives the option holder to delay the exercise for a lower strike price. However, coupon payments with lower interest rates can favour the early exercise. 

Determine the Value of a Callable Bond Since the option is held by the issuer, the option element decreases the value of the bond. Therefore, the value of a callable bond is found as an option-free bond less the option element according to Formula (11.3). For the hypothetical bond, the price is 106.13-2.31 or 103.82. This is shown in Figure 11.11. The binomial tree shows that at maturity the option free and callable bond have the same price, or 100. Before the maturity, if the interest rates go down, the callable bond s values are less than an option-free bond, and in particular when the embedded option is deeply in the money, the callable values equal the strike price according to the caU schedule. Conversely, when the interest rates go up, the option free and callable bonds have the same price. 

FIGURE 11.11 TTie binomial price tree for a callable bond. 

To explain the pricing methodology, we suppose a putable bond with the same characteristics of the callable bond. The putable bond can be given back to the issuer with the following put schedule shown in Table 11.4. 

The pricing of the conventional bond is the same than the one exposed for callable bonds in Figure 11.9. Therefore, the option-free bond is always equal to 106.13. The main difference consists in the estimation of the embedded option (put option rather than call option) and pricing of the putable bond. Thus, we illustrate these two steps ... 

Determine the Value of an Embedded Put Option Conversely to a callable bond, the embedded option of a putable bond is a put option. Therefore, the value is estimated as the maximum between 0 and the difference between the strike price and bond s price. The strike price is defined according to the put schedule, while the bond s price is the value of the option-free bond at each node as shown in Figure 11.9. The value at maturity of a putable option if exercised is given by Formula (11.10) ... 

Determine the Value of a Putable Bond As exposed in Formula (11.4), the value of a putable bond is the sum of an option-free bond and an embedded put optimi. Therefore, conversely to a callable bond, the embedded option increases the value of the bond. When the option is deeply in the money, the bond matches the values defined in the put schedule. When the option has no value, option free and putable bonds have the same price. The value of our hypothetical putable bond is 106.13 + 0.33 or 106.45. This is illustrated in Figure 11.14. 

FIGURE 11.16 The binomial price tree of a callable bond with step-up feature. 

As noted, a bond may contain an embedded option which permits the issuer to call or retire all or part of the issue before the maturity date. The bondholder, in effect, is the writer of the call option. From the bondholder s perspective, there are three disadvantages of the embedded call option. First, relative to bond that is option-free, the call option introduces uncertainty into the cash flow pattern. Second, since the issuer is more likely to call the bond when interest rates have fallen, if the bond is called, then the bondholder must reinvest the proceeds received at the lower interest rates. Third, a callable bond s upside potential is reduced because the bond price will not rise above the price at which the issuer can call the bond. Collectively, these three disadvantages are referred to as call risk. MBS and ABS that are securitized by loans where the borrower has the option to prepay are exposed to similar risks. This is called prepayment risk, which is discussed in Chapter 11. 

For callable bonds, the market convention is to calculate a yield to call in addition to a yield to maturity. A callable bond may be called at more than one price and these prices are specified in a call price schedule. The yield to call assumes that the issuer will call the bond at some call date and the call price is then specified in the call schedule... 

To illustrate the various yield to call measures, consider a callable bond with a 5.75% coupon issued by DZ Bank. The Security Description screen from Bloomberg is presented in Exhibit 3.12. The bond matures on 10 April 2012 and is callable on coupon anniversary dates until maturity at a call price of 100. Exhibit 3.13 present the Yields to Call screen. Using a settlement date of 22 July 2003, the various yield to call measures are presented. 

EXHBir 4.16 Price/Yield Relationship for a Callable Bond and an Option-Free Bond... 

In the discussion below, we will refer to a bond that may be called or is prepayable as a callable bond. Exhibit 4.16 shows the price/yield relationship for an option-free bond and a callable bond. The convex curve given by a-a" is the price/yield relationship for an option-free bond. The unusual shaped curve denoted by a-b in the exhibit is the price/yield relationship for the callable bond. 

The reason for the price/yield relationship for a callable bond is as follows. When the prevailing market yield for comparable bonds is higher than the coupon rate on the callable bond, it is unlikely that the issuer will call the issue. For example, if the coupon rate on a bond is 7% and the prevailing market yield on comparable bonds is 12%, it is highly unlikely that the issuer will call a 7% coupon bond so that it can issue a 12% coupon bond. Since the bond is unlikely to be called, the callable bond will have a similar price/yield relationship as an otherwise comparable option-free bond. Consequently, the callable bond is going to be valued as if it is an option-free bond. However, since there is still... 

