Lesson 8.2: Arbitrage-Free Valuation and Binomial Trees

Introduction

In this lesson, students will delve into the concepts of arbitrage-free valuation and the use of binomial trees for valuing fixed-income securities. Understanding these concepts is crucial for performing accurate valuations that reflect the true economic value of bonds and other fixed-income investments. This lesson will cover the law of one price, the construction of binomial trees, and their application in bond valuation.

Learning Objectives

By the end of this lesson, students will be able to:

• Explain arbitrage-free valuation using spot rates and the law of one price.
• Construct and utilize a binomial interest rate tree.
• Value a bond on an arbitrage-free basis.
• Use a binomial tree to value an option-free bond.
• Discuss the main ideas and terminology behind arbitrage-free valuation and binomial trees.

Section 1: Arbitrage-Free Valuation

Arbitrage-free valuation is a fundamental principle in financial economics that ensures that there are no arbitrage opportunities in a well-functioning market. An arbitrage opportunity arises when an investor can make a profit with no risk and no capital investment. In a properly functioning market, similar assets must sell for the same price, leading to the law of one price.

1.1 The Law of One Price

The law of one price states that in efficient markets, identical goods must sell for the same price when there are no transportation costs and no differential taxes applied to different markets. This principle can be extended to the valuation of fixed-income securities. For instance, if a bond with similar characteristics is priced differently in two markets, an arbitrageur can buy the cheaper bond and sell the more expensive one, pocketing the difference.

Let's consider an example:

Example 1: Law of One Price

Assume a bond that pays a semiannual coupon of 5% is trading at $1,000 in Market A and $1,050 in Market B. An investor could buy the bond in Market A, return it to Market B, and sell it for a profit of $50. Thus, the price must equalize due to market forces, confirming the law of one price.

1.2 Spot Rates and Forward Rates

In fixed-income securities, spot rates are the yields on zero-coupon bonds for different maturities. These spot rates are crucial for determining the present value of future cash flows. The present value of a bond's cash flows can be estimated using the formula:

$$PV = \sum_{t=1}^{n} \frac{C}{(1 + r_t)^t} + \frac{F}{(1 + r_n)^n}$$

where:

• $PV$ is the present value of the bond,
• $C$ is the coupon payment,
• $F$ is the face value of the bond,
• $r_t$ is the spot rate for period $t$,
• $n$ is the number of periods until maturity.

1.2.1 Example of Spot Rates

Let’s say a bond pays an annual coupon of $80$ and matures in $3$ years with a face value of $1,000$. Assuming the following spot rates:

• Year 1: $r_1 = 0.02$
• Year 2: $r_2 = 0.03$
• Year 3: $r_3 = 0.04$

We can calculate the bond's value as follows:

$$PV = \frac{80}{(1 + 0.02)^1} + \frac{80}{(1 + 0.03)^2} + \frac{1000}{(1 + 0.04)^3}$$

Calculating each term:

• For year 1: $\frac{80}{(1.02)} \approx 78.43$
• For year 2: $\frac{80}{(1.03^2)} \approx 75.49$
• For year 3: $\frac{1000}{(1.04^3)} \approx 889.00$

Thus,

$$PV \approx 78.43 + 75.49 + 889.00 = 1042.92$$

This bond is worth approximately $1,042.92 on an arbitrage-free basis.

Section 2: Building a Binomial Interest Rate Tree

A binomial tree is a visual and quantitative representation that helps in evaluating the future movements of interest rates. It is especially useful in options pricing and for bonds with embedded options.

2.1 Concept of a Binomial Interest Rate Tree

The idea behind the binomial interest rate tree is to model potential future movements in interest rates by creating a set of discrete time points (nodes). At each node, the interest rate can either increase or decrease, leading to multiple possible future rate paths.

To build a binomial interest rate tree:

1. Determine the initial interest rate (current spot rate).
2. Decide the up ($u$) and down ($d$) factors that depict how much the rates can move each period.
3. Establish the time steps (periods) until maturity.
4. Calculate future interest rates at each node.

Let's construct a simple binomial interest rate tree.

2.1.1 Example of a Binomial Tree

Assume an initial one-year spot rate of $5\%$ and set:

• Up factor, $u = 1.1$ (10% increase)
• Down factor, $d = 0.9$ (10% decrease)
• Time steps: 2 years.

The tree is constructed as follows:

• Year 0: $5\%$
• Year 1:
• Up: $5\% \times 1.1 = 5.5\%$
• Down: $5\% \times 0.9 = 4.5\%$
• Year 2:
• From $5.5\%$:
• Up: $5.5\% \times 1.1 = 6.05\%$
• Down: $5.5\% \times 0.9 = 4.95\%$
• From $4.5\%$:
• Up: $4.5\% \times 1.1 = 4.95\%$
• Down: $4.5\% \times 0.9 = 4.05\%$

The tree can be visualized as:

          5.5%
         /    \
       6.05%  4.95%
      /  \
   5%   4.5%  
       \    /
        4.05%

2.2 Using the Binomial Tree for Bond Valuation

Once the tree is built, you can use it to value bonds by following these steps:

1. Calculate cash flows at each node, based on the coupon payments and final face value.
2. Discount the cash flows back to the present value using the interest rates in the tree.
3. Sum the present values of the cash flows at each node to obtain the bond's value.

2.2.1 Example of Bond Valuation with a Binomial Tree

Assuming the following cash flows:

• Annual coupon payment: $80$
• Face value at maturity: $1,000$
• The bond matures in two years, with the rates previously defined at the nodes.

Calculating cash flows:

• At node $6.05\%$:
• Year 1 cash flow = $80$
• Year 2 cash flow = $1,000 + $80 = $1,080$
• At node $4.95\%$:
• Year 1 cash flow = $80$
• Year 2 cash flow = $1,000 + $80 = $1,080$
• At node $4.95\%$:
• Year 1 cash flow = $80$
• Year 2 cash flow = $1,080$
• At node $4.05\%$:
• Year 1 cash flow = $80$
• Year 2 cash flow = $1,080$

Now, we compute the present values:

• Discounting cash flows back from $6.05\%$:

$$PV = \frac{80}{1.055} + \frac{1080}{(1.055^2)}$$

Calculating gives

$$PV \approx 75.87 + 970.85 = 1046.72$$

• Similarly, compute for the other nodes (4.95% and 4.05%).

Summing the present values across corresponding nodes provides the value of the bond under an arbitrage-free framework.

Conclusion

In this lesson, students has learned the principles of arbitrage-free valuation and how to construct and use a binomial interest rate tree. These tools are essential for valuing fixed-income securities accurately and understanding the dynamics of interest rate movements. Familiarity with these concepts will enable students to approach bond valuation with confidence and precision.

Study Notes

• Arbitrage-free valuation ensures no risk-free profit opportunities exist in the market.
• The Law of One Price must uphold similar asset prices across markets.
• Spot rates determine the present value of future cash flows for bonds.
• A binomial tree models the movement of interest rates over time.
• The tree helps in the evaluation of future cash flows from bonds and assists in accurate pricing.