{"id":290,"date":"2026-07-19T16:20:31","date_gmt":"2026-07-19T16:20:31","guid":{"rendered":"https:\/\/eclass.co.ke\/class\/lesson\/8-6-valuation-and-analysis-of-callable-and-putable-bonds-with-interest-rate-volatility\/"},"modified":"2026-07-19T17:41:26","modified_gmt":"2026-07-19T17:41:26","slug":"8-6-valuation-and-analysis-of-callable-and-putable-bonds-with-interest-rate-volatility","status":"publish","type":"eclass_lesson","link":"https:\/\/eclass.co.ke\/class\/lesson\/8-6-valuation-and-analysis-of-callable-and-putable-bonds-with-interest-rate-volatility\/","title":{"rendered":"8.6 Valuation and Analysis of Callable and Putable Bonds with Interest Rate Volatility"},"content":{"rendered":"<h3>8.6.1 Determination of the Value of a Callable or Putable Bond from an Interest Rate Tree<\/h3>\n<p>The procedure to value a bond with an embedded option in the presence of interest rate volatility is as follows:<\/p>\n<ul>\n<li>Generate a tree of interest rates based on the given yield curve and interest rate volatility assumptions.<\/li>\n<li>At each node of the tree, determine whether the embedded options will be exercised.<\/li>\n<li>Apply the backward induction valuation methodology to calculate the bond&#x27;s present value.<\/li>\n<li>This methodology involves starting at maturity and working back from right to left to find the bond&#x27;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:<\/li>\n<\/ul>\n<p>2.5% Time 0 3.7041% 4.5245% 5.5258% 3.1681% 3.8695% Time 2 Time 1<\/p>\n<ul>\n<li>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.<\/li>\n<li>The model for interest-rate volatility is sqrt , where t is time in years between &#x27;time slices&#x27; (here it is one year).<\/li>\n<li>Callable Bond [where C = cash flow (% of par) and V = value of the callable bond&#x27;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.<\/li>\n<li>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.<\/li>\n<\/ul>\n<p>Valuation of Risky Callable and Putable Bonds The approach for default-free (sovereign) bonds can be extended to risky (corporate)<\/p>\n<p>bonds<\/p>\n<ul>\n<li>The industry-standard approach is to increase the discount rates above the default-free rates to reflect default risk.<\/li>\n<li>The second approach to valuing risky bonds is by making the default probabilities explicit &#8211; that is, by assigning a probability to each time period going forward.<\/li>\n<\/ul>\n<p>Notes:<\/p>\n<ul>\n<li>Information about default probabilities and recovery values may be accessible from credit default swaps.<\/li>\n<\/ul>\n<h3>8.6.2 Determination and use of option-adjusted spreads (OAS)<\/h3>\n<p>There are two standard approaches to construct a suitable yield curve for a risky bond:<\/p>\n<ul>\n<li>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<\/li>\n<li>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.<\/li>\n<\/ul>\n<p>Notes:<\/p>\n<ul>\n<li>If the bond&#x27;s price is given, the OAS is determined by trial and error.<\/li>\n<li>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.<\/li>\n<li>A larger OAS than that of a bond with similar characteristics and credit quality means that the bond is likely underpriced (cheap).<\/li>\n<\/ul>\n<h3>8.6.3 Effect of interest rate volatility on option-adjusted spreads<\/h3>\n<p>The dispersion of interest rates on the tree is volatility dependent, and so is the OAS<\/p>\n<ul>\n<li>As interest rate volatility increases, the OAS for the callable bond decreases.<\/li>\n<\/ul>\n<h3>Lesson Wrap-Up<\/h3>\n<p>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.<\/p>\n<h3>Review Prompts<\/h3>\n<ol>\n<li>Explain determination of the value of a callable or putable bond from an interest rate tree in your own words.<\/li>\n<li>Explain determination and use of option-adjusted spreads (oas) in your own words.<\/li>\n<li>Explain effect of interest rate volatility on option-adjusted spreads in your own words.<\/li>\n<li>State one exam-style risk, valuation, or market implication of the valuation and analysis of callable and putable bonds with interest rate volatility.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>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 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","template":"","class_list":["post-290","eclass_lesson","type-eclass_lesson","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson\/290","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson"}],"about":[{"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/types\/eclass_lesson"}],"author":[{"embeddable":true,"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/comments?post=290"}],"version-history":[{"count":2,"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson\/290\/revisions"}],"predecessor-version":[{"id":426,"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson\/290\/revisions\/426"}],"wp:attachment":[{"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/media?parent=290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}