Let s look at the difference in the price volatility properties relative to an option-free bond given the price/yield relationship for a callable bond shown in Exhibit 4.16. Exhibit 4.17 blows up the portion of the... 

The price volatility characteristic of a callable bond is important to understand. The characteristic of a callable bond—that its price appreciation is less than its price decline when rates change by a large number of basis points—is referred to as negative convexity. But notice from Exhibit 4.16 that callable bonds do not exhibit this characteristic at every yield level. When yields are high (relative to the issue s coupon... 

Note that the value for convexity given by the expressions above will always be positive—that is, the approximate price change due to convexity is positive for both yield increases and decreases, except for certain bonds such as callable bonds. 

Because a callable bond has more than one possible redemption date, its future cash flows are not clearly defined. To calculate the yield to maturity for such a bond, it is necessary to assume a particular redemption date. The market convention is to use the earliest possible one if the bond is priced above par and the latest possible one if it is priced below par. Yield calculated in this way is sometimes referred to as yield to worst (the Bloomberg term). 

A number of option-pricing models exist. Market participants often use variations on these models that they developed themselves or that were developed by their firms. The best-known of the pricing models is probably the Black-Scholes, whose fundamental principle is that a synthetic option can be created and valued by taking a position in the underlying asset and borrowing or lending funds in the market at the risk-free rate of interest. Although Black-Scholes is the basis for many other option models and is still used widely in the market, it is not necessarily appropriate for some interest rate instruments. Fabozzi (1997), for instance, states that the Black-Scholes model s assumptions make it unsuitable for certain bond options. As a result a number of alternatives have been developed to analyze callable bonds. 

A callable bond is essentially a conventional bond plus a short position in a call option, which acts as a cap on the bonds price and so reduces its value. If the value of the call option were to increase because of a fall in interest rates, therefore, the value of the callable bond would decrease. This is expressed in (11.3). 

The difference between the price of the option-free bond and the callable bond at any time is the price of the embedded call option. The behavior of the option element depends on the terms of the callable issue. 

That is not to say that probabilities do not have an impact on the option price. Far from it. If there is a very high probability that rates will increase, as in the example, an option s value to an investor will fall. This is reflected in the market value of the option or callable bond. When probabilities change, the market price changes as well. 

In period 1 there are two possible levels for the interest rate at period 2 there are four possible levels. After N periods, there will be 2 possible values for the interest rate. Calculating the current price of a 10-year callable bond that pays semiannual coupons involves generating more than one million possible values for the last period s set of nodes. For a 20-year bond, the number jumps to one trillion. (Note that the binomial models actually used in analyses have much shorter periods than six months, increasing the number of nodes.)... 

The tools discussed so far in this chapter are the building blocks of a simple model for pricing callable bonds. To illustrate how this model works, consider a hypothetical bond maturing in three years, with a 6 percent semiannual coupon and the call schedule shown in FIGURE 11.8. 

After calculating the prices for the conventional element of the callable bond, the next step is to compute the value of the option element. On the bond s maturity date, the option is worthless, because its strike is 100, which is the price the bond is redeemed at in any case. The option needs to be valued, however, at all the other node points. 

A number of factors dictate whether an option is exercised or not. The first is the asymmetric profit-loss profile of option holders their potential gain is theoretically unlimited when the price of the underlyir asset rises, but they lose only their initial investment if the price falls. This asymmetry fevors runnir an option position. Another consideration favorir holdir is the fact that the options time value is lost if it is exercised early. In callable bonds, the call price often decreases as the bond approaches mamriry. This creates an incentive to delay exercise until a lower strike price is available. Coupon payments, on the other hand, may fevor earlier exercise, since, in the case of a normal, nonembedded option, this allows the holder to earn interest sooner. 

It is now possible to complete the price tree for the callable bond. Remember that the option in the case of a callable bond is held by the issuer. Its value, given by the tree in figure 11.11, must therefore be subtracted from the conventional bond price, given by the tree in figure 11.10, to obtain the callable bond value. For instance, the current price of the callable bond is 105.875 — 0.76, or 105.115. FIGURE 11.12 shows the tree that results from this process. A tree constructed in this way, which is programmable into a spreadsheet or as a front-end application, can be used to price either a callable or a putable bond. 

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