Option-free bonds

In this chapter we present a discussion on convertible bonds, which have become popular hybrid financial instruments. Convertible bonds are financial instmments that give the bondholders the right, without imposing an obligation, to convert the bond into underlying security, usually common stocks, under conditions illustrate in the indenture at the time of issue. The hybrid characteristic defines the traditional valuation approach as the sum of two components the option-free bond and an embedded option (call option). The option element makes the valuation not easy, above all in pricing term sheets with specific contract clauses as the inclusion of soft calls, put options and reset features. The chapter shows practical examples of valuation in which financial advisors and investment banks adopts in different contexts. 

Convertible bonds give a risk premium above the bond floor, that is the excess of the convertible bond price above the option-free bond price (Figure 9.3). 

As explained in the introduction, the value of a convertible bond is the sum of two main components, the option-free bond and a call option on underlying security. The value of the option-free bond, or bond floor, is determined as the sum of future payments (coupon and principal at maturity). Therefore, the bond component is influenced by three main parameters, that is the maturity, the coupon percentage on par value and the yield to maturity (discount rate). Differently, the value of a call option can be found mainly through two option pricing models, Black Scholes model and binomial tree model. 

Determining the Value of an Option-Free Bond The fair value of an option-free bond is the sum of the present values of aU its cash flows in terms of coupon payments and principal repayment. The bond value is given by Equation (9.5) ... 

Craisider a hypothetical situation. Assume that an option-free bond paying a semi-annual coupon 5.5% on par value, with a maturity of 5 years and discount rate of 8.04% (EUR 5-year swap rate of 1.04% plus credit spread of 700 basis points). Therefore, the valuation of a conventional bond is performed as follows (Figure 9.4). 

The process is crmtinued until time to in which the value of an option-free bond is equal to the one obtained in Figure 9.4, or 89.71. This is shown in Figure 9.5. 

Determining the Value of a Convertible Bond The value of a convertible bond is the sum of the option-free bond and the conversion option element, or 107.5. 

Value of a Convertible Bond = Value of an Option-Free Bond... 

Option free bond — — Con edible bond — -Parit. ... 

The reason to use implied volatility is that market anticipates mean reversion and uses the implied volatility to gauge the volatility of individual assets relative to the market. Implied volatility represents a market option about the underlying asset and therefore is forward looking. However, the estimate of implied volatility is conditioned by the choice of other inputs in particular, the credit spread applied in the option-free bond and the conversion premium of the tmderlying asset (Example 9.2). 

Value of conversion option Value of an option free bond Value of a convertible bond... 

Consider the example of Beni Stabili SpA. On 11 July 2014 the convertible of Beni Stabili quoted to 112.229. Analysing Figure 9.13 we see that the option-free bond value is 98.5, while the option value is 613.7. In practical terms, the option value can be seen as the difference between market price of convertible and option-free bond (112.2 — 98.5). However, the option value is found through the implied volatility that is equal to 20.6%. 

The risk-free rate affects both elements, option-free bond and embedded option. Conversely, the credit spread is applied to the risk-free rate in order to find the price of the option-free bond. If the credit spread is also included into the option pricing model, the option value rises. For instance, consider the scenario in which the risk-free rate is 1.04% and the option value is 0.46. If the risk-free rate is 7.04%, then the option value increases to 0.66. Figure 9.16 shows the effect of a different interest rate level. 

The bond valuation is given by the value of an option-free bond less than the value of the embedded put option (Equation 9.15) ... 

Interest rates and credit spread A greater level of interest rates decreases the value of the option-free bond or bond floor. Because the credit spread is applied only into the bond floor valuation, a greater credit quality decreases the credit spread and interest rate, and increases the value of the option-free bond. Conversely, higher is the interest rates and credit spread, lower is the value of an option-free 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 ... 

P callable bond P option free bond P call option (11 - 3)... 

The value of a putable bond is the sum of an option-free bond and an embedded put option. It is attractive for investors because it works as a floor. Thus, greater the value of the option, greater the value of the putable bond. It is given by Formula (11.4) ... 

In the next section, we will show that the bond s value is estimated by assessing the value of the option-free bond and the value of the embedded option using the binomial tree. The same factors that are implemented into B S formula are used for the binomial tree. They are ... 

Determine the Value of an Option-Free Bond After determining the evolution of the interest rates, we calculate the value of the option-free bond. In this case, we develop a binomial tree by ignoring the call feature in which at maturity the value of bond is 100. Although the final value could be equal to 104 (principal plus coupons), we consider at maturity the bond s ex coupon value. In fact, at year 5 the bond s price is 100. 

FIGURE 11.9 The binomial price tree for an option-free bond. 

Determine the Value of an Embedded Call Option After determining the value of an option-free bond, we calculate the value of the option element. On the maturity, the value of the option is 0 because the bond s ex coupon value is 100 and equal to the strike price. In other nodes, the option has value if the strike price is less than bond s price. The strike price for each node is shown in Table 11.3. Consider also that the value of the option decreases as the bond approaches maturity due to the decreasing probability to redeem the bond. Figure 11.10 shows the value of a call option. The holder of the option has substantially the choice to exercise the option or wait a further period. Therefore, the value of the option if exercised is given by (11.7) ... 

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. 

Consider also that the optirai element can be included in the interest rate. In Section 11.1.3, we explained the concept of option-adjusted spread. Given the bond s price, we can calculate the spread including the option element. This is performed with a binomial tree used for pricing an option-free bond. 

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) ... 

The first effect that we have is that the value of an option-free bond increases with rising coupon payments. The pricing of the conventional bond performed in Figure 11.9 is modified with the step-up coupon payment. The pricing is shown in Figure 11.15. 

As shown in Figure 11.15, the added step-up feature increases the value from 106.13 to 108.5. However, the inclusion of this feature affects also the values of the embedded call option and callable bond. In practice, increasing the value of the conventional bond at each node increases the value of the embedded option. The call option is now 4.5. As a conventional callable bond, the value of a call option is then subtracted to the one of an option-free bond. Figure 11.16 shows that the value of a step-up callable note is 104. 

Bonds with embedded call and put options comprise a relatively small percentage of the European bond market. Exhibit 1.6 shows the percentage of the market value of the Euro Corporate Index and Pan-Euro Corporate Index attributable to bullets (i.e., option-free bonds), callable and putable bonds from the late 1990s through 31 May 2003. Accordingly, our discussion of bonds with embedded options in the remainder of the book will be confined to structured products. 

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