Options embedded

Making comparison between bonds could be difficult and several aspects must be considered. One of these is the bond s maturity. For instance, we know that the yield for a bond that matures in 10 years is not the same compared to the one that matures in 30 years. Therefore, it is important to have a reference yield curve and smooth that for comparison purposes. However, there are other features that affect the bond s comparison such as coupon size and structure, liquidity, embedded options and others. These other features increase the curve fitting and the bond s comparison analysis. In this case, the swap curve represents an objective tool to understand the richness and cheapness in bond market. According to O Kane and Sen (2005), the asset-swap spread is calculated as the difference between the bond s value on the par swap curve and the bond s market value, divided by the sensitivity of 1 bp over the par swap. 

The same bond structure (bullet as maturity type, no embedded options). 

From market observation we know that index-linked bonds can experience considerable volatility in prices, similar to conventional bonds, and therefore, there is an element of volatility in the real yield return of these bonds. Traditional economic theory states that the level of real interest rates is cmistant however, in practice they do vary over time. In addition, there are liquidity and supply and demand factors that affect the market prices of index-linked bonds. In this chapter, we present analytical techniques that can be applied to index-linked bonds, the duration and volatility of index-linked bonds and the concept of the real interest rate term structure. Moreover, we show the valuation of inflation-linked bonds with different cash flow structures and embedded options. 

Therefore, we propose an example in which we price an inflation-linked bond by using a binomial tree. Conventionally, this type of pricing model is not implemented in the reality, but it allows to understand the impact of the embedded option on bond s value. 

Option-adjusted spread The OAS is used for bonds with embedded options. This spread is calculated as the difference between the Z-spread and option value expressed in basis points. 

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. 

Determining the Value of an Embedded Option The value of an embedded option is found through the binomial tree model. The first step is to forecast the value of the underlying security in which the price S of a security can move, respectively, in the upstate and downstate with a probability of p and 1 p. The change in price occurs in discrete time interval At and will depend on the level of volatility assumed. An option written on the asset, with maturity T will move in discrete steps as the movements of the share prices. The process can be carried on for any number of time intervals (Figure 9.6). 

The embedded option component in convertible bonds makes the valuation sensitive from three main parameters share price, volatility and interest rate. These parameters affect the value of a convertible bond for both situations ... 

The second parameter that affects convertible value is the volatility. In fact, the volatility of the underlying asset is the main element that moves the value of the embedded option, in which pricing models are very sensitive from this parameter. Note that convertible price rises as the volatility increases. The chart shown in Figure 9.11 defines the value of the convertible bond with the volatilities of 25%, 35% and 45%. 

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. 

Conversely, at the lowest node, the hedge ratio is 0 because the option is out of money or 0. This means that in the first case the bond trade like the equity, while in the second case like a conventional bond. Therefore when the share price increases the delta approaches unity, implying that the option is deeply in the money. In contrast, when the share price is low relative to the conversion price, the sensitivity of the convertible and therefore of the embedded option is low. 

Bonds with embedded options are debt instruments that give the right to redeem the bond before maturity. As we know, the yield to maturity represents the key measure of bond s return (although, of course, it is an anticipated return that is seldom realised in practice). The calculation of the return is particularly easy for conventional bonds because the redemption date is known with certainty, as their value. In contrast, for callable bonds, but also for other bonds such as putable and sinking fund bonds, the redemption date is not known with certainty because the bonds can be redeemed before maturity. If we want to calculate... 

As introduced in Chapter 8, the most suitable measure of return for bonds with embedded options is known as option-adjusted spread or OAS. In this chapter, we show the analysis of bonds with embedded options, with particular focus on pricing methodology. 

Bonds with embedded options are instruments that give the option holder the right to redeem the bond before its maturity date. For callable bonds, this right is held by the issuer. The main reason for an issuer to issue these debt instruments is to get protection from the decline of interest rates or improvement of issuer s credit quahty. In other words, if interest rates fall or credit quality enhances, the issuer has convenience to retire the bond from the market in order to issue again another bond with lower interest rates. 

To calculate the internal rate of return of a bond with embedded option, we can have three main measures ... 

The duration shows the bond s price sensitivity to its yield to maturity. The change in bond s price is plotted in a curve in which the duration represents the slope of the tangent at any point of the curve. Conversely, the effective duration, or also known as curve duration, shows the price sensitivity to the change of the benchmark yield curve or market yield curve. This duration is more suitable than Macaulay or modified duration for bonds with embedded options because the latter ones have not a well-defined yield to maturity. The effective duration is given by (11.1) ... 

The convexity is a more correct measure of the price sensitivity. It measures the curvature of the price-yield relationship and the degree in which it diverges from the straight-line estimation. Like the duration, the standard measure of convexity does not consider the changes of market interest rates on bond s prices. Therefore, the conventional measure of price sensitivity used for bonds with embedded options is the effective convexity. It is given by (11.2) ... 

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. 

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. 

To calculate the value of these bonds, it is preferable to use the binomial tree model. The value of a straight bond is determined as the present values of expected cash flows in terms of coupon payments and principal repayment. For bonds with embedded options, since the main variable that drives their values is the interest rate, the binomial tree is the most suitable pricing model. 

In order to find a fair value of the embedded option, the Black Scholes model is not suitable for the following reasons ... 

Constant interest rate The main reason for which B S misprices the embedded option is that the model does not take into account the interest rate path ... 

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 ... 

The binomial tree model evaluates the return of a bond with embedded option by adding a spread to the risk-free yield curve. Generally, the price obtained by the model is compared to the one exchanged in the market. If the theoretical price is different, the model can be calibrated with three key elements. The first ones are the volatility and drift factor. They allow to calibrate the model interest rate path in order to obtain the equality with the market yield curve. The third one is the spread applied over the yield curve. Generally, when volatility and drift are correctly calibrated, the last element to select in order to obtain the market parity is the spread. Conventionally, banks define it in the following way ... 

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. 

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 embedded option has value when the interest rates go up. Figure 11.13 illustrates the value of a put option. As with callable bonds, the put option is worthless at maturity because the bond is given back in each case. Before the maturity, the put option decreases its value as the bond approaches maturity. The put option value is equal to 0.33. 

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. 

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. 

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