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Back-to-Basics

The following article is reprinted from the October, 1997 issue of
On the Edge, the Interactive Data Fixed Income Analytics bimonthly newsletter.

Back-to-Basics: Thinking in Terms of Options

Teri Geske
Senior Vice President, Product Development


Many fixed income securities have one or more types of embedded options, such as calls, puts, prepayments, rate caps/floors, etc. In previous “Back-to-Basics” articles we have discussed how Effective Duration and Convexity describe the price sensitivity of assets that contain these options to a change in interest rates, the goal being to provide some intuition as to why a security’s effective duration (and convexity) works out to be what it is. This kind of intuitive understanding allows us to see, for example, why the effective duration of an adjustable rate mortgage is not simply the time to its next reset date, or why the convexity of a puttable bond is large and positive. Developing that intuition means we can anticipate to some degree how a bond’s price will respond to a change in interest rates, without using complicated mathematical formulas (alas, even the most intuitive mind must still rely on those formulas to achieve the precision required for portfolio management). This article offers one approach to thinking about prices of bonds with embedded options which can be useful in understanding and explaining the duration and convexity the mathematical models produce.

The basic idea is to view a security with options as the sum of its parts - i.e., the value of its option-free components plus or minus the value of the embedded options. If we can form a reasonable prediction about the effect of a change in interest rates on the individual components, we can understand the security’s interest rate sensitivity. This way of thinking is more than just an educational exercise - it is one way of pricing contingent claims (e.g. securities with embedded options), based on the principal of “no arbitrage.” As a reminder, the “no arbitrage” argument states that if two investments (securities or portfolios) offer the same pay-off under all possible future states of the world, the two must have the same price today. Otherwise, investors would sell (short) the more expensive investment and use the proceeds to purchase the cheaper investment, locking in a guaranteed profit today (the price of the expensive security minus the price of the cheap one), with zero risk. This activity would depress the price of the overvalued investment and drive up the price of the undervalued one, re-establishing an equilibrium and eliminating the arbitrage. For example, consider the following investment possibilities, where only two possible pay-offs could occur:

Given these alternatives, an investor would be indifferent between an investment in Security A, or a portfolio containing one each of Security B and Security C, since the outcomes for B + C = the outcomes for Security A in all possible future states. Therefore, the prices of these two investment alternatives must be the same to prevent an arbitrage opportunity; therefore, the price (today) of Security A must be $0.80. The ability to replicate the payoff of Security A with some combination of Securities B and C allows us to price “A” using this “no arbitrage” approach.1

These concepts of replication and no arbit rage directly relate to our notion of how to intuitively understand a complicated bond’s price behavior by considering the behavior of its more basic components. For example, a callable bond may be viewed as a long position in a bond that pays a fixed coupon to the maturity date and short position in a call option that allows the issuer to truncate the coupon payments and return the principal early (perhaps with some call premium). We know that a decline in rates causes the price (present value) of a fixed coupon bond to increase; we also know that a decline in rates increases the value of the call option. Therefore, the change in the price of the callable bond in response to a decline in rates will be a function of the rising value of the “underlying” fixed maturity bond minus the rising value of the call option.

Consider a non-callable, 6.5% coupon bond maturing 7/31/07, priced at par. If interest rates decline 100 bps, the bond’s price rises to 107.541; the increase of $7.541 reflects the higher present value of the coupon payments and principal at maturity. An otherwise identical bond callable in five years at 102 (with a declining call premium thereafter) would be priced at 98.365, with the embedded option valued at $1.635. A 100 bp drop in rates would cause the callable bond’s price to rise to 104.204 - an increase of $5.84. So, the non-callable bond’s price increased by $1.69 more than the callable bond’s price. Why? Because the value of the call option increased by $1.69 (from $1.64 to $3.33), and the change in the callable bond’s price is equal to the change in the non-callable bond’s price minus the change in the option value. An additional decline in rates would cause an even smaller change in the value of the callable bond compared to the non-callable version, as the increase in the value of the call option would offset more of the increase in the present value of the non-callable bond’s cashflows to maturity. Ultimately, an increase in the present value of the cashflows to maturity would be completely offset by an increase in the value of the call option, resulting in a 0% change in the price of the callable bond - in other words, an effective duration of 0.00.

If we become accustomed to thinking about securities as a bundle of option-free cashflows and option components, we can intuitively understand the price behavior of more complicated securities. For example, an rise in interest rates increases the value of the reset cap and lifetime cap embedded in an adjustable rate mortgage; since the investor is short those options, the price of the ARM declines. Even the most complex securities can be understood using this thought process. Consider an inverse floating rate IO tranche of a CMO deal: A drop in rates would increase the coupon rate on the security, and would also cause the present value of those coupon payments to increase; however, a decline in rates simultaneously increases the value of the prepayment option of the underlying collateral, causing the value of the IO component of the security to decline (unless changes in prepayments are completely absorbed by other tranches in the deal). The new price of the security depends on the net effect of these changes.

We hope this discussion of how thinking in terms of options can be used to understand and evaluate the diverse security types involved in managing a fixed income portfolio. If you have any suggestions for future topics for this Back-to-Basics column, please let us know: you may contact marketing at (310) 479-9715 or e-mail us at fia.marketing@interactivedata.com.

1For a useful explanation of the theory behind arbitrage-free, risk-neutral pricing, interested readers may wish to refer to the article, “Equivalent Martingale Measures and Risk-Neutral Pricing: An Expository Note” by Rangarajan K. Sundaram; Journal of Derivatives - Fall 1997.