{"id":283,"date":"2026-07-19T16:20:31","date_gmt":"2026-07-19T16:20:31","guid":{"rendered":"https:\/\/eclass.co.ke\/class\/lesson\/7-5-zero-coupon-yield-curve-arbitrage-free-valuation-and-pathwise-valuation\/"},"modified":"2026-07-19T17:41:25","modified_gmt":"2026-07-19T17:41:25","slug":"7-5-zero-coupon-yield-curve-arbitrage-free-valuation-and-pathwise-valuation","status":"publish","type":"eclass_lesson","link":"https:\/\/eclass.co.ke\/class\/lesson\/7-5-zero-coupon-yield-curve-arbitrage-free-valuation-and-pathwise-valuation\/","title":{"rendered":"7.5 Zero-Coupon Yield Curve,Arbitrage-Free Valuation and Pathwise Valuation"},"content":{"rendered":"<h3>7.5.1 Pricing Using the Zero-Coupon Yield Curve and Pricing Using an Arbitrage-Free<\/h3>\n<p>Binomial Lattice If two valuation methods are arbitrage free, they should provide the same valuation result.<\/p>\n<p>Check using this process:<\/p>\n<p>1. Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond 2. Compare the pricing using the zero-coupon yield curve with the pricing using an arbitrage-free binomial lattice.<\/p>\n<p>Example. Consider an option-free bond with three-years remaining to maturity and a coupon rate of 5%. Spot rates are the following:<\/p>\n<p>Maturity (years) 1 2 3 Spot Rates 2.000% 3.015% 4.055% The interest rate tree is as follows:<\/p>\n<p>V<\/p>\n<ul>\n<li>Because the tree was calibrated to the same par curve (and spot curve) that was used to price this option-free bond using spot rates only, the tree gives the same price as the spot rate pricing.<\/li>\n<\/ul>\n<p>2.0% Time 0 4.482% 8.167% 6.051% 3.442% 4.646% Time 2 Time 1 V = 102.8101 C = 5 V = 102.8101 C = 5 V = 106.2668 C = 5 V = 102.0721 C = 5 V = 102.0721 C = 5 V = 102.0721 C = 5 V = 102.0721 C = 5 V = 102.0721 C = 5 V = 102.0721 C = 5 V = 102.0721 8.167% 4.646% 4.646% 3.442% 3.442% 8.167% 6.051% 2.0% 6.051% 2.0% 4.482% 4.482% Time 0 Time 1 Time 2 Time 4<\/p>\n<h3>7.5.2 Pathwise Valuation in a Binomial Interest Rate Framework and Computation of<\/h3>\n<p>the Value of a Fixed-Income Instrument Given Its Cash Flows along Each Path<\/p>\n<ul>\n<li>An alternative approach to backward induction in a binomial tree is called &quot;pathwise valuation.&quot;<\/li>\n<li>Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths.<\/li>\n<li>Pathwise valuation involves the following steps:<\/li>\n<li>Specify a list of all potential paths through the tree.<\/li>\n<li>Determine the present value of a bond along each potential path.<\/li>\n<li>Calculate the average across all possible paths.<\/li>\n<\/ul>\n<p>Example. Consider the same option-free bond as on slide 23, with three years remaining to maturity and a coupon rate of 5%.<\/p>\n<ul>\n<li>There are four potential paths of interest rates: HH, HL, LH, and LL. Using actual interest rates results in the following:<\/li>\n<\/ul>\n<p>Path Time 0 Time 1 Time 2 1 2% 4.602% 8.167% 2 2% 4.602% 6.051% 3 2% 3.409% 6.051% 4 2% 3.409% 4.482%<\/p>\n<ul>\n<li>Determining all potential paths is just like the following example.<\/li>\n<li>This example mirrors exactly the number of interest rate paths in our binomial interest rate tree.<\/li>\n<\/ul>\n<p>Present values:<\/p>\n<p>The result is the same as calculated using the binominal tree and spot rates.<\/p>\n<p>Path Time 0 1 100.5296 2 102.3449 3 103.4792 4 104.8876 Average 102.8103<\/p>\n<h3>Lesson Wrap-Up<\/h3>\n<p>This lesson should leave you able to explain the zero-coupon yield curve,arbitrage-free valuation and pathwise valuation 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 pricing using the zero-coupon yield curve and pricing using an arbitrage-free in your own words.<\/li>\n<li>Explain pathwise valuation in a binomial interest rate framework and computation of in your own words.<\/li>\n<li>State one exam-style risk, valuation, or market implication of the zero-coupon yield curve,arbitrage-free valuation and pathwise valuation.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>7.5.1 Pricing Using the Zero-Coupon Yield Curve and Pricing Using an Arbitrage-Free Binomial Lattice If two valuation methods are arbitrage free, they should provide the same valuation result. Check using this process: 1. Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond 2. Compare the pricing using the zero-coupon yield curve with the pricing [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","template":"","class_list":["post-283","eclass_lesson","type-eclass_lesson","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson\/283","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=283"}],"version-history":[{"count":2,"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson\/283\/revisions"}],"predecessor-version":[{"id":419,"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/eclass_lesson\/283\/revisions\/419"}],"wp:attachment":[{"href":"https:\/\/eclass.co.ke\/class\/wp-json\/wp\/v2\/media?parent=283"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}