Call options

Suspected correlations are displayed by calling option (Display Correlation Graph Pairwise) the coordinates can be marked either with a circle (default), or with the appropriate index so that outliers can be readily identified. 

Using the information in table 1.4, the used-car ad can be divided into five moves. In the first move (included in all six ads), the seller states the essential facts about the car (year, make, and model). In the second move (included in all but two ads), the seller highlights select features of the car (e.g., new tires, air bags, five speeds). In the third move (interchangeable with the second), the writer offers a subjective description of the car (e.g., good cond or fun ). In the fourth move, the writer states the price. Finally, in the fifth move, the seller provides contact information (1) a phone number, (2) a contact name (optional), and (3) when to call (optional). 

Genentech to remain independent company, Roche intended to exercise its call option and publicly reissue Genentech shares. 

Instead of assuming either that investments are reversible or that they cannot be delayed, recent research on investment stresses the fact that companies have opportunities to investment and that they must decide how to exploit those opportunities most effectively, analogous to financial options. A company with an opportunity to invest is holding something akin to a financial call option. When it decides to invest, the call option is exercised. 

In this thesis, we mainly focus on the derivation of call options (w = 1), keeping in mind that it is always easy to compute the appropriate probabilities for w = -1 via... 

Starting from the risk-neutral bond price dynamics (5.4), we derive the well known closed-form solution for the price of a zero-coupon bond option. Thus, as shown in section (2.1) the price of a call option on a discount bond is given by... 

Thus, we end up with the well known Black and Scholes -like formula for the price of a European call option on a zero-coupon bond... 

The equity can be assumed as a call option where the payoff is given by Equation (8.12) ... 

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. 

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. 

Therefore, at maturity T, the value of a call option is determined as the relationship between the stock price St of the underlying asset and the strike price X as follows (Equation 9.8) ... 

From Figure 9.8, the option value at time to is calculated starting at the final node (time T) by using the formula (9.8) and working backward in each node by applying the formula (9.9). The value of a call option at time to is equal to 0.46. 

The value of the conversion option is equal to the conversion ratio (number of common shares for which each bond may be exchanged) multiplied by the value of a call option, that is having a par value of 100 and a conversion price of 2.6, the conversion ratio is 38.5. The value of the conversion option at time to is 17.8 (0.46 X 38.5). This is the value of the option element. 

Interest rate (1.04%) - —Interest rate (7.04%) FIGURE 9.16 Value of a call option with a different interest rate. 

In 1973, Myron Scholes and Fisher Black developed a model known as B S model for valuing options. Like the binomial tree, in the B S model the option value depends mainly on the price of the underlying asset, volatility, interest rate, time to expiration and dividend yield. Because in this chapter, we propose the value of a cOTivertible as the sum of the straight bond and call option, the... 

Therefore, the model is easy to implement and gives similar results as the binomial tree. Because B S works in continuous compounding while the binomial tree in discrete time, the models give the same results only if the binomial tree has a high number of steps. The more periods in binomial tree are implemented, the nearer is the value that we get in both models. Consider the convertible bond pricing shown in Section 9.3.1. In that analysis we estimate the value of a call option using the binomial tree, obtaining a value per call of 0.46. 

In fact, in that nodes the maximum value of the call option is 0.78 (Figure 9.24). 

Multiplying the value of the call option (0.37) by the conversion ratio of 38.462, we obtain the value of conversion option (including soft call) which is equal to 14.3, lower than the one obtained without soft call of 17.8. As in... 

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 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. 

European call option For which the issuer can recall the bond only once on the call date ... 

American call option The bond can be called on the call date and any time thereafter ... 

Bermudan call option The bond is callable only on specified call dates. 

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) ... 

FIGURE 11.10 The binomial price tree for a call option. 

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 ... 

Step-up callable notes are a particular type of structured fixed income products. These bonds offer a coupon payment that increase during the bond s life. Moreover, they include a call option, that as we discussed earlier, the issuer has the right to redeem the bond early. The question, whether a callable step-up note will be called or not always depends on the evolution of interests rates. Therefore, the inclusion of these two characteristics makes the bond attractive to investors with higher performance than a conventional bond. The added variable coupon element acts for an investor as cushion compared to a conventional callable bond. In fact, the increasing coupon payment increases the value of a callable bond. However, if interest rates go down and coupon payments increase, the incentive of the issuer to redeem the bond early is greater than a simple callable. 

Value of an option free bond Value of a call option Value of a callable bond... 

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. 

Some bonds include a provision in their offer particulars that gives either the bondholder and/or the issuer an option to enforce early redemption of the bond. The most common type of option embedded in a bond is a call feature. A call provision grants the issuer the right to redeem all or part of the debt before the specified maturity date. An issuing company may wish to include such a feature as it allows it to replace an old bond issue with a lower coupon rate issue if interest rates in the market have declined. As a call feature allows the issuer to change the maturity date of a bond it is considered harmful to the bondholder s interests therefore the market price of the bond at any time will reflect this. A call option is included in all asset-backed securities based on mortgages, for obvious reasons. 

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. 

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