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8.6 Valuation and Analysis of Callable and Putable Bonds with Interest Rate Volatility

8.6.1 Determination of the Value of a Callable or Putable Bond from an Interest Rate Tree

The procedure to value a bond with an embedded option in the presence of interest rate volatility is as follows:

  • Generate a tree of interest rates based on the given yield curve and interest rate volatility assumptions.
  • At each node of the tree, determine whether the embedded options will be exercised.
  • Apply the backward induction valuation methodology to calculate the bond's present value.
  • This methodology involves starting at maturity and working back from right to left to find the bond's present value Example. Consideradefault-free three-year 4.25% annual coupon bond using the interest rate tree below (10% volatility) if in years 1 and 2 they are 1) callable and 2) putable at par:

2.5% Time 0 3.7041% 4.5245% 5.5258% 3.1681% 3.8695% Time 2 Time 1

  • The yield curve remains the same as in the example on slides 9-11 with one-year, two- year, and three-year par yields of 2.500%, 3.000%, and 3.500%, respectively. But we now assume an interest rate volatility of 10% instead of 0% and interest-rate tree is calibrated respectively.
  • The model for interest-rate volatility is sqrt , where t is time in years between 'time slices' (here it is one year).
  • Callable Bond [where C = cash flow (% of par) and V = value of the callable bond's future cash flows (% of par).+ Notes: At 10% volatility, call option value is 102.114-101.540=0.574, which is greater than 0.407 when zero volatility is assumed.
  • Putable Bond C=4.25 V=98.791 Put at 100 C=4.25 V=99.738 Put at 100 C=4.25 V=100.526 C=4.25 V=101.304 C=4.25 V=100.366 V=102.522 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 2.5% 3.1891% 3.1681% 3.3695% 3.3695% 2.5% Time 0 3.7041% 3.7041% 4.5242% 5.5258% 5.5258% 4.5242% Time 1 Time 2 Time 3 C=4.25 V=98.791 C=4.25 V=99.738 C=4.25 V=100.526 Called at 100 C=4.25 V=100.022 Called at 100 C=4.25 V=99.658 V=101.540 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 C=4.25 V=100 2.5% 3.1891% 3.1681% 3.3695% 3.3695% 2.5% 3.7041% 3.7041% 4.5242% 5.5258% 5.5258% 4.5242% Time 0 Time 1 Time 2 Time 3 Notes: At 10% volatility, put option value is 102.522-102.114=0.408, which is greater than 0.283 when zero volatility is assumed.

Valuation of Risky Callable and Putable Bonds The approach for default-free (sovereign) bonds can be extended to risky (corporate)

bonds

  • The industry-standard approach is to increase the discount rates above the default-free rates to reflect default risk.
  • The second approach to valuing risky bonds is by making the default probabilities explicit – that is, by assigning a probability to each time period going forward.

Notes:

  • Information about default probabilities and recovery values may be accessible from credit default swaps.

8.6.2 Determination and use of option-adjusted spreads (OAS)

There are two standard approaches to construct a suitable yield curve for a risky bond:

  • Use an issuer-specific curve (might be impossible due to cost and availability of data).Raise the one-year forward rates derived from the default-free benchmark yield curve by a fixed Z-spread
  • A second approach can be used for risky bonds with embedded options:Option-adjusted spread (OAS) is the constant spread that, when added to all the one-period forward rates on the interest rate tree, makes the arbitrage-free value of the bond equal to its market price.

Notes:

  • If the bond's price is given, the OAS is determined by trial and error.
  • An OAS lower than that for a bond with similar characteristics and credit quality indicates that the bond is likely overpriced (rich) and should be avoided.
  • A larger OAS than that of a bond with similar characteristics and credit quality means that the bond is likely underpriced (cheap).

8.6.3 Effect of interest rate volatility on option-adjusted spreads

The dispersion of interest rates on the tree is volatility dependent, and so is the OAS

  • As interest rate volatility increases, the OAS for the callable bond decreases.

Lesson Wrap-Up

This lesson should leave you able to explain the valuation and analysis of callable and putable bonds with interest rate volatility in a fixed-income context and connect it to the decisions made by issuers, investors, or analysts.

Review Prompts

  1. Explain determination of the value of a callable or putable bond from an interest rate tree in your own words.
  2. Explain determination and use of option-adjusted spreads (oas) in your own words.
  3. Explain effect of interest rate volatility on option-adjusted spreads in your own words.
  4. State one exam-style risk, valuation, or market implication of the valuation and analysis of callable and putable bonds with interest rate volatility.