Bonds with embedded options

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. 

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 ... 

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. 

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 ... 

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. 

Now let s turn to the price volatility characteristics of bonds with embedded options. As explained in previous chapters, the price of a bond with an embedded option is comprised of two components. The first is the value of the same bond if it had no embedded option. That is, the price if the bond is option free. The second component is the value of the embedded option. 

There are valuation models that can be used to value bonds with embedded options. These models take into account how changes in yield will affect the expected cash flows. Thus, when V and V+ are the values produced from these valuation models, the resulting duration takes into account both the discounting at different interest rates and how the expected cash flows may change. When duration is calculated in this manner, it is referred to as effective duration or option-adjusted duration or OAS duration. Below we explain how effective duration is calculated based on the lattice model and the Monte Carlo model. 

As with duration, there is little difference between modified convexity and effective convexity for option-free bonds. However, for bonds with embedded options there can be quite a difference between the calculated modified convexity and effective convexity measures. In fact, for all option-free bonds, either convexity measure will have a positive value. For bonds with embedded options, the calculated effective convexity measure can be negative when the calculated modified convexity measure is positive. 

This discussion covers the main factors affecting bond returns in the European fixed income market, namely, the random fluctuations of interest rates and bond yield spreads, the risk of an obligor defaulting on its debt, or issuer-specific risk, and currency risk. There are also other, more subtle sources of risk. Some bonds such as mortgage-backed and asset-backed securities are exposed to prepayment risk, but such instruments still represent a small fraction of the total outstanding European debt. Bonds with embedded options are exposed to volatility risk. However, it is not apparent that this risk is significant outside derivatives markets. 

The yield calculation for conventional bonds is relatively straightforward. This is because their redemption dates are fixed, so their total cash flows—the data required to calculate yield to maturity— are known with certainty. Less straightforward to analyze are bonds with embedded options—calls, puts, or sinking funds—so called because the option element cannot be separated from the bond itself The difficulty in analyzing these bonds lies in the fact that some aspects of their cash flows, such as the timing or value of their future payments, are uncertain. 

If a bond s actual redemption date differs from the assumed one, its return computed this way is meaningless. The market, therefore, prefers to use other methods to calculate the return of callable bonds. The most common method is option-adjusted spread, or OAS, analysis. Although the discussion in this chapter centers on callable bonds, the principles enunciated apply to all bonds with embedded options. 

The market quotes bonds with embedded options in terms of yield spreads. A cheap bond trades at a high spread, a dear one at a low spread. The usual convention is to quote the spread between the redemption yield of the bond being analyzed and that of a government bond having an equivalent maturity. This is not an accurate measure of the actual difference in value between the two bonds, however. The reason is that, as explained in chapter 1, the redemption yield computation unrealistically discounts all a bond s cash flows at a single rate. 

Price VolatilMy of Bonds with Embedded Options... 

Effective duration recognizes that yield changes may effect the future cash flow of a bond and so its price. For bonds with embedded options the difference between traditional duration and effective duration can be significant. The effective duration of a callable bond, for example, is sometimes half its traditional duration. As noted in chapter l4, for mortgage-backed securities, the difference is sometimes greater still. 

Just as standard duration is not appropriate for bonds with embedded options, neither is traditional convexity. This is because traditional convexity, like traditional duration, fails to take into account the impact on a bond s future cash flows of a change in market interest rates. As discussed in chapter l4, the approximate convexity of any bond may be derived, following in Fabozzi (1997), using equation (11.14). 

If the prices used for P f, P, and Po are calculated assuming that the bond s remaining cash flows will not change when market rates do, the convexity computed is for an option-free bond. For bonds with embedded options, the prices used in the equation should be derived using a binomial model, in which the cash flows do change with interest rates. The result is effective or option-adjusted convexity. 

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