Introduction

In the realm of fixed income securities, the construction of replicating portfolios is a fundamental concept. It enables us to match the cash flows of a given fixed-income security using multiple fixed income instruments. This technique is not only essential for understanding bond pricing but is also rooted in the concept of arbitrage, which is a core principle in finance. In this chapter, we will delve into the mechanics of constructing replicating portfolios, explain the arbitrage argument underlying this concept, and demonstrate how it can be applied to bond pricing scenarios.

Constructing a Replicating Portfolio

A replicating portfolio is a carefully selected combination of other fixed income securities that mirrors the cash flows of a target fixed-income security. The objective is to create a portfolio that produces the same cash flows at the same times as the target security. This technique is based on the principle of no-arbitrage, which states that identical cash flows must have identical prices.

Using Arbitrage as the Basis

The concept of replicating portfolios is deeply rooted in arbitrage, which is the process of exploiting price differences for profit without incurring any risk. When there is a discrepancy in the prices of two securities with identical cash flows, arbitrageurs step in to capitalize on this difference. By creating a replicating portfolio, we essentially eliminate these arbitrage opportunities.

Application to Bond Pricing

Let’s illustrate how constructing a replicating portfolio applies to bond pricing:

Consider two bonds: Bond A and Bond B. Bond A has a known set of cash flows at specific time intervals, and we want to determine its price. Bond B is a collection of zero-coupon bonds or other fixed income instruments.

To price Bond A, we create a replicating portfolio using Bond B. We choose the quantities of the components in Bond B in such a way that the portfolio’s cash flows match those of Bond A. In other words, we replicate the cash flows of Bond A using Bond B.

Using the law of no-arbitrage, the price of Bond A should be equal to the price of the replicating portfolio made up of Bond B’s components. If there is a difference in price, arbitrageurs will step in to take advantage of the price disparity until equilibrium is reached.

Example: Let’s say Bond A pays USD 100 in one year and USD 100 in two years. We create a replicating portfolio using two zero-coupon bonds: one maturing in one year and another maturing in two years. By purchasing the right quantities of these zero-coupon bonds, we can ensure that our replicating portfolio has cash flows of USD 100 in one year and USD 100 in two years, matching Bond A.

If the replicating portfolio’s price is $190 today, according to the no-arbitrage principle, Bond A’s price should also be USD 190. If Bond A were priced higher or lower, arbitrage opportunities would exist until the prices align.