Pathwise Valuation

Pathwise valuation involves discounting a bond’s cash flows for each likely interest rate path and calculating the average of these values across all the paths. It is an alternative method to the backward induction approach.

The following steps are involved in pathwise valuation:

• Step 1: Come up with a list of all the possible interest rate paths across the tree;
• Step 2: Discount the cash flows of the bond along each possible path; and
• Step 3: Determine the average across all the possible paths.

Given a binomial interest rate tree with \(\text{n}\) periods, there will be \(2^{\text{n – 1}}\) unique paths. With a three-period tree, for example, there will be four possible paths, i.e.,  \(2^{\text{2}}\)

Example: Pathwise Valuation

Given the following interest rate tree, determine the value of a three-year option-free bond that pays 5% annual coupons with a face value of $100.

$$ \textbf{One-Year Implied Forward Rates} \\ \begin{array}{c|c|c} \textbf{Time 0} & \textbf{Time 1} & \textbf{Time 2} \\ \hline & & 10.693\% \\ \hline & 6.886\% & \\ \hline 4.000\% & & 7.861\% \\ \hline & 5.123\% & \\ \hline & & 5.899\% \\ \end{array} $$

Solution

The four possible interest rate paths for this bond are as per the following table:

$$ \begin{array}{c|cccc} \textbf{Interest rate path} & & \textbf{Year} & \\ \hline & \bf{1} & \bf{2} & \bf{3} & \textbf{Present Value} \\ 1 (uu) & 4.000\% & 6.886\% & 10.693\% & 94.64 \\ \hline 2 (ud) & 4.000\% & 6.886\% & 7.861\% & 96.88 \\ \hline 3 (du) & 4.000\% & 5.123\% & 7.861\% & 98.42 \\ \hline 4 (dd) & 4.000\% & 5.123\% & 5.899\% & 100.07 \end{array} $$

$$ \begin{align*} \text{Present Value for path 1} & =\frac{5}{1.04}+\frac{5}{1.04\times1.06886} \\ & +\frac{105}{1.04\times1.06886\times1.10693} \\ & =$94.64 \\ \text{Present Value for path 2} & =\frac{5}{1.04}+\frac{5}{1.04\times1.06886} \\ & +\frac{105}{1.04\times1.06886\times1.07861} \\ & =$96.88 \end{align*} $$

And so on.

The average of the path values is:

$$ \frac{1}{4}\times\left(94.64+96.88+98.42+100.07\right)=$97.50 $$

Notice that this value is similar to the one generated using the arbitrage-free binomial lattice and the zero-coupon yield curve.

Question

The following table shows the interest rate paths and the possible forward rates along the paths of a three-year option-free 4% annual coupon bond with a face value of $100.

$$ \begin{array}{c|ccc} \textbf{Interest rate path} & & \textbf{Year} & \\ \hline & \bf{1} & \bf{2} & \bf{3} \\ 1 & 2.50\% & 5.81\% & 11.36\% \\ \hline 2 & 2.50\% & 5.81\% & 8.42\% \\ \hline 3 & 2.50\% & 4.30\% & 8.42\% \\ \hline 4 & 2.50\% & 4.30\% & 6.24\% \end{array} $$

Using the pathwise valuation approach, the present value for the third path of the bond is closest to:

1. $79.88.
2. $82.78.
3. $97.37.

Solution

The correct answer is C.

$$ \begin{align*} \text{Present Value for path 3} & =\frac{4}{1.0250}+\frac{4}{1.0250\times1.0430} \\ & +\frac{104}{1.0250\times1.0430\times1.0842} \\ & =$97.37 \end{align*} $$

Reading 29: The Arbitrage-Free Valuation Framework

LOS 29 (g) Describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixed income instrument given its cash flows along each path.