Bonds call options, example

Example of a Zero-Coupon Bond Call Option with Ibslcok... 

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. 

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. 

As yields in the market decline, the concern is that the issuer will call the bond. The issuer won t necessarily exercise the call option as soon as the market yield drops below the coupon rate. Yet, the value of the embedded call option increases as yields approach the coupon rate from higher yield levels. For example, if the coupon rate on a bond is 7% and the market yield declines to 7.5%, the issuer will most likely not call the issue. However, market yields are at a level at which the investor is concerned that the issue may eventually be called if market yields decline further. Cast in terms of the value of the embedded call option, that option becomes more valuable to the issuer and therefore it reduces the price relative to an otherwise comparable option-free bond. In Exhibit 4.16, the value of the embedded call option at a given yield can be measured by the difference between the price of an option-free bond (the price shown on the curve a-a ) and the price on the curve a-b. Notice that at low yield levels (below y on the horizontal axis), the value of the embedded call option is high. 

Moneyness—Is the option worth exercising If so, it is said to be in-the-money (ITM). Our call option struck at 98 would be in-the-money if the underlying bond was trading above 98. If the bond were trading below 98, the call would instead be out-of-the-money (OTM). Finally, if the current price of the underlying asset was the same as the strike price, 98 in this example, the option would be at-the-money (ATM). Premium—The amount paid by the buyer of an option is called the premium. This is normally paid up-front. 

Intrinsic Value—This is the value that would be realised if the option were exercised right now at prevailing market prices, provided that exercise was worthwhile. For example, a call option struck at 98 would have an intrinsic value of 3 if the underlying bond were trading at 101. The put option at the same strike price would have an intrinsic value of 0, however, not -3, as it would not be worth exercising. 

Time Value—Option premiums normally exceed their intrinsic value. For example, the premium for a call option struck at 98 on a bond trading at 101 might be 4, not 3. The time value of an option is the... 

Taking the same example as that developed to demonstrate the Vasicek model earlier, we now price the 3-year European call option on a 10-year pure discount bond using the CIR model for the short interest rates. Recall that face value is 1 and exercise price K is equal to 0.5. As in the example with the Vasicek model, we consider that o = 2% and tq = 3.75%. The CIR model overcomes the problem of negative interest rates (acknowledged as a problem for the Vasicek model) as long as 2a > o. This is true, for example, if we take a = 0.0189 and P = 0.24. Feeding this information into the above formulae is relatively tedious. A spreadsheet application is provided by Jackson and Staunton, After some work we get that the price of the call is... 

We now revisit the earlier Vasicek example for short interest rates to consider the case where the underlying bond pays an annual coupon at a 5% rate (p = 0.05), all the other characteristics remain as before. In order to calculate the call price of the coupon-bond European option first we need to calculate the interest rate such that the present value at the maturity of the option of all later cash flows on the bond equals the strike price. This is done by trial and error using equation (18.48) and the value we get here is = 22.30%. Next, we map the strike price into a series of strike prices via equation (18.50) that are then associated with coupon payments considered as zero-coupon bonds and calculate the value of the European call options contingent on those zero-coupon bonds as in the above example. The calculations are described in Exhibit 18.7. 

The option premium has two constituents intrinsic value and time value. Intrinsic value equals the difference berween the strike price and the underlying assets current price. It is what the options holders would realize if they were to exercise it immediately. Say a call option on a bond futures contract has a strike price of 100 and the contract is trading at 105. A holder who exercises the option, buying the futures at 100 and selling the contract immediately at 105, earns a profit of 5 that is the options intrinsic value. A put options intrinsic value is the amount by which the current underlying assets price is below the strike. Because an option holder will exercise it only if there is a benefit to so doing, the intrinsic value will never be less than zero. Thus, if the bond future in the example were trading at 95, the intrinsic value of the call option would be zero, not —5. 

Assume now that the one-year zero-coupon bond in the example has a call option written on it that matures in six months (at period 1) and has a strike price of 97.40. FIGURE 11.5 is the binomial tree for this option, based on the binomial lattice for the one-year bond in figure 11.4. The figure shows that at period 1, if the six-month rate is 5.50 percent, the call option has no value, because the bond s price is below the strike price. If, on the other hand, the six-month rate is at the lower level, the option has a value of 97.5562 - 94.40, or 0.1562. 

Other derivatives, such as forward-rate agreements and swaps, have similar profiles, as, of course, do cash instruments such as bonds and stocks. Options break the pattern. Because these contracts confer a right but impose no obligation on their holders and impose an obligation but confer no right on their sellers, the payoff profiles for the two parties are different. If, instead of the futures contract itself, the traders in the previous example take long and short positions in a call option on the contract at a strike price of 114, their payoff profiles will be those shown in FIGURE 8.2. 

The term embedded is used because the option element of the bond cannot be stripped out and traded separately, for example, the call option inherent in a callable bond. 

Convertible instruments are usually issued with attached call or put options. Such features can be implemented into the valuation model. If a soft call feature has been implemented, it enables the issuer to force the conversion when the share price overcomes a percentage or trigger level above the conversion price. However, this option cannot be called in the first years hard call . Differently, after the protection period, the issuer can exercise the option. This second time is referred to soft call . Using the same example shown in Section 9.3.1, we assume that the bond may be redeemed in whole but not in part at their principal amount plus accrued interest on the last 2 years, in which the maturity date is at 20 February 2019. On and after this call date , if the share price exceeds 130% of the conversion price the issuer can force the conversion. Figure 9.23 shows the stock price tree in which at years 4 and 5 the stock price is above the threshold. 

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... 

Strike Price (or Exercise Price)— The price, fixed at the outset, at which the underlying asset may be bought (for a call) or sold (for a put). In the previous example, the option was a call struck at 98, giving the holder the right to buy the bond at this price. 

---