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7.5 Zero-Coupon Yield Curve,Arbitrage-Free Valuation and Pathwise Valuation

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 using an arbitrage-free binomial lattice.

Example. Consider an option-free bond with three-years remaining to maturity and a coupon rate of 5%. Spot rates are the following:

Maturity (years) 1 2 3 Spot Rates 2.000% 3.015% 4.055% The interest rate tree is as follows:

V

  • 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.

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

7.5.2 Pathwise Valuation in a Binomial Interest Rate Framework and Computation of

the Value of a Fixed-Income Instrument Given Its Cash Flows along Each Path

  • An alternative approach to backward induction in a binomial tree is called "pathwise valuation."
  • Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths.
  • Pathwise valuation involves the following steps:
  • Specify a list of all potential paths through the tree.
  • Determine the present value of a bond along each potential path.
  • Calculate the average across all possible paths.

Example. Consider the same option-free bond as on slide 23, with three years remaining to maturity and a coupon rate of 5%.

  • There are four potential paths of interest rates: HH, HL, LH, and LL. Using actual interest rates results in the following:

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%

  • Determining all potential paths is just like the following example.
  • This example mirrors exactly the number of interest rate paths in our binomial interest rate tree.

Present values:

The result is the same as calculated using the binominal tree and spot rates.

Path Time 0 1 100.5296 2 102.3449 3 103.4792 4 104.8876 Average 102.8103

Lesson Wrap-Up

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.

Review Prompts

  1. Explain pricing using the zero-coupon yield curve and pricing using an arbitrage-free in your own words.
  2. Explain pathwise valuation in a binomial interest rate framework and computation of in your own words.
  3. State one exam-style risk, valuation, or market implication of the zero-coupon yield curve,arbitrage-free valuation and pathwise valuation.